We compute the mapping class group of the manifolds ♯g(S2k+1×S2k+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\sharp ^g(S^{2k+1}\times S^{2k+1})$$\end{document} for k>0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$k>0$$\end{document} in terms of the automorphism group of the middle homology and the group of homotopy (4k+3)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(4k+3)$$\end{document}-spheres. We furthermore identify its Torelli subgroup, determine the abelianisations, and relate our results to the group of homotopy equivalences of these manifolds.
Mathematics Subject Classification57R5055N2257R60University of Cambridgepublisher-imprint-nameBirkhäuservolume-issue-count5issue-article-count17issue-toc-levels0issue-pricelist-year2020issue-copyright-holderSpringer Nature Switzerland AGissue-copyright-year2020article-contains-esmNoarticle-numbering-styleContentOnlyarticle-registration-date-year2020article-registration-date-month9article-registration-date-day23article-toc-levels0toc-levels0volume-typeRegularjournal-productNonStandardArchiveJournalnumbering-styleContentOnlyarticle-grants-typeOpenChoicemetadata-grantOpenAccessabstract-grantOpenAccessbodypdf-grantOpenAccessbodyhtml-grantOpenAccessbibliography-grantOpenAccessesm-grantOpenAccessonline-firstfalsepdf-file-referenceBodyRef/PDF/29_2020_Article_600.pdfpdf-typeTypesettarget-typeOnlinePDFissue-online-date-year2020issue-online-date-month12issue-online-date-day4issue-print-date-year2020issue-print-date-month12issue-print-date-day4issue-typeRegulararticle-typeOriginalPaperjournal-subject-primaryMathematicsjournal-subject-secondaryMathematics, generaljournal-subject-collectionMathematics and Statisticsopen-accesstrue
The classical mapping class group Γg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _g$$\end{document} of a genus g surface naturally generalises to all even dimensions 2n as the group of isotopy classesof orientation-preserving diffeomorphisms of the g-fold connected sum Wg=♯g(Sn×Sn)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g=\sharp ^g(S^n\times S^n)$$\end{document}. Its action on the middle cohomology H(g):=Hn(Wg;Z)≅Z2g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$ H(g)\,{:}{=}\,\mathrm {H}^n(W_g;\mathbf {Z})\cong \mathbf {Z}^{2g}$$\end{document} provides a homomorphism Γgn→GL2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_g\rightarrow \mathrm {GL}_{2g}(\mathbf {Z})$$\end{document} whose image is the symplectic group Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} in the surface case 2n=2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2n=2$$\end{document}, and a certain arithmetic subgroup Gg⊂Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g\subset \mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} or Gg⊂Og,g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g\subset \mathrm {O}_{g,g}(\mathbf {Z})$$\end{document} in general, the description of which we shall recall later. The kernel Tgn⊂Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_g^n\subset \Gamma _g^n$$\end{document} of the resulting extension0⟶Tgn⟶Γgn⟶Gg⟶0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0 \longrightarrow \mathrm {T}^n_g \longrightarrow \Gamma ^n_g \longrightarrow G_g \longrightarrow 0 \end{aligned}$$\end{document}is known as the Torelli group—the subgroup of isotopy classes acting trivially on homology. In contrast to the surface case, the Torelli group in high dimensions 2n≥6\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2n\ge 6$$\end{document} is comparatively manageable: there is an extension0⟶Θ2n+1⟶Tgn⟶H(g)⊗SπnSO(n)⟶0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0 \longrightarrow \Theta _{2n+1} \longrightarrow \mathrm {T}^n_g \longrightarrow H(g)\otimes S\pi _n\mathrm {SO}(n) \longrightarrow 0 \end{aligned}$$\end{document}due to Kreck [35], which relates Tgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_g^n$$\end{document} to the finite abelian group of homotopy spheres Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}$$\end{document} and the image of the stabilisation map S:πnSO(n)→πnSO(n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S:\pi _n\mathrm {SO}(n)\rightarrow \pi _n\mathrm {SO}(n+1)$$\end{document} whose isomorphism class is shown in Table 1.
The description of Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _g^n$$\end{document} up to these two extension problems has found a variety of applications [2, 5–7, 18, 23, 29, 33, 38, 39], especially in relation to the study of moduli spaces of manifolds [22]. The remaining extensions (1) and (2) have been studied more closely for particular values of g and n [15, 19, 21, 36, 37, 48] but are generally not well-understood (see e.g. [15, p.1189], [21, p.873], [2, p.425]). In the present work, we resolve the remaining ambiguity for n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} odd, resulting in a complete description of the mapping class group Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _g^n$$\end{document} and the Torelli group Tgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_g^n$$\end{document} in terms of the arithmetic group Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document} and the group of homotopy spheres Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}$$\end{document}.
To explain our results, note that (1) and (2) induce further extensions0⟶Θ2n+1⟶Γgn⟶Γgn/Θ2n+1⟶0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned}&\displaystyle 0 \longrightarrow \Theta _{2n+1} \longrightarrow \Gamma ^n_g \longrightarrow \Gamma ^n_g/\Theta _{2n+1} \longrightarrow 0\quad \end{aligned}$$\end{document}0⟶H(g)⊗SπnSO(n)⟶Γgn/Θ2n+1⟶Gg⟶0,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned}&\displaystyle 0 \longrightarrow H(g)\otimes S\pi _n\mathrm {SO}(n) \longrightarrow \Gamma ^n_g/\Theta _{2n+1} \longrightarrow G_g \longrightarrow 0, \end{aligned}$$\end{document}which express Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _g^n$$\end{document} in terms of Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document} and Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}$$\end{document} up to two extension problems that are similar to (1) and (2), but are more convenient to analyse as both of their kernels are abelian. We resolve these two extension problems completely, beginning with an algebraic description of the second one in Sect. 2, which enables us in particular to decide when it splits.
Theorem A
For n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} odd and g≥1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 1$$\end{document}, the extension0⟶H(g)⊗SπnSO(n)⟶Γgn/Θ2n+1⟶Gg⟶0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0 \longrightarrow H(g)\otimes S\pi _n\mathrm {SO}(n) \longrightarrow \Gamma ^n_g/\Theta _{2n+1} \longrightarrow G_g \longrightarrow 0 \end{aligned}$$\end{document}splits for n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document}. For n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}, it splits if and only if g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}.
Unlike (4), the extension (3) is central and thus classified by a class in H2(Γgn/Θ2n+1;Θ2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\Gamma ^n_g/\Theta _{2n+1};\Theta _{2n+1})$$\end{document}, which our main result, Theorem B below, identifies in terms of three cohomology classes
which can be expressed algebraically, using our description of Γgn/Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_g/\Theta _{2n+1}$$\end{document} mentioned above. Referring to Sect. 3 for the precise definition of these classes, we encourage the reader to think of them geometrically in terms of their pullbacks along the compositioninduced by taking path components and quotients: the pullback of the first class, which is closely related to Meyer’s signature cocycle [43], evaluates a class represented by an oriented smooth fibre bundle π:E→S\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi :E\rightarrow S$$\end{document} over a closed oriented surface S with fibre Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g$$\end{document} to an eighth of the signature sgn(E)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {sgn}(E)$$\end{document} of its total space, the pullback of the second one assigns such a bundle the Pontryagin number p(n+1)/42(E)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$p_{(n+1)/4}^2(E)$$\end{document} up to a fixed constant, and the pullback of the third class evaluates [π]\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[\pi ]$$\end{document} to a certain linear combination of sgn(E)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {sgn}(E)$$\end{document} and p(n+1)/42(E)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$p_{(n+1)/4}^2(E)$$\end{document}. In addition to these three classes, our identification of the extension (3) for n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} odd involves two particular homotopy spheres: the first one, ΣP∈Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _P\in \Theta _{2n+1}$$\end{document}, is the Milnor sphere—the boundary of the E8\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$E_8$$\end{document}-plumbing [8, Sect. V], and the second one, ΣQ∈Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q\in \Theta _{2n+1}$$\end{document}, arises as the boundary of the plumbing of two copies of a linear Dn+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^{n+1}$$\end{document}-bundles over Sn+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^{n+1}$$\end{document} classified by a generator of SπnSO(n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S\pi _n\mathrm {SO}(n)$$\end{document}. We write (-)·Σ:H2(Γgn/Θ2n+1;Z)→H2(Γgn/Θ2n+1;Θ2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(-)\cdot \Sigma :\mathrm {H}^2(\Gamma ^n_g/\Theta _{2n+1};\mathbf {Z})\rightarrow \mathrm {H}^2(\Gamma ^n_g/\Theta _{2n+1};\Theta _{2n+1})$$\end{document} for the change of coefficients induced by Σ∈Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma \in \Theta _{2n+1}$$\end{document}.
Moreover, this extension splits if and only if n≡1(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 1\ (\mathrm {mod}\ 4)$$\end{document} and g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}.
The extension (2) describing the Torelli group Tgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}^n_{g}$$\end{document} is the pullback of the extension determined in Theorem B along the map H(g)⊗SπnSO(n)→Γgn/Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$H(g)\otimes S\pi _n\mathrm {SO}(n)\rightarrow \Gamma _g^n/\Theta _{2n+1}$$\end{document}, so the combination of the previous result with our identification of Γgn/Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g}^n/\Theta _{2n+1}$$\end{document} provides an algebraic description of both Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _g^n$$\end{document} and Tgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_g^n$$\end{document} in terms of Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document} and Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}$$\end{document}. We derive several consequences from this, beginning with deciding when the more commonly considered extensions (1) and (2) split.
The second application of our description of the groups Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g}$$\end{document} and Tgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}^n_g$$\end{document} is a computation of their abelianisations.
Although Theorem A exhibits the extension (4) as nontrivial in some cases, its abelianisation turns out to split (see Corollary 2.4), so there exists a splittingH1(Γgn/Θ2n+1)≅H1(Gg)⊕(H(g)⊗SπnSO(n))Gg,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {H}_1(\Gamma _g^n/\Theta _{2n+1})\cong \mathrm {H}_1(G_g)\oplus ( H(g)\otimes S\pi _n\mathrm {SO}(n))_{G_g}, \end{aligned}$$\end{document}which participates in the following identification of the abelianisation of Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g}^n$$\end{document} and Tgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_{g}^n$$\end{document}.
The extension (4) induces a split short exact sequence of Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document}-modules 0⟶Θ2n+1/⟨ΣQ⟩⟶H1(Tgn)⟶ρ∗H(g)⊗SπnSO(n)⟶0.\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\longrightarrow \Theta _{2n+1}/\langle \Sigma _Q\rangle \longrightarrow \mathrm {H}_1(\mathrm {T}_{g}^n)\overset{\rho _*}{\longrightarrow } H(g)\otimes S\pi _n\mathrm {SO}(n)\longrightarrow 0. \end{aligned}$$\end{document} In particular, the commutator subgroup of Tgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_{g}^n$$\end{document} is generated by ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document}.
These splittings of H1(Γgn)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\Gamma _g^n)$$\end{document} and H1(Tgn)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\mathrm {T}_g^n)$$\end{document} are constructed abstractly, but can often be made more concrete by means of a refinement of the mapping torus as a mapt:Γgn⟶Ω2n+1τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} t:\Gamma _g^n\longrightarrow \Omega ^{\tau _{>n}}_{2n+1} \end{aligned}$$\end{document}to the bordism group of closed (2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(2n+1)$$\end{document}-manifolds M equipped with a lift of their stable normal bundle M→BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$M\rightarrow \mathrm {BO}$$\end{document} to the n-connected cover τ>nBO→BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}\rightarrow \mathrm {BO}$$\end{document}. To state the resulting more explicit description of the abelianisations of Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _g^n$$\end{document} and Tgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_g^n$$\end{document}, we write σn′\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\sigma _n'$$\end{document} for the minimal positive signature of a closed smooth n-connected (2n+2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(2n+2)$$\end{document}-dimensional manifold. For n≠1,3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 1,3,7$$\end{document} odd, the intersection form of such a manifold is unimodular and even, so σn′\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\sigma _n'$$\end{document} is divisible by 8.
The morphism t∗⊕p∗:H1(Γgn)⟶Ω2n+1τ>n⊕H1(Gg)⊕(H(g)⊗SπnSO(n))Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} t_*\oplus p_*:\mathrm {H}_1(\Gamma ^n_g)\longrightarrow \Omega ^{\tau _{>n}}_{2n+1}\oplus \mathrm {H}_1(G_g) \oplus \big (H(g)\otimes S\pi _n\mathrm {SO}(n)\big )_{G_g} \end{aligned}$$\end{document} is an isomorphism for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document}, and for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} if n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}. For g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} and n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document}, it is surjective, has kernel of order σn′/8\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\sigma _n'/8$$\end{document} generated by ΣP\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _P$$\end{document}, and splits for n≡1(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 1\ (\mathrm {mod}\ 4)$$\end{document}.
The morphism t∗⊕ρ∗:H1(Tgn)⟶Ω2n+1τ>n⊕(H(g)⊗SπnSO(n))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} t_*\oplus \rho _*:\mathrm {H}_1(\mathrm {T}_g^n)\longrightarrow \Omega ^{\tau _{>n}}_{2n+1}\oplus \big ( H(g)\otimes S\pi _n\mathrm {SO}(n)\big ) \end{aligned}$$\end{document} is an isomorphism for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}. For n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document}, it is surjective, has kernel of order σn′/8\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\sigma _n'/8$$\end{document} generated by ΣP\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _P$$\end{document}, and splits Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document}-equivariantly for n≡1(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 1\ (\mathrm {mod}\ 4)$$\end{document}.
Remark
In Theorem G below, we determine the abelianisation of Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1}$$\end{document} and Tg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}^n_{g,1}$$\end{document} for n≥4\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 4$$\end{document} even in which case the morphisms t∗⊕p∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_*\oplus p_*$$\end{document} and t∗⊕ρ∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_*\oplus \rho _*$$\end{document} are isomorphisms for allg≥1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 1$$\end{document}.
As shown in [34, Prop. 2.15], the minimal signature σn′\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\sigma _n'$$\end{document} is nontrivial for n odd, grows very quickly with n, and can be expressed in terms of Bernoulli numbers.
For some values of g and n, Theorem E leaves open whether t∗⊕p∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_*\oplus p_*$$\end{document} and t∗⊕ρ∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_*\oplus \rho _*$$\end{document} split. The morphisms p∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$p_*$$\end{document} and ρ∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\rho _*$$\end{document} always split by Theorem D, and in Sect. 4.1 we relate the question of whether there exist compatible splittings of t∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_*$$\end{document} to a known open problem in the theory of highly connected manifold, showing in particular that such splittings do exist when assuming a conjecture of Galatius and Randal-Williams.
Work of Thurston [52] shows that the component of the identity diffeomorphism is perfect as a discrete group, so the abelianisation of the full diffeomorphism group considered as a discrete group agrees with H1(Γgn)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\Gamma _g^n)$$\end{document}.
In view of Theorem E, it is of interest to determine the bordism groups Ω2n+1τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Omega ^{\tau _{>n}}_{2n+1}$$\end{document}, the abelianisation H1(Gg)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(G_g)$$\end{document}, and the coinvariants (H(g)⊗SπnSO(n))Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(H(g)\otimes S\pi _n\mathrm {SO}(n))_{G_g}$$\end{document}. The computation(H(g)⊗SπnSO(n))Gg≅0g≥2orn=3,6,7orn≡5(mod8)Z/22g=1andn≡0(mod8)Z/2otherwise\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} (H(g)\otimes S\pi _n\mathrm {SO}(n))_{G_g}\cong {\left\{ \begin{array}{ll} 0&{}g\ge 2\text { or }n=3,6,7\text { or }n\equiv 5\ (\mathrm {mod}\ 8)\\ \mathbf {Z}/2^2&{}g=1\text { and }n\equiv 0\ (\mathrm {mod}\ 8)\\ \mathbf {Z}/2&{}\text {otherwise} \end{array}\right. } \end{aligned}$$\end{document}is straightforward (see Lemma A.2 and Table 1). The abelianisation of Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document} is known and summarised in Table 2 (see Lemma A.1). Finally, the bordism groups Ω2n+1τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Omega ^{\tau _{>n}}_{2n+1}$$\end{document} are closely connected to the stable homotopy groups of spheres: the canonical mapπ2n+1s≅Ω2n+1fr⟶Ω2n+1τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \pi ^s_{2n+1}\cong \Omega ^{\mathrm {fr}}_{2n+1}\longrightarrow \Omega ^{\tau _{>n}}_{2n+1} \end{aligned}$$\end{document}factors through the cokernel of the J-homomorphism and work of Schultz and Wall [50, 55] implies that the induced morphism is often an isomorphism (see Corollary 3.6).
The abelianisation of Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document} for n odd
(Schultz, Wall). For n odd, the natural morphism coker(J)2n+1→Ω2n+1τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {coker}(J)_{2n+1}\rightarrow \Omega ^{\tau _{>n}}_{2n+1}$$\end{document} is surjective with cyclic kernel. It is an isomorphism for n≡1(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 1\ (\mathrm {mod}\ 4)$$\end{document} and for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}.
Combined with Theorem E, this reduces the computation of the abelianisation of Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _g^n$$\end{document} and Tgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}^n_g$$\end{document} in many cases to determining the cokernel of the J-homomorphism—a well-studied problem in stable homotopy theory. Table 3 shows the resulting calculation of the abelianisations of the groups Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _g^n$$\end{document} and Tgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}^n_g$$\end{document} for the first few values of n .
Some abelianisations of Tgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_g^n$$\end{document} and Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _g^n$$\end{document}
After the completion of this work, Burklund–Hahn–Senger [11] and Burklund–Senger [12] showed that for n odd, the homotopy sphere ΣQ∈Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q\in \Theta _{2n+1}$$\end{document} bounds a parallelisable manifold if and only if n≠11\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 11$$\end{document}. This implies in particular that aside from the case n=11\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=11$$\end{document}
the canonical map coker(J)2n+1→Ω2n+1τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {coker}(J)_{2n+1}\rightarrow \Omega _{2n+1}^{\tau _{>n}}$$\end{document} is an isomorphism, which extends the theorem attributed to Schultz and Wall above,
the conjecture of Galatius–Randal-Williams mentioned in the third part of the previous remark holds, and
the minimal signature σn′\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\sigma _n'$$\end{document} appearing in Theorem E is computable from [34, Prop. 2.15].
Homotopy equivalences
As an additional application of our results, we briefly discuss the group of homotopy classes of orientation-preserving homotopy equivalences.
The natural map can be seen to factor through the quotient Γgn/Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _g^n/\Theta _{2n+1}$$\end{document} and to induce a commutative diagram of the form
which exhibits the lower row—an extension describing due to Baues [1]—as the extension pushout of the extension (4) along the left vertical morphism, which is induced by the restriction J:SπnSO(n)→Sπ2nSn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$J:S\pi _n\mathrm {SO}(n)\rightarrow S\pi _{2n}S^{n}$$\end{document} of the unstable J-homomorphism, where Sπ2nSn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S\pi _{2n}S^{n}$$\end{document} is the image of the suspension map S:π2nSn→π2n+1Sn+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S:\pi _{2n}S^{n}\rightarrow \pi _{2n+1}S^{n+1}$$\end{document}. By Theorem A, the upper row splits in most cases and thus induces a compatible splitting of Baues’ extension. In the cases in which the upper row does not split, we show that Baues’ extension cannot split either.
For n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document}, the two extensions in (6) admit compatible splittings. For n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}, either of the extensions splits if and only if g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}.
The induced morphism is an isomorphism for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document}, and a split epimorphism with kernel the coinvariants (H(g)⊗Sπ2nSn)Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(H(g)\otimes S\pi _{2n}S^{n})_{G_g}$$\end{document}, which vanish for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document} or n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}, and are isomorphic to the group Sπ2nSn/(2·Sπ2nSn)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S\pi _{2n}S^{n}/(2\cdot S\pi _{2n}S^{n})$$\end{document} otherwise.
The groups Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g,1}^n$$\end{document} for n even
Some parts in our analysis of Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g,1}^n$$\end{document} go through when n≥4\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 4$$\end{document} is even as well, but a few key steps do not and would require new arguments. For instance, a different approach to the extension problem (4) would be necessary, as well as an extension of Theorem 3.12 to incorporate the Arf invariant. The abelianisation of the groups Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1}$$\end{document} and Tg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}^n_{g,1}$$\end{document}, however, can be determined without fully solving the extensions (3) and (4) if n is even. It turns out that in this case, the morphisms considered in Theorem E are isomorphisms for allg≥1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 1$$\end{document}, which we shall prove as part of Sect. 4.2.
Theorem G
For n≥4\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 4$$\end{document} even and g≥1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 1$$\end{document}, the morphismst∗⊕p∗:H1(Γg,1n)⟶Ω2n+1τ>n⊕H1(Gg)⊕(H(g)⊗SπnSO(n))Ggt∗⊕ρ∗:H1(Tg,1n)⟶Ω2n+1τ>n⊕(H(g)⊗SπnSO(n))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned}&t_*\oplus p_*:\mathrm {H}_1(\Gamma _{g,1}^n) \longrightarrow \Omega ^{\tau _{>n}}_{2n+1}\oplus \mathrm {H}_1(G_g) \oplus \big ( H(g)\otimes S\pi _n\mathrm {SO}(n)\big )_{G_g}\\&\quad t_*\oplus \rho _*:\mathrm {H}_1(\mathrm {T}_{g,1}^n) \longrightarrow \Omega ^{\tau _{>n}}_{2n+1}\oplus \big (H(g)\otimes S\pi _n\mathrm {SO}(n)\big ) \end{aligned}$$\end{document}are isomorphisms for g≥1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 1$$\end{document}. The group of coinvariants (H(g)⊗SπnSO(n))Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\big ( H(g)\otimes S\pi _n\mathrm {SO}(n)\big )_{G_g}$$\end{document} is described in (5).
Other highly connected manifolds
Instead of restricting to Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g$$\end{document}, one could consider any (n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(n-1)$$\end{document}-connected almost parallelisable manifold M of dimension 2n≥6\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2n\ge 6$$\end{document}. Baues’ and Kreck’s work [1, 35] applies in this generality, so there are analogues of the sequences (1)–(6) describing and . However, for n odd—the case of our main interest—Wall’s classification of highly connected manifolds [55] implies that any such manifold is diffeomorphic to a connected sum Wg♯Σ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g\sharp \Sigma $$\end{document} with an exotic sphere Σ∈Θ2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma \in \Theta _{2n}$$\end{document}, aside from those of Kervaire invariant 1, which only exist in dimensions 6, 14, 30, 62, and possibly 126 by work of Hill–Hopkins–Ravenel [25]. The mapping class group for Σ∈Θ2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma \in \Theta _{2n}$$\end{document} and n odd in turn is understood in terms of Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _g^n$$\end{document}: Kreck’s work [35, Lem. 3, Thm 3] shows that the former is a quotient of the latter by a known element Σ′∈Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma '\in \Theta _{2n+1}$$\end{document} of order at most 2, which is trivial if and only if η·[Σ]∈coker(J)2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\eta \cdot [\Sigma ]\in \mathrm {coker}(J)_{2n+1}$$\end{document} vanishes.
Previous results
The extensions (1) and (2) and their variants (3) and (4) have been studied by various authors before, and some special cases of our results were already known:
As an application of their programme on moduli spaces of manifolds, Galatius–Randal-Williams [21] determined the abelianisation of Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _g^n$$\end{document} for g≥5\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 5$$\end{document} and used this to determine the extension (3) for n≡5(mod8)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 5\ (\mathrm {mod}\ 8)$$\end{document} up to automorphisms of Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}$$\end{document} as long g≥5\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 5$$\end{document}. Our work recovers and extends their results, also applies to low genera g<5\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g<5$$\end{document}, and does not rely on their work on moduli spaces of manifolds.
Theorems A and F for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document} reprove results due to Crowley [15].
Baues [1, Thm 8.14, Thm 10.3] showed that the lower extension in (6) splits for n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document} odd, which we recover as part of the first part of Corollary F.
The case (g,n)=(1,3)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(g,n)=(1,3)$$\end{document} of Theorem A and Theorem C (ii) can be deduced from work of Krylov [37] and Fried [19], who also showed that the extension of Theorem C (i) does not split in this case. Krylov [36, Thms 2.1, 3.2, 3.3] moreover established the case n≡5(mod8)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 5\ (\mathrm {mod}\ 8)$$\end{document} of Theorem C (i) for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}. For n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document}, he also proved the case n≡3(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 3\ (\mathrm {mod}\ 4)$$\end{document} of Theorem A and the case n≡3(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 3\ (\mathrm {mod}\ 4)$$\end{document} of Theorem C (ii) for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}.
Further applications
Our main result Theorem B has been used in [32] in conjunction with Galatius–Randal-Williams’ work on moduli spaces of manifolds [22] to compute the second stable homology of the theta-subgroup of Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} (see Sect. 1.2), or equivalently, the second quadratic symplectic algebraic K-theory group of the integers KSp2q(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {KSp}^q_2(\mathbf {Z})$$\end{document}.
Outline
Section 1 serves to recall foundational material on diffeomorphism groups and their classifying spaces, as well as to introduce different variants of the extensions (1) and (2) and to establish some of their basic properties. In Sect. 2, we study the action of Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _g^n$$\end{document} on the set of stable framings of Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g$$\end{document} to identify the extension (3) and prove Theorem A. Section 3 aims at the proof of our main result Theorem B, which requires some preparation. We recall the relation between relative Pontryagin classes and obstruction theory in Sect. 3.1, discuss aspects of Wall’s classification of highly connected manifolds in Sect. 3.2, relate this class of manifolds to Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}$$\end{document}-bundles over surfaces with certain boundary conditions in Sect. 3.3 (which incidentally is the key geometric insight to prove Theorem B), construct the cohomology classes appearing in the statement of Theorem B in Sects. 3.4 and 3.5, and finish with the proof of Theorem B in Sect. 3.6. In Sect. 4, we analyse the extensions (1) and (2) and compute the abelianisation of Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g}$$\end{document} and Tgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_g^n$$\end{document}, proving Corollaries C–E and Theorem G. Section 5 briefly discusses the group of homotopy equivalences and proves Corollary F. In the appendix, we compute various low-degree (co)homology groups of the symplectic group Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} and its arithmetic subgroup Gg⊂Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g\subset \mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document}.
Variations on two extensions of KreckDifferent flavours of diffeomorphisms
Throughout this work, we writeWg=♯g(Sn×Sn)andWg,1=♯g(Sn×Sn)\int(D2n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} W_g=\sharp ^{g}(S^n\times S^n)\quad \text {and}\quad W_{g,1}=\sharp ^{g}(S^n\times S^n)\backslash \mathrm {int}(D^{2n}) \end{aligned}$$\end{document}for the g-fold connected sum of Sn×Sn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^n\times S^n$$\end{document}, including W0=S2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_0=S^{2n}$$\end{document}, and the manifold obtained from Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g$$\end{document} removing the interior of an embedded disc D2n⊂Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^{2n}\subset W_{g}$$\end{document}. Occasionally, we view the manifold Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}$$\end{document} alternatively as the iterated boundary connected sum Wg,1=♮gW1,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}=\natural ^gW_{1,1}$$\end{document} of W1,1=Sn×Sn\int(D2n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{1,1}=S^n\times S^n\backslash \mathrm {int}(D^{2n})$$\end{document}. We call g the genus of Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g$$\end{document} or Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}$$\end{document} and denote by and the groups of orientation-preserving diffeomorphisms, not necessarily fixing the boundary in the case of Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}$$\end{document}. We shall also consider the subgroupsof diffeomorphisms required to fix a neighbourhood of the boundary ∂Wg,1≅S2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\partial W_{g,1}\cong S^{2n-1}$$\end{document} or a neighbourhood of a chosen disc D2n-1⊂∂Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^{2n-1}\subset \partial W_{g,1}$$\end{document} in the boundary, respectively. All groups of diffeomorphisms are equipped with the smooth topology so that
BDiff∂(Wg,1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {BDiff}_{\partial }(W_{g,1})$$\end{document} classifies (Wg,1,S2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(W_{g,1},S^{2n-1})$$\end{document}-bundles, i.e. smooth Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}$$\end{document}-bundles with a trivialisation of their S2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^{2n-1}$$\end{document}-bundle of boundaries, and
BDiff∂/2(Wg,1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {BDiff}_{\partial /2}(W_{g,1})$$\end{document} classifies (Wg,1,D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(W_{g,1},D^{2n-1})$$\end{document}-bundles, that is, smooth Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}$$\end{document}-bundles with a trivialised D2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^{2n-1}$$\end{document}-subbundle of its S2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^{2n-1}$$\end{document}-bundle of boundaries.
Taking path components, we obtain various groups of isotopy classesExtending diffeomorphisms by the identity provides a map , which induces an isomorphism on path components by work of Kreck as long as n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document}.
Lemma 1.1
(Kreck) The canonical map Γg,1n→Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1}\rightarrow \Gamma ^n_g$$\end{document} is an isomorphism for n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document}.
Proof
Taking the differential at the centre of the disc induces a fibration to the oriented frame bundle of Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g$$\end{document}. Its fibre is the subgroup of diffeomorphisms that fix a point and its tangent space, so it is equivalent to the subgroup of diffeomorphisms fixing a small disc around that point, which is in turn equivalent to Diff∂(Wg,1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Diff}_\partial (W_{g,1})$$\end{document}. We thus have fibration sequences of the formwhose long exact sequences show that the morphism in question is surjective and also that its kernel is generated by a single isotopy class given by “twisting” a collar [0,1]×S2n-1⊂Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[0,1]\times S^{2n-1}\subset W_{g,1}$$\end{document} using a smooth based loop in SO(2n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {SO}(2n)$$\end{document} that represents a generator of π1SO(2n)≅Z/2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _1\mathrm {SO}(2n)\cong \mathbf {Z}/2$$\end{document} (see [35, p. 647]). It follows from [35, Lem. 3 b), Lem. 4] that this isotopy class is trivial as Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g$$\end{document} bounds the, handlebody ♮g(Dn+1×Sn)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\natural ^g(D^{n+1}\times S^n)$$\end{document}, which is parallelisable. □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
For the purpose of studying the mapping class group Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _g^n$$\end{document}, we can thus equally well work with Diff∂(Wg,1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Diff}_\partial (W_{g,1})$$\end{document} instead of , which is advantageous since there is a stabilisation map Diff∂/2(Wg,1)→Diff∂/2(Wg+1,1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Diff}_{\partial /2}(W_{g,1})\rightarrow \mathrm {Diff}_{\partial /2}(W_{g+1,1})$$\end{document} by extending diffeomorphisms over an additional boundary connected summand via the identity, which restricts to a map Diff∂(Wg,1)→Diff∂(Wg+1,1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Diff}_{\partial }(W_{g,1})\rightarrow \mathrm {Diff}_{\partial }(W_{g+1,1})$$\end{document} and thus induces stabilisation maps of the formΓg,1/2n⟶Γg,1/2nandΓg,1n⟶Γg+1,1n,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \Gamma ^n_{g,1/2}\longrightarrow \Gamma ^n_{g,1/2}\quad \text { and } \quad \Gamma ^n_{g,1}\longrightarrow \Gamma ^n_{g+1,1}, \end{aligned}$$\end{document}that allow us to compare mapping class groups of different genera. The group Γ0,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{0,1}$$\end{document} has a convenient alternative description: gluing two closed d-discs along their boundaries via a diffeomorphism supported in a disc Dd⊂∂Dd+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^{d}\subset \partial D^{d+1}$$\end{document} gives a morphism π0Diff∂(Dd)⟶Θd+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _0\mathrm {Diff}_\partial (D^{d})\longrightarrow \Theta _{d+1}$$\end{document} to Kervaire–Milnor’s [30] finite abelian group Θd\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _d$$\end{document} of oriented homotopy d-spheres up to h-cobordism. By work of Cerf [13], this is an isomorphism for d≥5\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d\ge 5$$\end{document}, so we identify these two groups henceforth. Iterating the stabilisation map yields a sequence of mapsΘ2n+1=π0Diff∂(D2n)=Γ0,1n⟶Γg,1n⟶Γg,1/2n.\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \Theta _{2n+1}=\pi _0\mathrm {Diff}_{\partial }(D^{2n})=\Gamma ^n_{0,1}\longrightarrow \Gamma _{g,1}^n\longrightarrow \Gamma _{g,1/2}^n. \end{aligned}$$\end{document}
Lemma 1.2
For n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document}, the image of Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}$$\end{document} in Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g,1}^n$$\end{document} is central and becomes trivial in Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g,1/2}^n$$\end{document}. The induced morphism(Γg,1n/(im(Θ2n+1→Γg,1n))⟶Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \Big (\Gamma ^n_{g,1}/(\mathrm {im}(\Theta _{2n+1}\rightarrow \Gamma ^n_{g,1})\Big )\longrightarrow \Gamma _{g,1/2}^n \end{aligned}$$\end{document}is an isomorphism.
Proof
Every diffeomorphism of Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}$$\end{document} supported in a disc is isotopic to one that is supported in an arbitrary small neighbourhood of the boundary and thus commutes with any diffeomorphism in Diff∂(Wg,1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Diff}_\partial (W_{g,1})$$\end{document} up to isotopy, which shows the first part of the claim. For the others, we consider the sequence of topological groupsDiff∂(Wg,1)⟶Diff∂/2(Wg,1)⟶Diff∂(D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {Diff}_ \partial (W_{g,1})\longrightarrow \mathrm {Diff}_{\partial /2}(W_{g,1})\longrightarrow \mathrm {Diff}_{\partial }(D^{2n-1}) \end{aligned}$$\end{document}induced by restricting diffeomorphisms in Diff∂/2(Wg,1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Diff}_{\partial /2}(W_{g,1})$$\end{document} to the moving part of the boundary. This is a fibration sequence by the parametrised isotopy extension theorem. Mapping this sequence for g=0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=0$$\end{document} into (1.3) via the iterated stabilisation map, we see that the looped map ΩDiff∂(D2n-1)→Diff∂(Wg,1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Omega \mathrm {Diff}_\partial (D^{2n-1})\rightarrow \mathrm {Diff}_{\partial }(W_{g,1})$$\end{document} induced by the fibration sequence (1.3) factors as the compositionΩDiff∂(D2n-1)⟶Diff∂(D2n)⟶Diff∂(Wg,1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \Omega \mathrm {Diff}_\partial (D^{2n-1})\longrightarrow \mathrm {Diff}_\partial (D^{2n})\longrightarrow \mathrm {Diff}_{\partial }(W_{g,1}) \end{aligned}$$\end{document}of the map defining the Gromoll filtration with the iterated stabilisation map. Since the first map in this factorisation is surjective on path components by Cerf’s work [13], the claim will follow from the long exact sequence on homotopy groups of (1.3) once we show that the map Γg,1/2n→π0Diff∂(D2n-1)=Θ2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g,1/2}^n\rightarrow \pi _0\mathrm {Diff}_{\partial }(D^{2n-1})=\Theta _{2n}$$\end{document} has trivial image. Using that any orientation preserving diffeomorphism fixes any chosen oriented codimension 0 disc up isotopy by the isotopy extension theorem, one sees that this image agrees with the inertia group of Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g$$\end{document}, which vanishes by work of Kosinski and Wall [31, 54]. □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
Wall’s quadratic form
We recall Wall’s quadratic from associated to an (n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(n-1)$$\end{document}-connected 2n-manifold [55], specialised to the case of our interest—the iterated connected sums Wg=♯g(Sn×Sn)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g=\sharp ^g(S^n\times S^n)$$\end{document} in dimension 2n≥6\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2n\ge 6$$\end{document}.
The intersection form λ:H(g)⊗H(g)→Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\lambda :H(g){\otimes }H(g)\rightarrow \mathbf {Z}$$\end{document} on the middle cohomology H(g):=Hn(Wg;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$ H(g) \,{:}{=}\,\mathrm {H}^n(W_g;\mathbf {Z})$$\end{document} is a nondegenerate (-1)n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(-1)^{n}$$\end{document}-symmetric bilinear form. We use Poincaré duality to identify H(g) with πn(Wg;Z)≅Hn(Wg)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _n(W_g;\mathbf {Z})\cong \mathrm {H}_n(W_g)$$\end{document} and a result of Haefliger [24] to represent classes in πn(Wg)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _n(W_g)$$\end{document} by embedded spheres e:Sn↪Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$e:S^n \hookrightarrow W_g$$\end{document}, unique up to isotopy as long as n≥4\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 4$$\end{document}. As Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g$$\end{document} is stably parallelisable, the normal bundle of such e is stably trivial and hence gives a class q([e]) inker(πn(BSO(n))→πn(BSO(n+1))≅Z/Λn=ZifnevenZ/2ifnodd,n≠3,70ifn=3,7.,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \ker \big (\pi _n(\mathrm {BSO}(n))\rightarrow \pi _n(\mathrm {BSO}(n+1)\big )\cong \mathbf {Z}/\Lambda _n={\left\{ \begin{array}{ll} \mathbf {Z}&{}\quad \text{ if } n\text { even}\\ \mathbf {Z}/2&{}\quad \text{ if } n\text { odd},n\ne 3,7\\ 0&{}\text{ if } n=3,7. \end{array}\right. }, \end{aligned}$$\end{document}where Λn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Lambda _n$$\end{document} is the image of the usual map πn(SO(n+1))→πn(Sn)≅Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _n(\mathrm {SO}(n+1))\rightarrow \pi _n(S^n)\cong \mathbf {Z}$$\end{document} (see e.g. [41, §1.B]).1 This defines a function q:H(g)⟶Z/Λn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$q:H(g)\longrightarrow \mathbf {Z}/\Lambda _n$$\end{document}, which Wall [55] showed to satisfy the following two properties.
Note that for n even, (i) and (ii) imply q([e])=12λ([e],[e])∈Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$q([e])=\frac{1}{2}\lambda ([e],[e])\in \mathbf {Z}$$\end{document}, so q can in this case be recovered from λ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\lambda $$\end{document}. The triple (H(g),λ,q)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(H(g),\lambda , q)$$\end{document} is the quadratic form associated to Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g$$\end{document}. The decomposition Wg=♯g(Sn×Sn)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g=\sharp ^g (S^n\times S^n)$$\end{document} into connected summands induces a basis (e1,…,eg,f1,…fg)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(e_1,\ldots ,e_g,f_1,\ldots f_g)$$\end{document} of H(g)≅Z2g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$ H(g)\cong \mathbf {Z}^{2g}$$\end{document} with respect to which q and λ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\lambda $$\end{document} have the formq:Z2g⟶Z/Λn(x1,…xg,y1,…,yg)⟼∑i=1gxiyiandJg,(-1)n=0I(-1)nI0,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} q:\begin{array}{rcl} \mathbf {Z}^{2g} &{} \longrightarrow &{} \mathbf {Z}/\Lambda _n \\ (x_1,\ldots x_g,y_1,\ldots ,y_g) &{} \longmapsto &{} \sum _{i=1}^gx_iy_i \end{array}\quad \text {and}\quad J_{g,(-1)^n}=\left( \begin{matrix} 0 &{} I \\ (-1)^n I &{} 0 \end{matrix}\right) , \end{aligned}$$\end{document}so the automorphism group of the quadratic form can be identified asGg:=Aut(H(g),λ,q)≅Og,g(Z)nevenSp2gq(Z)nodd,n≠3,7Sp2g(Z)n=3,7,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} G_g\,{:}{=}\,\mathrm {Aut}\big (H(g),\lambda ,q\big )\cong {\left\{ \begin{array}{ll} \mathrm {O}_{g,g}(\mathbf {Z})&{}\quad n\text { even}\\ \mathrm {Sp}_{2g}^q(\mathbf {Z})&{}\quad n\text { odd},n\ne 3,7\\ \mathrm {Sp}_{2g}(\mathbf {Z})&{}\quad n=3,7, \end{array}\right. } \end{aligned}$$\end{document}whereOg,g(Z)={A∈Z2g×2g∣ATJg,1A=Jg,1},Sp2g(Z)={A∈Z2g×2g∣ATJg,-1A=Jg,-1},Sp2gq(Z)={A∈Sp2g(Z)∣qA=q}.\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned}&\mathrm {O}_{g,g}(\mathbf {Z})=\{A\in \mathbf {Z}^{2g\times 2g}\mid A^T J_{g,1} A=J_{g,1}\},\\&\mathrm {Sp}_{2g}(\mathbf {Z})=\{A\in \mathbf {Z}^{2g\times 2g}\mid A^T J_{g,-1} A=J_{g,-1}\},\\&\mathrm {Sp}_{2g}^q(\mathbf {Z})=\{A\in \mathrm {Sp}_{2g}(\mathbf {Z})\mid qA=q\}. \end{aligned}$$\end{document}In the theory of theta-functions, the finite index subgroup Sp2gq(Z)≤Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}^q(\mathbf {Z})\le \mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} is known as the theta group; it is the stabiliser of the standard theta-characteristic with respect to the transitive Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document}-action on the set of even characteristics (see e.g. [58]). Using this description, it is straightforward to compute its index in Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} to be 22g-1+2g-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2^{2g-1}+2^{g-1}$$\end{document}.
Kreck’s extensions
To recall Kreck’s extensions [35, Prop. 3] describing Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g,1}^n$$\end{document} for n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document}, note that an orientation-preserving diffeomorphism of Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g$$\end{document} induces an automorphism of the quadratic form (H(g),λ,q)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$( H(g),\lambda ,q)$$\end{document}. This provides a morphism Γg,1n→Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g,1}^n\rightarrow G_g$$\end{document}, which Kreck proved to be surjective using work of Wall [56].2 This explains the first extension0⟶Tg,1n⟶Γg,1n⟶Gg⟶0.\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\longrightarrow \mathrm {T}_{g,1}^n\longrightarrow \Gamma _{g,1}^n\longrightarrow G_g\longrightarrow 0. \end{aligned}$$\end{document}The second extension describes the Torelli subgroup Tg,1n⊂Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_{g,1}^n\subset \Gamma _{g,1}^n$$\end{document} and has the form0⟶Θ2n+1⟶Tg,1n⟶ρH(g)⊗SπnSO(n)⟶0,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\longrightarrow \Theta _{2n+1}\longrightarrow \mathrm {T}_{g,1}^n\overset{\rho }{\longrightarrow } H(g)\otimes S\pi _n\mathrm {SO}(n)\longrightarrow 0, \end{aligned}$$\end{document}where SπnSO(n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S\pi _n\mathrm {SO}(n)$$\end{document} denotes the image of the morphism S:πnSO(n)→πnSO(n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S:\pi _n\mathrm {SO}(n)\rightarrow \pi _n\mathrm {SO}(n+1)$$\end{document} induced by the usual inclusion SO(n)⊂SO(n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {SO}(n)\subset \mathrm {SO}(n+1)$$\end{document}. The isomorphism type of this image can be extracted from work of Kervaire [28] to be as shown in Table 1. As a diffeomorphism supported in a disc acts trivially on cohomology, the morphism Θ2n+1→Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}\rightarrow \Gamma _{g,1}^n$$\end{document} in (1.2) has image in Tg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_{g,1}^n$$\end{document}, which explains first map in the extension (1.5). To define the second one, we canonically identify H(g)⊗SπnSO(n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$ H(g)\otimes S\pi _n\mathrm {SO}(n)$$\end{document} with Hom(Hn(Wg;Z),SπnSO(n))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Hom}(\mathrm {H}_n(W_g;\mathbf {Z}),S\pi _n\mathrm {SO}(n))$$\end{document} and note that for a given isotopy class [ϕ]∈Tgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[\phi ]\in \mathrm {T}_g^n$$\end{document} and a class [e]∈Hn(Wg;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[e]\in \mathrm {H}_n(W_g;\mathbf {Z})$$\end{document} represented by an embedded sphere e:Sn↪Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$e:S^n\hookrightarrow W_g$$\end{document}, the embedding ϕ∘e\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\phi \circ e$$\end{document} is isotopic to e, so we can assume that ϕ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\phi $$\end{document} fixes e pointwise by the isotopy extension theorem. The derivative of ϕ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\phi $$\end{document} thus induces an automorphism of the once stabilised normal bundle ϑ(e)⊕ε\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\vartheta (e)\oplus \varepsilon $$\end{document}, which after choosing a trivialisation ϑ(e)⊕ε≅εn+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\vartheta (e)\oplus \varepsilon \cong \varepsilon ^{n+1}$$\end{document} defines the image ρ([ϕ])([e])∈πnSO(n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\rho ([\phi ])([e])\in \pi _n\mathrm {SO}(n+1)$$\end{document} of [e] under the morphism ρ([ϕ])∈Hom(Hn(Wg;Z),SπnSO(n))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\rho ([\phi ])\in \mathrm {Hom}(\mathrm {H}_n(W_g;\mathbf {Z}),S\pi _n\mathrm {SO}(n))$$\end{document}, noting that ρ([ϕ])([e])\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\rho ([\phi ])([e])$$\end{document} is independent of all choices and actually lies in the subgroup SπnSO(n)⊂πnSO(n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S\pi _n\mathrm {SO}(n)\subset \pi _n\mathrm {SO}(n+1)$$\end{document} (see [35, Lem. 1]).
Instead of the extensions (1.4) and (1.5), we shall mostly be concerned with two closely related variants which we describe now. By Kreck’s result, the morphism Θ2n+1→Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}\rightarrow \Gamma _{g,1}^n$$\end{document} is injective, so gives rise to an extension 0→Θ2n+1→Γg,1n→Γg,1n/Θ2n+1→0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$0\rightarrow \Theta _{2n+1}\rightarrow \Gamma ^n_{g,1}\rightarrow \Gamma ^n_{g,1}/\Theta _{2n+1}\rightarrow 0$$\end{document}, which combined with the canonical identification Γg,1n/Θ2n+1≅Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1}/\Theta _{2n+1}\cong \Gamma ^n_{g,1/2}$$\end{document} of Lemma 1.2 has the form0⟶Θ2n+1⟶Γg,1n⟶Γg,1/2n⟶0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\longrightarrow \Theta _{2n+1}\longrightarrow \Gamma ^n_{g,1}\longrightarrow \Gamma ^n_{g,1/2}\longrightarrow 0 \end{aligned}$$\end{document}and agrees with the extension induced by taking path components of the chain of inclusions Diff∂(D2n)⊂Diff∂(Wg,1)⊂Diff∂/2(Wg,1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Diff}_{\partial }(D^{2n})\subset \mathrm {Diff}_{\partial }(W_{g,1})\subset \mathrm {Diff}_{\partial /2}(W_{g,1})$$\end{document}. The action of Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1/2}$$\end{document} on H(g) preserves the quadratic form as Γg,1n→Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1}\rightarrow \Gamma ^n_{g,1/2}$$\end{document} is surjective by Lemma 1.2, so this action yields an extension0⟶H(g)⊗SπnSO(n)⟶Γg,1/2n⟶pGg⟶0,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\longrightarrow H(g)\otimes S\pi _n\mathrm {SO}(n)\longrightarrow \Gamma ^n_{g,1/2}\overset{p}{\longrightarrow } G_g\longrightarrow 0, \end{aligned}$$\end{document}which, via the isomorphism Tg,1n/Θ2n+1≅H(g)⊗SπnSO(n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_{g,1}^n/\Theta _{2n+1}\cong H(g)\otimes S\pi _n\mathrm {SO}(n)$$\end{document} induced by (1.5), corresponds to the quotient of the extension (1.4) by Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}$$\end{document}, using Γg,1n/Θ2n+1≅Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1}/\Theta _{2n+1}\cong \Gamma ^n_{g,1/2}$$\end{document} once more.
Lemma 1.3
The action of Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document} on H(g)⊗SπnSO(n)≅Hom(Hn(Wg;Z),SπnSO(n))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$H(g)\otimes S\pi _n\mathrm {SO}(n)\cong \mathrm {Hom}(\mathrm {H}_n(W_{g};\mathbf {Z}),S\pi _n\mathrm {SO}(n))$$\end{document} induced by the extension (1.7) is through the standard action of Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document} on Hn(Wg;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_n(W_{g};\mathbf {Z})$$\end{document}.
Proof
In view of the commutative diagram
it suffices to establish the identity ρ(ϕ∘ψ∘ϕ-1)=p(ϕ)·ρ(ψ)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\rho (\phi \circ \psi \circ \phi ^{-1})=p(\phi )\cdot \rho (\psi )$$\end{document} for all ϕ∈Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\phi \in \Gamma ^n_{g,1}$$\end{document} and ψ∈Tg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\psi \in \mathrm {T}_{g,1}^n$$\end{document}. Unwrapping the definition of ρ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\rho $$\end{document}, the image of p(ϕ)·ρ(ψ)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$p(\phi )\cdot \rho (\psi )$$\end{document} on a homology class in Hn(Wg,1;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_n(W_{g,1};\mathbf {Z})$$\end{document} is given by the automorphismεn+1⟶F-1ϑ(ϕ-1∘e)⊕ε⟶d(ψ)ϑ(ϕ-1∘e)⊕ε⟶Fεn+1,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \varepsilon ^{n+1}\overset{F^{-1}}{\longrightarrow }\vartheta (\phi ^{-1}\circ e)\oplus \varepsilon \overset{d(\psi )}{\longrightarrow }\vartheta (\phi ^{-1}\circ e)\oplus \varepsilon \overset{F}{\longrightarrow }\varepsilon ^{n+1}, \end{aligned}$$\end{document}where e is an embedded sphere which represents the homology class and is pointwise fixed by ϕ∘ψ∘ϕ-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\phi \circ \psi \circ \phi ^{-1}$$\end{document} and F is any choice of framing of ϑ(ϕ-1∘e)⊕ε\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\vartheta (\phi ^{-1}\circ e)\oplus \varepsilon $$\end{document}. For the particular choice of framingϑ(e)⊕ε⟶d(ϕ-1)⊕εϑ(ϕ-1∘e)⊕ε⟶Fεn+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \vartheta (e)\oplus \varepsilon \overset{d(\phi ^{-1})\oplus \varepsilon }{\longrightarrow }\vartheta (\phi ^{-1}\circ e)\oplus \varepsilon \overset{F}{\longrightarrow }\varepsilon ^{n+1} \end{aligned}$$\end{document}to compute the image of [e]∈Hn(Wg,1;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[e]\in \mathrm {H}_n(W_{g,1};\mathbf {Z})$$\end{document} under ρ(ϕ∘ψ∘ϕ-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\rho (\phi \circ \psi \circ \phi ^{-1})$$\end{document}, the claimed identity is a consequence of the chain rule for the differential. □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
Stabilisation
Iterating the stabilisation map (1.1) induces a morphism
of group extensions for h≤g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$h\le g$$\end{document}, which exhibits the upper row as the pullback of the lower row, so the extension (1.6) for a fixed genus g determines those for all h≤g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$h\le g$$\end{document}. The situation for the extension (1.7) is similar: writing Wg,1≅Wh,1♮Wg-h,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}\cong W_{h,1}\natural W_{g-h,1}$$\end{document}, we obtain a decomposition H(g)≅H(h)⊕H(g-h)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$H(g)\cong H(h)\oplus H(g-h)$$\end{document}, which yields a stabilisation maps:=(-)⊕idH(g-h):Gh⟶Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} s\,{:}{=}\,(-)\oplus \mathrm {id}_{H(g-h)}:G_h\longrightarrow G_{g} \end{aligned}$$\end{document}and two morphisms of Gh\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_h$$\end{document}-modules, an inclusion H(h)→s∗H(g)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$H(h)\rightarrow s^*H(g)$$\end{document} and a projection s∗H(g)→H(h)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$s^*H(g)\rightarrow H(h)$$\end{document}. These morphisms express the extension (1.7) for genus h≤g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$h\le g$$\end{document} as being obtained from that for genus g by pulling back along s:Gh→Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$s:G_h\rightarrow G_{g}$$\end{document} followed by forming the extension pushout along s∗H(g)→H(h)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$s^*H(g)\rightarrow H(h)$$\end{document}. They also induce a morphism of extensions of the form
The action on the set of stable framings and Theorem A
This section serves to resolve the extension problem0⟶H(g)⊗SπnSO(n)⟶Γg,1/2n⟶Gg⟶0,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\longrightarrow H(g)\otimes S\pi _n\mathrm {SO}(n)\longrightarrow \Gamma _{g,1/2}^n\longrightarrow G_g\longrightarrow 0, \end{aligned}$$\end{document}described in the previous section. Our approach is in parts inspired by work of Crowley [15], who identified this extension in the case n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}.
The group Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1/2}$$\end{document} acts on the set of equivalence classes of stable framingsF:TWg,1⊕εk≅ε2n+kfork≫0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} F:TW_{g,1}\oplus \varepsilon ^k\cong \varepsilon ^{2n+k}\quad \text {for }k\gg 0 \end{aligned}$$\end{document}that extend the standard stable framing on TD2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$TD^{2n-1}$$\end{document}, by pulling back stable framings along the derivative. As the equivalence classes of such framings naturally form a torsor for the group of pointed homotopy classes [Wg,1,SO]∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[W_{g,1},\mathrm {SO}]_*$$\end{document}, the action of Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1/2}$$\end{document} on a fixed choice of stable framing F as above yields a functionsF:π0Diff∂/2(Wg,1)⟶[Wg,1,SO]∗≅Hom(Hn(Wg;Z),πnSO)≅H(g)⊗πnSO,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} s_F:\pi _0\mathrm {Diff}_{\partial /2}(W_{g,1})\longrightarrow [W_{g,1},\mathrm {SO}]_*\cong \mathrm {Hom}(\mathrm {H}_n(W_g;\mathbf {Z}),\pi _n\mathrm {SO})\cong H(g)\otimes \pi _n\mathrm {SO}, \end{aligned}$$\end{document}where the first isomorphism is induced by πn(-)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _n(-)$$\end{document} and the Hurewicz isomorphism, and the second one by the universal coefficient theorem. This function is a 1-cocycle (or crossed homomorphism) for the canonical action of Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1/2}$$\end{document} on H(g)⊗πnSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$ H(g)\otimes \pi _n\mathrm {SO}$$\end{document} (cf. [15, Prop. 3.1]) and as this action factors through the map p:Γg,1/2n→Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$p:\Gamma ^n_{g,1/2}\rightarrow G_g$$\end{document}, we obtain a morphism of the form(sF,p):Γg,1/2n→(H(g)⊗πnSO)⋊Gg,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} (s_F,p):\Gamma ^n_{g,1/2}\rightarrow ( H(g)\otimes \pi _n\mathrm {SO})\rtimes G_g, \end{aligned}$$\end{document}which is independent of F up to conjugation in the target by a straightforward check. This induces a morphism from the extension (2.1) to the trivial extension of Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document} by the Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document}-module H(g)⊗πnSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$H(g)\otimes \pi _n\mathrm {SO}$$\end{document},
The left vertical map is induced by the natural map SπnSO(n)→πnSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S\pi _n\mathrm {SO}(n)\rightarrow \pi _n\mathrm {SO}$$\end{document} originating from the inclusion SO(n)⊂SO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {SO}(n)\subset \mathrm {SO}$$\end{document} and is an isomorphism for n≠1,3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 1,3,7$$\end{document} odd as a consequence of the following lemma whose proof is standard (see e.g. [41, §1B)]).
Lemma 2.1
For n odd, the morphism SπnSO(n)→πnSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S\pi _n\mathrm {SO}(n)\rightarrow \pi _n\mathrm {SO}$$\end{document} induced by the inclusion SO(n)⊂SO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {SO}(n)\subset \mathrm {SO}$$\end{document} is an isomorphism for n≠1,3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 1,3,7$$\end{document} odd. For n=1,3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=1,3,7$$\end{document}, it is injective with cokernel Z/2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}/2$$\end{document}.
As a result, the diagram (2.2) induces a splitting of (2.1) for n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document} odd, since all vertical maps are isomorphisms. This proves the cases n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document} of the following reformulation of Theorem A (see Sect. 1.3). We postpone the proof of the cases n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document} to Sect. 3.6.
Theorem 2.2
For n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} odd, the extension0⟶H(g)⊗SπnSO(n)⟶Γg,1/2n⟶pGg⟶0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\longrightarrow H(g)\otimes S\pi _n\mathrm {SO}(n)\longrightarrow \Gamma ^n_{g,1/2}\overset{p}{\longrightarrow } G_g\longrightarrow 0 \end{aligned}$$\end{document}splits for n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document}. For n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}, it splits if and only if g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}.
Even though the extension does not split for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}, the morphism (2.2) is still injective by Lemma 2.1 and thus expresses the extension in question as a subextension of the trivial extension of Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document} by the Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document}-module H(g)⊗πnSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$H(g)\otimes \pi _n\mathrm {SO}$$\end{document}. Crowley [15, Cor. 3.5] gave an algebraic description of this subextension and concluded that it splits if and only if g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}. We proceed differently and prove this fact in Sect. 3.3 directly, which can in turn be used to determine the extension in the following way: by the discussion in Sect. 1.4, it is sufficient to determine its extension class in H2(Gg;H(g)⊗SπnSO(n))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(G_g;H(g)\otimes S\pi _n\mathrm {SO}(n))$$\end{document} for g≫0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\gg 0$$\end{document}. For n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}, we have Gg≅Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g\cong \mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} with its usual action on H(g)⊗SπnSO(n)≅Z2g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$H(g)\otimes S\pi _n\mathrm {SO}(n)\cong \mathbf {Z}^{2g}$$\end{document}. Using work of Djament [16, Thm 1], one can compute H2(Sp2g(Z);Z2g)≅Z/2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z}^{2g})\cong \mathbf {Z}/2$$\end{document} for g≫0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\gg 0$$\end{document}, so there is only one nontrivial extension of Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document} by H(g)⊗SπnSO(n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$H(g)\otimes S\pi _n\mathrm {SO}(n)$$\end{document}, which must be the one in consideration because of the second part of Theorem 2.2. Note that this line of argument gives a geometric proof for the following useful fact on the twisted cohomology of Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} as a byproduct, which can also be derived algebraically (see for instance [15, Sect. 2]).
Corollary 2.3
The pullback of the unique nontrivial class in H2(Sp2g(Z);Z2g)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z}^{2g})$$\end{document} for g≫0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\gg 0$$\end{document} to H2(Sp2h(Z);Z2h)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\mathrm {Sp}_{2h}(\mathbf {Z});\mathbf {Z}^{2h})$$\end{document} for h≤g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$h\le g$$\end{document} is trivial if and only if h=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$h=1$$\end{document}.
We close this section by relating the abelianisation of Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1/2}$$\end{document} to that of Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document}. The latter is content of Lemma A.1.
Corollary 2.4
For n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} odd, the morphismH1(Γg,1/2n)⟶H1(Gg)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {H}_1(\Gamma ^n_{g,1/2})\longrightarrow \mathrm {H}_1(G_g) \end{aligned}$$\end{document}is split surjective and has the coinvariants (H(g)⊗SπnSO(n))Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(H(g)\otimes S\pi _n\mathrm {SO}(n))_{G_g}$$\end{document} as its kernel, which vanish for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document}. For g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} it vanishes if and only if n≡5(mod8)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 5\ (\mathrm {mod}\ 8)$$\end{document} or n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}, and has order 2 otherwise.
Proof
The claim regarding the coinvariants follows from Lemma A.2 and Table 1. Since they vanish for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document}, the remaining statement follows from the exact sequenceH2(Γg,1/2n)→H2(Gg)→(H(g)⊗SπnSO(n))Gg→H1(Γg,1/2n)→H1(Gg)→0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {H}_2(\Gamma ^n_{g,1/2})\rightarrow \mathrm {H}_2(G_g)\rightarrow \big (H(g)\otimes S\pi _n\mathrm {SO}(n)\big )_{G_g}\rightarrow \mathrm {H}_1(\Gamma ^n_{g,1/2})\rightarrow \mathrm {H}_1(G_g)\rightarrow 0 \end{aligned}$$\end{document}induced by (2.1), combined with the fact that this extension splits for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} by Theorem 2.2. □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
Signatures, obstructions, and Theorem B
By Lemma 1.2, the extension0⟶Θ2n+1⟶Γg,1n⟶Γg,1/2n⟶0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\longrightarrow \Theta _{2n+1}\longrightarrow \Gamma ^n_{g,1}\longrightarrow \Gamma ^n_{g,1/2}\longrightarrow 0 \end{aligned}$$\end{document}discussed in Sect. 1.3 is central and is as such classified by a class in H2(Γg,1/2n;Θ2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\Gamma ^n_{g,1/2};\Theta _{2n+1})$$\end{document} with Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1/2}$$\end{document} acting trivially on Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}$$\end{document}. In this section, we identify this extension class in terms of the algebraic description of Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1/2}$$\end{document} provided in the previous section, leading to a proof of our main result Theorem B. Our approach is partially based on ideas of Galatius–Randal-Williams [21, Sect. 7], who determined the extension for n≡5(mod8)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 5\ (\mathrm {mod}\ 8)$$\end{document} and g≥5\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 5$$\end{document} up to automorphisms of Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}$$\end{document}.
We begin with an elementary recollection on the relation between Pontryagin classes and obstructions to extending trivialisations of vector bundles, mainly to fix notation.
Obstructions and Pontryagin classes
Let k≥1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$k\ge 1$$\end{document} and ξ:X→τ>4k-1BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\xi :X\rightarrow \tau _{>4k-1}\mathrm {BO}$$\end{document} be a map to the (4k-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(4k-1)$$\end{document}st connected cover of BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {BO}$$\end{document} with a lift ξ¯:A→τ>4kBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\bar{\xi }:A\rightarrow \tau _{>4k}\mathrm {BO}$$\end{document} over a subspace A⊂X\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$A\subset X$$\end{document} along the canonical map τ>4kBO→τ>4k-1BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>4k}\mathrm {BO}\rightarrow \tau _{>4k-1}\mathrm {BO}$$\end{document}. Such data has a relative Pontryagin class pk(ξ,ξ¯)∈H4k(X,A;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$p_k(\xi ,\bar{\xi })\in \mathrm {H}^{4k}(X,A;\mathbf {Z})$$\end{document} given as the pullback along the map (ξ,ξ¯):(X,A)→(τ>4k-1BO,τ>4kBO)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\xi ,\bar{\xi }):(X,A)\rightarrow (\tau _{>4k-1}\mathrm {BO},\tau _{>4k}\mathrm {BO})$$\end{document} of the unique lift to H4k(τ>4k-1BO,τ>4kBO;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^{4k}(\tau _{>4k-1}\mathrm {BO},\tau _{>4k}\mathrm {BO};\mathbf {Z})$$\end{document} of the pullback pk∈H4k(τ>4k-1BO;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$p_k\in \mathrm {H}^{4k}(\tau _{>4k-1}\mathrm {BO};\mathbf {Z})$$\end{document} of the usual Pontryagin class pk∈H4k(BSO;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$p_k\in \mathrm {H}^{4k}(\mathrm {BSO};\mathbf {Z})$$\end{document}. The class pk(ξ,ξ¯)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$p_k(\xi ,\bar{\xi })$$\end{document} is related to the primary obstruction χ(ξ,ξ¯)∈H4k(X,A;π4k-1SO)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi (\xi ,\bar{\xi })\in \mathrm {H}^{4k}(X,A;\pi _{4k-1}\mathrm {SO})$$\end{document} to solving the lifting problem
by the equalitypk(ξ,ξ¯)=±ak(2k-1)!·χ(ξ,ξ¯),\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} p_k(\xi ,\bar{\xi })=\pm a_k(2k-1)!\cdot \chi (\xi ,\bar{\xi }), \end{aligned}$$\end{document}up to the choice of a generator π4k-1SO≅Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _{4k-1}\mathrm {SO}\cong \mathbf {Z}$$\end{document} (cf. [44, Lem. 2]). We suppress the lift ξ¯\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\bar{\xi }$$\end{document} from the notation whenever there is no source of confusing. For us, X=M\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$X=M$$\end{document} will usually be a compact oriented 8k-manifold and A=∂M\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$A=\partial M$$\end{document} its boundary, in which case we can evaluate χ2(ξ,ξ¯)∈H8k(M,∂M;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2(\xi ,\bar{\xi })\in \mathrm {H}^{8k}(M,\partial M;\mathbf {Z})$$\end{document} against the relative fundamental class [M,∂M]\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[M,\partial M]$$\end{document} to obtain a number χ2(ξ,ξ¯)∈Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2(\xi ,\bar{\xi })\in \mathbf {Z}$$\end{document}. The following two sources of manifolds are relevant for us.
For a compact oriented n-connected (2n+2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(2n+2)$$\end{document}-manifold whose boundary is a homotopy sphere, there is a (up to homotopy) unique lift M→τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$M\rightarrow \tau _{>n}\mathrm {BO}$$\end{document} of the stable oriented normal bundle. On the boundary ∂M\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\partial M$$\end{document}, this lifts uniquely further to τ>n+1BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n+1}\mathrm {BO}$$\end{document}, so we obtain a canonical class χ(M)∈Hn+1(M,∂M;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi (M)\in \mathrm {H}^{n+1}(M,\partial M;\mathbf {Z})$$\end{document} and a characteristic number χ2(M)∈Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2(M)\in \mathbf {Z}$$\end{document}.
Consider a (Wg,1,D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(W_{g,1},D^{2n-1})$$\end{document}-bundle π:E→B\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi :E\rightarrow B$$\end{document}, i.e. a smooth Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}$$\end{document}-bundle with a trivialised D2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^{2n-1}$$\end{document}-subbundle of its ∂Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\partial W_{g,1}$$\end{document}-bundle ∂π:∂E→B\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\partial \pi :\partial E\rightarrow B$$\end{document} of boundaries. The standard framing of D2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^{2n-1}$$\end{document} induces a trivialisation of stable vertical tangent bundle TπE:E→BSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$T_\pi E:E\rightarrow \mathrm {BSO}$$\end{document} over the subbundle B×D2n-1⊂∂E\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$B\times D^{2n-1}\subset \partial E$$\end{document}, which extends uniquely to a τ>n+1BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n+1}\mathrm {BO}$$\end{document}-structure on TπE|∂E\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$T_\pi E|_{\partial E}$$\end{document} by obstruction theory. Using that Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_g$$\end{document} is n-parallelisable, another application of obstruction theory shows that the induced τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}$$\end{document}-structure on TπE|∂E\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$T_\pi E|_{\partial E}$$\end{document} extends uniquely to a τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}$$\end{document}-structure on TπE\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$T_\pi E$$\end{document}, so the above discussion provides a class χ(TπE)∈Hn+1(E,∂E;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi (T_\pi E)\in \mathrm {H}^{n+1}(E,\partial E;\mathbf {Z})$$\end{document}, and, assuming B is an oriented closed surface, a number χ2(TπE)∈Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2(T_\pi E)\in \mathbf {Z}$$\end{document}.
Highly connected almost closed manifolds
As a consequence of Theorem 3.12, we shall see that (Wg,1,D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(W_{g,1},D^{2n-1})$$\end{document}-bundles over surfaces are closely connected to n-connected almost closed (2n+2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(2n+2)$$\end{document}-manifolds. These manifolds were classified by Wall [55], which we now recall for n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} in a form tailored to later applications, partly following [34, Sect. 2].
A compact manifold M is almost closed if its boundary is a homotopy sphere. We write Adτ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$A_{d}^{\tau _{>n}}$$\end{document} for the abelian group of almost closed oriented n-connected d-manifolds up to oriented n-connected bordism restricting to an h-cobordism on the boundary. Recall that Ωdτ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Omega _{d}^{\tau _{>n}}$$\end{document} denotes the bordism group of closed d-manifolds M equipped with a τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}$$\end{document}-structure on their stable normal bundle M→BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$M\rightarrow \mathrm {BO}$$\end{document}, i.e. a lift M→τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$M\rightarrow \tau _{>n}\mathrm {BO}$$\end{document} to the n-connected cover. By classical surgery, the group Ωdτ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Omega _{d}^{\tau _{>n}}$$\end{document} is canonically isomorphic to the bordism group of closed oriented n-connected d-manifolds up to n-connected bordism as long as d≥2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d\ge 2n+1$$\end{document}, so we will use both descriptions interchangeably. There is an exact sequenceΘ2n+2⟶Ω2n+2τ>n⟶A2n+2τ>n⟶∂Θ2n+1⟶Ω2n+1τ>n⟶0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \Theta _{2n+2}\longrightarrow \Omega _{2n+2}^{\tau _{>n}}\longrightarrow A_{2n+2}^{\tau _{>n}}\overset{\partial }{\longrightarrow }\Theta _{2n+1}\longrightarrow \Omega _{2n+1}^{\tau _{>n}}\longrightarrow 0 \end{aligned}$$\end{document}due to Wall [57, p. 293] in which the two outer morphisms are the obvious ones, noting that homotopy d-spheres n-connected for n<d\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n<d$$\end{document}. The morphisms Ω2n+2τ>n→A2n+2τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Omega _{2n+2}^{\tau _{>n}}\rightarrow A_{2n+2}^{\tau _{>n}}$$\end{document} and ∂:A2n+2τ>n→Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\partial :A_{2n+2}^{\tau _{>n}}\rightarrow \Theta _{2n+1}$$\end{document} are given by cutting out an embedded disc and by assigning to an almost closed manifold its boundary, respectively. By surgery theory, the subgroupbA2n+2:=im(A2n+2τ>n⟶∂Θ2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {bA}_{2n+2}\,{:}{=}\,\mathrm {im}(A_{2n+2}^{\tau _{>n}}\overset{\partial }{\longrightarrow }\Theta _{2n+1}) \end{aligned}$$\end{document}of homotopy (2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(2n+1)$$\end{document}-spheres bounding n-connected manifolds contains the cyclic subgroup bP2n+2⊂Θ2n+2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bP}_{2n+2}\subset \Theta _{2n+2}$$\end{document} of homotopy (2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(2n+1)$$\end{document}-spheres bounding parallelisable manifolds, so the right end of (3.1) receives canonical a map from Kervaire–Milnor’s exact sequence [30],
which in particular induces a morphism coker(J)2n+1→Ω2n+1τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {coker}(J)_{2n+1}\rightarrow \Omega ^{\tau _{>n}}_{2n+1}$$\end{document}, concretely given by representing a class in coker(J)2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {coker}(J)_{2n+1}$$\end{document} by a stably framed manifold and restricting its stable framing to a τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}$$\end{document}-structure.
Wall’s classification
For our purposes, Wall’s computation [55, 57] of A2n+2τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$A_{2n+2}^{\tau _{>n}}$$\end{document} is for n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} odd is most conveniently stated in terms of two particular almost closed n-connected (2n+2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(2n+2)$$\end{document}-manifolds, namely
Milnor’sE8\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$E_8$$\end{document}-plumbingP, arising from plumbing together 8 copies of the disc bundle of the tangent bundle of the standard (n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(n+1)$$\end{document}-sphere such that the intersection form of P agrees with the E8\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$E_8$$\end{document}-form (see e.g.[8, Ch. V.2]), and
the manifold Q, obtained from plumbing together two copies of a linear Dn+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^{n+1}$$\end{document}-bundle over the (n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(n+1)$$\end{document}-sphere representing a generator of SπnSO(n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S\pi _n\mathrm {SO}(n)$$\end{document}.
The following can be derived from Wall’s work (see e.g. [34, Thm 2.1]).
Theorem 3.2
(Wall) For n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} odd, the bordism group A2n+2τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$A_{2n+2}^{\tau _{>n}}$$\end{document} satisfiesA2n+2τ>n≅Z⊕Z/2ifn≡1(mod8)Z⊕Zifn≡3(mod4)Zifn≡5(mod8).\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} A_{2n+2}^{\tau _{>n}}\cong {\left\{ \begin{array}{ll} \mathbf {Z}\oplus \mathbf {Z}/2&{}\quad \text{ if } n\equiv 1\ (\mathrm {mod}\ 8)\\ \mathbf {Z}\oplus \mathbf {Z}&{}\quad \text{ if } n\equiv 3\ (\mathrm {mod}\ 4)\\ \mathbf {Z}&{}\quad \text{ if } n\equiv 5\ (\mathrm {mod}\ 8). \end{array}\right. } \end{aligned}$$\end{document}The first summand is generated by P in all cases but n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document} where it is generated by HP2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {H}P^2$$\end{document} and OP2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {O}P^2$$\end{document}. The second summand for n≡5(mod8)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 5\ (\mathrm {mod}\ 8)$$\end{document} is generated by Q.
From a consultation of Table 1, one sees that the group SπnSO(n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S\pi _n\mathrm {SO}(n)$$\end{document} vanishes for n≡5(mod8)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 5\ (\mathrm {mod}\ 8)$$\end{document}, so Q∈A2n+2τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$Q\in A_{2n+2}^{\tau _{>n}}$$\end{document} is trivial in this case, which shows that the subgroup bA2n+2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}$$\end{document} is for all n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} odd generated by the boundariesΣP:=∂P∈bA2n+2andΣQ:=∂Q∈bA2n+3.\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \Sigma _P\,{:}{=}\,\partial P\in \mathrm {bA}_{2n+2}\quad \text {and}\quad \Sigma _Q\,{:}{=}\,\partial Q\in \mathrm {bA}_{2n+3}. \end{aligned}$$\end{document}In the cases n≡1(mod8)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 1\ (\mathrm {mod}\ 8)$$\end{document} in which Q defines a Z/2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}/2$$\end{document}-summand, its boundary ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document} is trivial by a result of Schultz [50, Cor. 3.2, Thm 3.4 iii)]. For n≡3(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 3\ (\mathrm {mod}\ 4)$$\end{document} on the other hand, it is nontrivial by a calculation of Kosinski [31, p. 238–239].
For n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} odd, the Milnor sphere ΣP∈Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _P\in \Theta _{2n+1}$$\end{document} is well-known to be nontrival and to generate the cyclic subgroup bP2n+2⊂Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bP}_{2n+2}\subset \Theta _{2n+1}$$\end{document} whose order can be expressed in terms of numerators of divided Bernoulli numbers (see e.g. [41, Lem. 3.5 (2), Cor. 3.20]), so Theorem 3.3 has the following corollary.
Corollary 3.5
For n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} odd, the subgroup bA2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+1}$$\end{document} is nontrivial. It is generated by ΣP\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _P$$\end{document} for n≡1(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 1\ (\mathrm {mod}\ 4)$$\end{document}, by ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document} for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}, and by ΣP\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _P$$\end{document} and ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document} for n≡3(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 3\ (\mathrm {mod}\ 4)$$\end{document}.
Combining the previous results with the diagram (3.2), we obtain the following result, which we already mentioned in the introduction.
Corollary 3.6
The natural morphism coker(J)2n+1→Ω2n+2τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {coker}(J)_{2n+1}\rightarrow \Omega ^{\tau _{>n}}_{2n+2}$$\end{document} is an isomorphism for n≡1(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 1\ (\mathrm {mod}\ 4)$$\end{document} and for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}. For n≡3(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 3\ (\mathrm {mod}\ 4)$$\end{document}, it is an epimorphism whose kernel is generated by the class [ΣQ]∈coker(J)2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[\Sigma _Q]\in \mathrm {coker}(J)_{2n+1}$$\end{document}.
[ΣQ]∈coker(J)2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[\Sigma _Q]\in \mathrm {coker}(J)_{2n+1}$$\end{document} is conjecturally trivial [21, Conj. A] for all n odd. Until recently (see Remark 3.8), this was only known for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document} and n≡1(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 1\ (\mathrm {mod}\ 4)$$\end{document} by the results above.
As mentioned in the introduction, after the completion of this work, Burklund–Hahn–Senger [11] and Burklund–Senger [12] showed that [ΣQ]\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[\Sigma _Q]$$\end{document} vanishes in coker(J)2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {coker}(J)_{2n+1}$$\end{document} for n odd if and only if n≠11\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 11$$\end{document}, confirming Conjecture 3.7 for n≠11\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 11$$\end{document} and disproving it for n=11\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=11$$\end{document}. This has as a consequence that, for n≠11\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 11$$\end{document} odd, the subgroup bA2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+1}$$\end{document} is generated by ΣP\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _P$$\end{document} even for n≡3(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 3\ (\mathrm {mod}\ 4)$$\end{document} and that the morphism coker(J)2n+1→Ω2n+2τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {coker}(J)_{2n+1}\rightarrow \Omega _{2n+2}^{\tau _{>n}}$$\end{document} discussed in Corollary 3.6 is an isomorphism.
Invariants
It follows from Theorems 3.2 and 3.3 that the boundary of an n-connected almost closed (2n+2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(2n+2)$$\end{document}-manifold M is determined by at most two integral bordism invariants of M. Concretely, we consider the signature sgn:A2n+2τ>n→Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {sgn}:A_{2n+2}^{\tau _{>n}}\rightarrow \mathbf {Z}$$\end{document} and for n≡3(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 3\ (\mathrm {mod}\ 4)$$\end{document} the characteristic number χ2:A2n+2τ>n→Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2:A_{2n+2}^{\tau _{>n}}\rightarrow \mathbf {Z}$$\end{document}, explained in Example 3.1. As discussed for example in [34, Sect. 2.1], these functionals evaluate tosgn(P)=8sgn(Q)=0χ2(P)=0χ2(Q)=8forn=3,72otherwise,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {sgn}(P)=8 \quad \quad \mathrm {sgn}(Q)=0\quad \chi ^2(P)=0\quad \chi ^2(Q)={\left\{ \begin{array}{ll}8&{}\quad \text {for }n=3,7\\ 2&{}\quad \text {otherwise}\end{array}\right. }, \end{aligned}$$\end{document}and on the closed manifolds HP2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {H}P^2$$\end{document} and OP2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {O}P^2$$\end{document} tosgn(HP2)=sgn(OP2)=1χ2(HP2)=χ2(OP2)=1,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {sgn}(\mathbf {H}P^2)=\mathrm {sgn}(\mathbf {O}P^2)=1\quad \quad \chi ^2(\mathbf {H}P^2)=\chi ^2(\mathbf {O}P^2)=1, \end{aligned}$$\end{document}which results in the following formula for boundary spheres of highly connected manifolds when combined with the discussion above.
As in the introduction, we denote by σn′\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\sigma _n'$$\end{document} the minimal positive signature of a smooth closed n-connected (2n+2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(2n+2)$$\end{document}-manifold. This satisfies σn′=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\sigma _n'=1$$\end{document} for n=1,3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=1,3,7$$\end{document} as witnessed by CP2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {C}P^2$$\end{document}, HP2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {H}P^2$$\end{document}, and OP2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {O}P^2$$\end{document}, and in all other cases, it can be expressed in terms of the subgroup bA2n+2⊂Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}\subset \Theta _{2n+1}$$\end{document} as follows.
Lemma 3.10
For n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} odd, the quotient bA2n+2/⟨ΣQ⟩\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}/\langle \Sigma _Q \rangle $$\end{document} is a cyclic group generated by the class of ΣP\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _P$$\end{document}. It is trivial if n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document} and of order σn′/8\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\sigma _n'/8$$\end{document} otherwise.
Proof
For n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}, the claim is a consequence of Theorem 3.2 and Lemma 3.4. In the case n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document}, it follows from taking vertical cokernels in the commutative diagram
with exact rows, obtained from a combination of Theorem 3.2 with (3.1) and (3.3). □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
Remark 3.11
In [34, Prop. 2.15], the minimal signature σn′\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\sigma _n'$$\end{document} was expressed in terms of Bernoulli numbers and the order of [ΣQ]∈coker(J)2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[\Sigma _Q]\in \mathrm {coker}(J)_{2n+1}$$\end{document}, from which one can conclude that for n≠1,3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 1,3,7$$\end{document}, the signature of such manifolds is divisible by 2n+3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2^{n+3}$$\end{document} if (n+1)/2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(n+1)/2$$\end{document} is odd and by 2n-2ν2(n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2^{n-2\nu _2(n+1)}$$\end{document} otherwise, where ν2(-)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\nu _2(-)$$\end{document} denotes the 2-adic valuation (see [34, Cor. 2.18]).
Bundles over surfaces and almost closed manifolds
In order to identify the cohomology class in H2(Γg,1/2n;Θ2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\Gamma _{g,1/2}^n;\Theta _{2n+1})$$\end{document} that classifies the central extension0⟶Θ2n+1⟶Γg,1n⟶Γg,1/2n⟶0,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\longrightarrow \Theta _{2n+1}\longrightarrow \Gamma ^n_{g,1}\longrightarrow \Gamma ^n_{g,1/2}\longrightarrow 0, \end{aligned}$$\end{document}we first determine how it evaluates against homology classes, i.e. identify its imaged2:H2(Γg,1/2n;Z)⟶Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} d_2:\mathrm {H}_2(\Gamma ^n_{g,1/2};\mathbf {Z})\longrightarrow \Theta _{2n+1} \end{aligned}$$\end{document}under the map h participating in the universal coefficient theorem0→Ext(H1(Γg,1/2n;Z),Θ2n+1)→H2(Γg,1/2n;Θ2n+1)⟶hHom(H2(Γg,1/2n;Z);Θ2n+1)→0,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned}&0\rightarrow \mathrm {Ext}(\mathrm {H}_1(\Gamma ^n_{g,1/2};\mathbf {Z}),\Theta _{2n+1})\rightarrow \mathrm {H}^2(\Gamma ^n_{g,1/2};\Theta _{2n+1})\\&\quad \overset{h}{\longrightarrow }\mathrm {Hom}(\mathrm {H}_2(\Gamma ^n_{g,1/2};\mathbf {Z});\Theta _{2n+1})\rightarrow 0, \end{aligned}$$\end{document}followed by resolving the remaining ambiguity originating from the Ext-term. Indicated by our choice of notation, the morphism d2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_2$$\end{document} can be viewed alternatively as the first possibly nontrivial differential in the E2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$E_2$$\end{document}-page of the Serre spectral sequence of the extension (3.5) (cf. [26, Thm 4]). Before identifying this differential, we remind the reader of two standard facts we shall make frequent use of.
The canonical map of spectra MSO→HZ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {MSO}\rightarrow \mathbf {HZ}$$\end{document} is 4-connected, so pushing forward fundamental classes induces an isomorphism Ω∗SO(X)→H∗(X;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Omega ^{\mathrm {SO}}_*(X)\rightarrow \mathrm {H}_*(X;\mathbf {Z})$$\end{document} for ∗≤3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$*\le 3$$\end{document} and any space X, and
the 1-truncation of a connected space X (in particular the natural map BG→Bπ0G\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {B}G\rightarrow \mathrm {B}\pi _0G$$\end{document} for a topological group G) induces a surjection H2(X;Z)↠H2(K(π1X,1);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(X;\mathbf {Z}) \twoheadrightarrow \mathrm {H}_2(K(\pi _1X,1);\mathbf {Z})$$\end{document}, whose kernel agrees with the image of the Hurewicz homomorphism π2X→H2(X;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _2X\rightarrow \mathrm {H}_2(X;\mathbf {Z})$$\end{document}.
The key geometric ingredient to identify the differential d2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_2$$\end{document} is the following result.
Theorem 3.12
Let n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} be odd and π:E→S\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi :E\rightarrow S$$\end{document} a (Wg,1,D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(W_{g,1},D^{2n-1})$$\end{document}-bundle over an oriented closed surface S. There exists a class E′∈A2n+2τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$E'\in A_{2n+2}^{\tau _{>n}}$$\end{document} such that
its boundary ∂E′∈Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\partial E'\in \Theta _{2n+1}$$\end{document} is the image of the class [π]∈H2(BDiff∂/2(Wg,1);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[\pi ]\in \mathrm {H}^2(\mathrm {BDiff}_{\partial /2}(W_{g,1});\mathbf {Z})$$\end{document} under the composition
By the isotopy extension theorem, the restriction map to the moving part of the boundary Diff∂/2(Wg,1)→Diff∂(D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Diff}_{\partial /2}(W_{g,1})\rightarrow \mathrm {Diff}_\partial (D^{2n-1})$$\end{document} is a fibration. As its image is contained in the component of the identity (see the proof of Lemma 1.2), this fibration induces the upper row in a map of fibrations
whose bottom row is induced by the extension (3.5). The two right vertical maps are induced by taking components and the left vertical map is the induced map on homotopy fibres. The latter agrees with the delooping of the Gromoll map ΩDiff∂(D2n-1)→Diff∂(D2n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Omega \mathrm {Diff}_\partial (D^{2n-1})\rightarrow \mathrm {Diff}_\partial (D^{2n})$$\end{document} followed by taking components, which one checks by looping the fibre sequences and using thatΩDiff∂id(D2n-1)⟶Diff∂(Wg,1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \Omega \mathrm {Diff}^{\mathrm {id}}_\partial (D^{2n-1})\longrightarrow \mathrm {Diff}_{\partial }(W_{g,1}) \end{aligned}$$\end{document}is given by “twisting” a collar [0,1]×S2n-1⊂Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$ [0,1]\times S^{2n-1}\subset W_{g,1}$$\end{document}, meaning that it sends a smooth loop γ∈ΩDiff∂id(D2n-1)⊂ΩDiff∂id(S2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\gamma \in \Omega \mathrm {Diff}^{\mathrm {id}}_\partial (D^{2n-1})\subset \Omega \mathrm {Diff}^{\mathrm {id}}_\partial (S^{2n-1})$$\end{document} to the diffeomorphism that is the identity outside the collar and is given by (t,x)↦(t,γ(t)·x)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(t,x)\mapsto (t,\gamma (t)\cdot x)$$\end{document} on the collar. Now consider the commutative square
obtained from delooping (3.6) once to the right and using H2(B2Θ2n+1)≅Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(B^2\Theta _{2n+1})\cong \Theta _{2n+1}$$\end{document}. By transgression, the bottom arrow agrees with the differential d2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_2$$\end{document} in the statement. Combining this with the Hurewicz theorem, the square (3.7) gives a factorisationH2(BDiff∂/2(Wg,1))→H2(BDiff∂id(D2n-1))≅π2(BDiff∂id(D2n-1))→Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {H}_2(\mathrm {BDiff}_{\partial /2}(W_{g,1}))\rightarrow \mathrm {H}_2(\mathrm {BDiff}_{\partial }^{\mathrm {id}}(D^{2n-1}))\cong \pi _2(\mathrm {BDiff}_{\partial }^{\mathrm {id}}(D^{2n-1}))\rightarrow \Theta _{2n+1} \end{aligned}$$\end{document}of the map in the first part of the statement, which thus has the following geometric description: a smooth (Wg,1,D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(W_{g,1},D^{2n-1})$$\end{document}-bundle π:E→S\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi :E\rightarrow S$$\end{document} represents a class [π]∈H2(BDiff∂/2(Wg,1);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[\pi ]\in \mathrm {H}^2(\mathrm {BDiff}_{\partial /2}(W_{g,1});\mathbf {Z})$$\end{document} and its image under the first map in the composition is the class [π+]∈H2(BDiff∂id(D2n-1);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[\pi _+]\in \mathrm {H}_2(\mathrm {BDiff}^{\mathrm {id}}_{\partial }(D^{2n-1});\mathbf {Z})$$\end{document} of its (∂Wg,1,D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\partial W_{g,1},D^{2n-1})$$\end{document}-bundle π+:E+→S\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _+:E_+\rightarrow S$$\end{document} of boundaries, which in turn maps under the inverse of the Hurewicz homomorphism to a (∂Wg,1,D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\partial W_{g,1},D^{2n-1})$$\end{document}-bundle π-:E-→S2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _-:E_-\rightarrow S^2$$\end{document} over the 2-sphere that is bordant, as a bundle, to π-\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _-$$\end{document}. That is, there exists a (∂Wg,1,D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\partial W_{g,1},D^{2n-1})$$\end{document}-bundle π¯:E¯→K\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\bar{\pi }:\bar{E}\rightarrow K$$\end{document} over an oriented bordism K between S and S2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^2$$\end{document} that restricts to π+\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _+$$\end{document} over S and to π-\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _-$$\end{document} over S2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^2$$\end{document}. We claim that the image of the (∂Wg,1,D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\partial W_{g,1},D^{2n-1})$$\end{document}-bundle π-\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _-$$\end{document} under the final map in the composition is the homotopy sphere Σπ∈Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _\pi \in \Theta _{2n+1}$$\end{document} obtained by doing surgery on the total space E-\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$E_-$$\end{document} along the trivialised subbundle D2n-1×S2⊂E-\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^{2n-1}\times S^2\subset E_-$$\end{document}. This is most easily seen by thinking of a class in πkBDiff∂(Dd)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _k\mathrm {BDiff}_\partial (D^d)$$\end{document} as a smooth bundle Dd→P→Dk\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^d\rightarrow P\rightarrow D^k$$\end{document} together with a trivialisation φ:Dd×∂Dk≅P|∂Dk\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\varphi :D^d\times \partial D^k\cong P|_{\partial D^k}$$\end{document} and a trivialised ∂Dk\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\partial D^k$$\end{document}-subbundle ψ:∂Dd×Dk↪P\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\psi :\partial D^d\times D^k\hookrightarrow P$$\end{document} such that φ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\varphi $$\end{document} and ψ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\psi $$\end{document} agree on ∂Dd×∂Dk\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\partial D^d\times \partial D^k$$\end{document}. From this point of view, the morphism πkBDiff∂(Dd)→πk-1BDiff∂(Dd×D1)≅πk-1BDiff∂(Dd+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _k\mathrm {BDiff}_\partial (D^d)\rightarrow \pi _{k-1}\mathrm {BDiff}_\partial (D^{d}\times D^1)\cong \pi _{k-1}\mathrm {BDiff}_\partial (D^{d+1})$$\end{document} induced by the Gromoll map is given by sending such a bundle p:P→Dk≅Dk-1×D1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$p:P\rightarrow D^k\cong D^{k-1}\times D^1$$\end{document} to the (Dd×D1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(D^d\times D^1)$$\end{document}-bundle (pr1∘p):P→Dk-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\mathrm {pr}_1\circ p):P\rightarrow D^{k-1}$$\end{document}, and the isomorphism π1BDiff∂(Dd)≅Θd+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _1\mathrm {BDiff}_\partial (D^{d})\cong \Theta _{d+1}$$\end{document} is given by assigning to a disc bundle Dd→P→D1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^d\rightarrow P\rightarrow D^1$$\end{document} the manifold P∪∂Dd×D1∪Dd×∂D1Dd×D1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$P\cup _{\partial D^d\times D^1\cup D^d\times \partial D^1} D^d\times D^1$$\end{document}. Deccomposing the sphere into half-discs Sn=D+n∪D-n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^n=D^n_+\cup D^n_-$$\end{document}, we see from this description that the composition πkBDiff∂(Dd)→Θd+k\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _k\mathrm {BDiff}_\partial (D^d)\rightarrow \Theta _{d+k}$$\end{document} of the iterated Gromoll map with the isomorphism π1BDiff∂(Dd+k-1)≅Θd+k\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _1\mathrm {BDiff}_\partial (D^{d+k-1})\cong \Theta _{d+k}$$\end{document} maps a class represented by an Sd\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^{d}$$\end{document}-bundle Sd→Q→Sk\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^d\rightarrow Q\rightarrow S^k$$\end{document} with a trivialisation φ:D+k×Sd≅Q|D+k\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\varphi :D_+^k\times S^d\cong Q|_{D_+^k}$$\end{document} and a trivialised subbundle ψ:Sk×D+d↪Q\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\psi :S^k\times D_+^d\hookrightarrow Q$$\end{document} that agree on D+k×D+d\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^k_+\times D^d_+$$\end{document} to the homotopy sphere(Q\int(φ(D+k×Sd)∪ψ(Sk×D+d)))∪(Dk×Dd)≅(Q\int(ψ(Sk×D+d)))∪Sk×∂Dd(Dk+1×∂Dd)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned}&\Big (Q\backslash \mathrm {int}\big (\varphi (D_+^k\times S^d)\cup \psi (S^k\times D_+^d)\big )\Big )\cup (D^k\times D^d)\\&\quad \cong \Big (Q\backslash \mathrm {int}\big (\psi (S^k\times D^d_+)\big )\Big )\cup _{S^k\times \partial D^d}(D^{k+1}\times \partial D^d) \end{aligned}$$\end{document}obtained by doing surgery along the trivialised Dd\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^d$$\end{document}-subbundle, where Dk×Dd\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^k\times D^d$$\end{document} is glued to Q\int(φ(D+k×Sd)∪∂(Dk×Sd)ψ(Sk×D+d))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$Q\backslash \mathrm {int}\big (\varphi (D_+^k\times S^d)\cup _{\partial (D^k\times S^d)} \psi (S^k\times D_+^d)\big )$$\end{document} along the embedding∂(Dk×Dd)=∂Dk×Dd∪Dk×∂Dd→φ|∂D+k×D-d∪ψ|D-k×∂D+d∂(Q\int(φ(D+k×Sd)∪ψ(Sk×D+d)).\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned}&\partial (D^k\times D^d)=\partial D^k\times D^d\cup D^k\times \partial D^d\\&\quad \xrightarrow {\varphi |_{\partial D^k_+\times D^d_-}\cup \psi |_{D^k_-\times \partial D^d_+}}\partial \Big (Q\backslash \mathrm {int}\big (\varphi (D_+^k\times S^d)\cup \psi (S^k\times D_+^d)\Big ). \end{aligned}$$\end{document}This in particular implies the claim we made above in the case k=2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$k=2$$\end{document} and d=2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d=2n-1$$\end{document}.
As a consequence of this description of the morphism in consideration, the image Σπ∈Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _\pi \in \Theta _{2n+1}$$\end{document} of the class [π]\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[\pi ]$$\end{document} comes equipped with a nullbordism, namely N:=E∪E+E¯∪E-W,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$N\,{:}{=}\,E\cup _{E_+}\bar{E}\cup _{E_-}W,$$\end{document} where W is the trace of the performed surgery. Omitting the trivialised D2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^{2n-1}$$\end{document}-subbundles, the situation can be summarised schematically as follows
A choice of a stable framing of K induces stable framings on S and S2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^2$$\end{document} and thus a stable isomorphism TE≅TπE⊕π∗TS≅sTπE\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$TE\cong T_\pi E\oplus \pi ^*TS\cong _sT_\pi E$$\end{document} using which the canonical τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}$$\end{document}-structure on TπE\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$T_\pi E$$\end{document} and the τ>n+1BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n+1}\mathrm {BO}$$\end{document}-one on TπE|E+\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$T_\pi E|_{E_+}$$\end{document} (see Example 3.1) induce a τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}$$\end{document}-structure on TE and a τ>n+1BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n+1}\mathrm {BO}$$\end{document}-structure on TE|E+≅sTE+\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$TE|_{E_+}\cong _sTE_+$$\end{document}. With these choices, we have χ(TπE)=χ(TE,TE+)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi (T_\pi E)=\chi (TE,TE_+)$$\end{document}. By construction, the restriction of this τ>n+1BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n+1}\mathrm {BO}$$\end{document}-structure to TE+|S×D2n-1≅sTS\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$TE_+|_{S\times D^{2n-1}}\cong _s TS$$\end{document} agrees with the τ>n+1BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n+1}\mathrm {BO}$$\end{document}-structure on TS obtained from the stable framing of K, so we obtain a τ>n+1BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n+1}\mathrm {BO}$$\end{document}-structure on TE¯|E+∪K×D2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$T\bar{E}|_{E_+\cup K\times D^{2n-1}}$$\end{document}, which by obstruction theory extends to one on T(E¯∪E-W)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$T(\bar{E}\cup _{E_-}W)$$\end{document}: the relative Serre spectral sequence shows that H∗(E¯,E+∪K×D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$H^*(\bar{E},E_+\cup K\times D^{2n-1})$$\end{document} vanishes for ∗≤2n-2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$*\le 2n-2$$\end{document} and thus that Hi+1(E¯∪E-W,E+∪K×D2n-1;πi(τ≤nSO))≅Hi+1(E¯∪E-W,E¯;πi(τ≤nSO))≅Hi+1(W,E-;πi(τ≤nSO))≅Hi+1(D3,S2;πi(τ≤nSO))=0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$H^{i+1}(\bar{E}\cup _{E_-}W,E_+\cup K\times D^{2n-1};\pi _i(\tau _{\le n}\mathrm {SO}))\cong H^{i+1}(\bar{E}\cup _{E_-}W,\bar{E};\pi _i(\tau _{\le n}\mathrm {SO}))\cong H^{i+1}(W,E_-;\pi _i(\tau _{\le n}SO))\cong H^{i+1}(D^3,S^2;\pi _i(\tau _{\le n}SO))=0$$\end{document} for i≤2n-3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$i\le 2n-3$$\end{document}, using W≃E-∪S2D3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W\simeq E_{-}\cup _{S^2}D^3$$\end{document} and π2SO=0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _2SO=0$$\end{document}. The restriction of this τ>n+1BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n+1}\mathrm {BO}$$\end{document}-structure on TE¯|E+∪K×D2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$T\bar{E}|_{E_+\cup K\times D^{2n-1}}$$\end{document} to a τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}$$\end{document} and the canonical τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}$$\end{document}-structure on TE≅TπE\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$TE\cong T_\pi E$$\end{document} (see Example 3.1) assemble to a τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}$$\end{document}-structure on N. By construction, the canonical restriction map (using excision)H∗(E,E+;Z)≅H∗(N,E¯∪E-W;Z)⟶H∗(N,Σπ;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} H^*(E,E_+;\mathbf {Z})\cong H^*(N,\bar{E}\cup _{E_-}W; \mathbf {Z})\longrightarrow H^*(N,\Sigma _\pi ; \mathbf {Z}) \end{aligned}$$\end{document}sends χ(TπE)=χ(TE,TE+)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi (T_\pi E)=\chi (TE,TE_+)$$\end{document} to χ(TN,TΣπ)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi (TN,T\Sigma _\pi )$$\end{document}, so we conclude χ2(TπE)=χ2(TN,TΣπ)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2(T_\pi E)=\chi ^2(TN,T\Sigma _\pi )$$\end{document}. To finish the proof, note that the τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}$$\end{document}-structure on TN allows us to do surgery away from the boundary on N to obtain an n-connected manifold E′\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$E'$$\end{document}, which gives a class in A2n+2τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$A_{2n+2}^{\tau _{>n}}$$\end{document} as aimed for: ∂E′=Σπ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\partial E'=\Sigma _\pi $$\end{document} holds by construction, χ2(E′)=χ2(TN,TΣπ)=χ2(TπE)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2(E')=\chi ^2(TN,T\Sigma _\pi )=\chi ^2(T_\pi E)$$\end{document} by the bordism invariance of Pontryagin numbers (see Example 3.1), and sgn(E′)=sgn(N)=sgn(E)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {sgn}(E')=\mathrm {sgn}(N)=\mathrm {sgn}(E)$$\end{document} by the additivity and bordism invariance of the signature.□\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
Combining the previous result with Proposition 3.9, we conclude thatsends a homology class [π]\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[\pi ]$$\end{document} represented by a bundle π:E→S\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi :E\rightarrow S$$\end{document} to a certain linear combination of ΣP\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _P$$\end{document} and ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document} whose coefficients involve the invariants sgn(E)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {sgn}(E)$$\end{document} and χ2(TπE)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2(T_\pi E)$$\end{document}. In the following two subsections, we shall see that these functionalssgn:H2(BDiff∂/2(Wg,1);Z)⟶Zandχ2:H2(BDiff∂/2(Wg,1);Z)⟶Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {sgn}:\mathrm {H}_2(\mathrm {BDiff}_{\partial /2}(W_{g,1});\mathbf {Z})\longrightarrow \mathbf {Z}\quad \text {and}\quad \chi ^2:\mathrm {H}_2(\mathrm {BDiff}_{\partial /2}(W_{g,1});\mathbf {Z})\longrightarrow \mathbf {Z}\end{aligned}$$\end{document}factor through the compositionand have a more algebraic description in terms of H(g)⊗πnSO)⋊Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$H(g)\otimes \pi _n\mathrm {SO})\rtimes G_g$$\end{document}. This uses(sF,p):Γg,1/2n⟶(H(g)⊗πnSO)⋊Gg,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} (s_F,p):\Gamma ^n_{g,1/2}\longrightarrow (H(g)\otimes \pi _n\mathrm {SO})\rtimes G_g, \end{aligned}$$\end{document}induced by acting on a stable framing of Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}$$\end{document} (agreeing with the usual framing on D2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^{2n-1}$$\end{document}) as explained in Sect. 2.
Signatures of bundles of symplectic lattices
The standard action of the symplectic group Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} on Z2g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}^{2g}$$\end{document} gives rise to a local system H(g)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathcal {H}(g)$$\end{document} over BSp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {BSp}_{2g}(\mathbf {Z})$$\end{document} and the usual symplectic form on Z2g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}^{2g}$$\end{document} gives a morphism λ:H(g)⊗H(g)→Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\lambda :\mathcal {H}(g)\otimes \mathcal {H}(g)\rightarrow \mathbf {Z}$$\end{document} of local systems to the constant system. To an oriented closed surface S with a map f:S→BSp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$f:S\rightarrow \mathrm {BSp}_{2g}(\mathbf {Z})$$\end{document}, we can associate a bilinear form⟨-,-⟩f:H1(S;f∗H(g))⊗H1(S;f∗H(g))→Z,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \langle -,-\rangle _f:\mathrm {H}^1(S; f^*\mathcal {H}(g))\otimes \mathrm {H}^1(S; f^*\mathcal {H}(g))\rightarrow \mathbf {Z}, \end{aligned}$$\end{document}defined as the compositionH1(S;f∗H(g))⊗H1(S;f∗H(g))→⌣H2(S;f∗H(g)⊗H(g))⟶λH2(S;Z)→[S]Z.\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {H}^1(S; f^*\mathcal {H}(g))\otimes \mathrm {H}^1(S; f^*\mathcal {H}(g))\xrightarrow {\smile } \mathrm {H}^2(S; f^*\mathcal {H}(g)\otimes \mathcal {H}(g))\overset{\lambda }{\longrightarrow }\mathrm {H}^2(S;\mathbf {Z})\xrightarrow {[S]}\mathbf {Z}. \end{aligned}$$\end{document}As both the cup-product and the symplectic pairing λ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\lambda $$\end{document} are antisymmetric, the form ⟨-,-⟩f\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\langle -,-\rangle _f$$\end{document} is symmetric. The usual argument for the bordism invariance of the signature shows that its signature sgn(⟨-,-⟩f)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {sgn}(\langle -,-\rangle _f)$$\end{document} depends only on the bordism class [f]∈Ω2SO(BSp2g(Z))≅H2(Sp2g(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[f]\in \Omega ^{\mathrm {SO}}_{2}(\mathrm {BSp}_{2g}(\mathbf {Z}))\cong \mathrm {H}_2(\mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z})$$\end{document} and thus induces a morphismsgn:H2(Sp2g(Z);Z)⟶Z,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {sgn}:\mathrm {H}_2(\mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z})\longrightarrow \mathbf {Z}, \end{aligned}$$\end{document}which is compatible with the usual inclusion Sp2g(Z)⊂Sp2g+2(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})\subset \mathrm {Sp}_{2g+2}(\mathbf {Z})$$\end{document}.
Remark 3.13
As Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} is perfect for g≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 3$$\end{document} (see Lemma A.1), the morphism (3.8) determines a unique cohomology class sgn∈H2(Sp2g(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {sgn}\in \mathrm {H}^2(\mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z})$$\end{document}. There is a well-known purely algebraically defined cocycle representative of this class due to Meyer [43], known as the Meyer cocycle.
The morphism (3.8) measures signatures of total spaces of smooth bundles over surfaces (even of fibrations of Poincaré complexes). More precisely, for a compact oriented (4k+2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(4k+2)$$\end{document}-manifold M, the action of its group of diffeomorphisms on the middle cohomology induces a morphism for 2g=rk(H2k+1(M))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2g=\mathrm {rk}(\mathrm {H}^{2k+1}(M))$$\end{document} and the resulting compositioncan be shown to map a homology class represented by a smooth bundle over a surface to the signature of its total space. This fact can either be proved along the lines of [14] or extracted from [42] and it has in particular the following consequence.
Lemma 3.14
For n odd, the compositionH2(BDiff∂/2(Wg,1);Z)⟶H2(Sp2g(Z);Z)→sgnZ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {H}_2(\mathrm {BDiff}_{\partial /2}(W_{g,1});\mathbf {Z})\longrightarrow \mathrm {H}_2(\mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z})\xrightarrow {\mathrm {sgn}}\mathbf {Z}\end{aligned}$$\end{document}sends the class of an (Wg,1,D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(W_{g,1},D^{2n-1})$$\end{document}-bundle π:E→S\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi :E\rightarrow S$$\end{document} to the signature sgn(E)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {sgn}(E)$$\end{document} of its total space.
We proceed by computing the image of the signature morphism (3.8) and of its pullback to the theta-subgroup Sp2gq(Z)⊂Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}^q(\mathbf {Z})\subset \mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} as defined in Sect. 1.2.
The signatures realised by classes in H2(Sp2g(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z})$$\end{document} are well-known (see e.g. [3, Lem 6.5, Thm 6.6 (vi)]). To prove that the signature of classes in H2(Sp2gq(Z))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathrm {Sp}_{2g}^q(\mathbf {Z}))$$\end{document} is divisible by 8, recall from Sects. 1.2 and 1.3 that for n odd the morphism Diff∂/2(Wg,1)→Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Diff}_{\partial /2}(W_{g,1})\rightarrow \mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} lands in the subgroup Sp2gq(Z)⊂Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}^q(\mathbf {Z})\subset \mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} as long as n≠1,3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 1,3,7$$\end{document}, so we have a compositionH2(BDiff∂/2(Wg,1);Z)⟶H2(Γg,1/2n;Z)⟶H2(Sp2gq(Z);Z)→sgnZ,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {H}_2(\mathrm {BDiff}_{\partial /2}(W_{g,1});\mathbf {Z})\longrightarrow \mathrm {H}_2(\Gamma ^n_{g,1/2};\mathbf {Z})\longrightarrow \mathrm {H}_2(\mathrm {Sp}_{2g}^q(\mathbf {Z});\mathbf {Z})\xrightarrow {\mathrm {sgn}}\mathbf {Z}, \end{aligned}$$\end{document}which maps the class of a bundle π:E→S\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi :E\rightarrow S$$\end{document} by Lemma 3.14 to sgn(E)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {sgn}(E)$$\end{document}. The latter agrees by Theorem 3.12 with the signature of an almost closed n-connected (2n+2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(2n+2)$$\end{document}-manifold, so it is divisible by 8 as the intersection form of such manifolds is unimodular and even (see e.g. [55]). This proves the claimed divisibility, since the first two morphisms in the composition are surjective, the first one because of the second reminder at the beginning of Sect. 3.3 and the second one by Corollary 2.4. As the signature morphism vanishes on H2(Sp2(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathrm {Sp}_{2}(\mathbf {Z});\mathbf {Z})$$\end{document} by the first part, it certainly vanishes on H2g(Sp2q(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_{2g}(\mathrm {Sp}^{q}_{2}(\mathbf {Z});\mathbf {Z})$$\end{document}. Consequently, by the compatibility of the signature with the inclusion Sp2g(Z)⊂Sp2g+2(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})\subset \mathrm {Sp}_{2g+2}(\mathbf {Z})$$\end{document}, the remaining claim follows from constructing a class in H2(Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathrm {Sp}_{2g}^q(\mathbf {Z});\mathbf {Z})$$\end{document} of signature 8 for g=2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=2$$\end{document}. Using H2(Sp4(Z);Z)≅Z⊕Z/2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathrm {Sp}_{4}(\mathbf {Z});\mathbf {Z})\cong \mathbf {Z}\oplus \mathbf {Z}/2$$\end{document} (see e.g. [3, Lem. A.1(iii)]) and the first part of the claim, the existence of such a class is equivalent to the image of H2(Sp4q(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathrm {Sp}^q_4(\mathbf {Z});\mathbf {Z})$$\end{document} in the torsion free quotient H2(Sp4(Z);Z)free≅Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathrm {Sp}_{4}(\mathbf {Z});\mathbf {Z})_{\mathrm {free}}\cong \mathbf {Z}$$\end{document} containing 2. That it contains 10 is ensured by transfer, since the index of Sp4q(Z)⊂Sp4(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{4}^q(\mathbf {Z})\subset \mathrm {Sp}_{4}(\mathbf {Z})$$\end{document} is 10 (see Sect. 1.2). As H1(Sp4(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\mathrm {Sp}_4(\mathbf {Z});\mathbf {Z})$$\end{document} and H1(Sp4q(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\mathrm {Sp}^q_4(\mathbf {Z});\mathbf {Z})$$\end{document} are 2-torsion by Lemma A.1, it therefore suffices to show that H2(Sp4q(Z);F5)→H2(Sp4(Z);F5)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathrm {Sp}^q_4(\mathbf {Z});\mathbf {F}_5)\rightarrow \mathrm {H}_2(\mathrm {Sp}_{4}(\mathbf {Z});\mathbf {F}_5)$$\end{document} is nontrivial, for which we consider the level 2 congruence subgroup Sp4(Z,2)⊂Sp4(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{4}(\mathbf {Z},2)\subset \mathrm {Sp}_4(\mathbf {Z})$$\end{document}, i.e. the kernel of the reduction map Sp4(Z)→Sp4(Z/2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_4(\mathbf {Z})\rightarrow \mathrm {Sp}_4(\mathbf {Z}/2)$$\end{document}, which is surjective (see e.g. [45, Thm 1] for an elementary proof). From the explicit description of Sp2gq(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}^q(\mathbf {Z})$$\end{document} presented in Sect. 1.2, one sees that it contains the congruence subgroup Sp4(Z,2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{4}(\mathbf {Z},2)$$\end{document}. As a result, it is enough to prove that H2(Sp4(Z,2);F5)→H2(Sp4(Z);F5)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathrm {Sp}_4(\mathbf {Z},2);\mathbf {F}_5)\rightarrow \mathrm {H}_2(\mathrm {Sp}_{4}(\mathbf {Z});\mathbf {F}_5)$$\end{document} is nontrivial, which follows from an application of the Serre spectral sequence of the extension0⟶Sp4(Z,2)⟶Sp4(Z)⟶Sp4(Z/2)⟶0,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\longrightarrow \mathrm {Sp}_4(\mathbf {Z},2)\longrightarrow \mathrm {Sp}_4(\mathbf {Z})\longrightarrow \mathrm {Sp}_4(\mathbf {Z}/2)\longrightarrow 0, \end{aligned}$$\end{document}using that H1(Sp4(Z,2);F5)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\mathrm {Sp}_{4}(\mathbf {Z},2);\mathbf {F}_5)$$\end{document} vanishes by a result of Sato [49, Cor. 10.2] and that the groups H∗(Sp4(Z/2);F5)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_*(\mathrm {Sp}_4(\mathbf {Z}/2);\mathbf {F}_5)$$\end{document} are trivial in low degrees, because of the exceptional isomorphism between Sp4(Z/2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_4(\mathbf {Z}/2)$$\end{document} and the symmetric group in 6 letters as explained for instance in [46, p. 37]. □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
Remark 3.16
There are at least two other proofs for the divisibility of the signature of classes in H2(Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathrm {Sp}_{2g}^q(\mathbf {Z});\mathbf {Z})$$\end{document} by 8. One can be extracted from the proof of [21, Lem. 7.5 i)] and another one is given in [4, Thm 12.1]. The proof in [21] shows actually something stronger, namely that the form ⟨-,-⟩f\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\langle -,-\rangle _f$$\end{document} associated to a class [f]∈H2(Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[f]\in \mathrm {H}_2(\mathrm {Sp}^q_{2g}(\mathbf {Z});\mathbf {Z})$$\end{document} is always even. We shall give a different proof of this fact as part of the second part of Lemma 3.19 below.
For g≥4\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 4$$\end{document}, the existence of a class in H2(Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathrm {Sp}_{2g}^q(\mathbf {Z});\mathbf {Z})$$\end{document} of signature 8 was shown as part of the proof of [21, Thm 7.7], using that the image of H2(Sp2g(Z,2);Z)⟶H2(Sp2g(Z);Z)≅Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {H}_2(\mathrm {Sp}_{2g}(\mathbf {Z},2);\mathbf {Z})\longrightarrow \mathrm {H}_2(\mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z})\cong \mathbf {Z}\end{aligned}$$\end{document} for g≥4\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 4$$\end{document} is known to be 2·Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2\cdot \mathbf {Z}$$\end{document} by a result of Putman [47, Thm F]. However, this argument breaks for small values of g, in which case the image of sgn:H2(Sp2gq(Z);Z)→Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {sgn}:\mathrm {H}_2(\mathrm {Sp}_{2g}^q(\mathbf {Z});\mathbf {Z})\rightarrow \mathbf {Z}$$\end{document} was not known before, at least to the knowledge of the author.
By Lemma 3.15, the signatures of classes in H2(Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathrm {Sp}_{2g}^q(\mathbf {Z});\mathbf {Z})$$\end{document} are divisible by 8, so we obtain a morphism of the formsgn/8:H2(Sp2gq(Z);Z)⟶Z.\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {sgn}/8:\mathrm {H}_2(\mathrm {Sp}_{2g}^{q}(\mathbf {Z});\mathbf {Z})\longrightarrow \mathbf {Z}. \end{aligned}$$\end{document}To lift this morphism to a cohomology class in H2(Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\mathrm {Sp}_{2g}^q(\mathbf {Z});\mathbf {Z})$$\end{document}, we considera:Z/4⟶Sp2q(Z)⊂Sp2gq(Z),\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} a:\mathbf {Z}/4\longrightarrow \mathrm {Sp}_2^q(\mathbf {Z})\subset \mathrm {Sp}_{2g}^q(\mathbf {Z}), \end{aligned}$$\end{document}induced by the matrix 0-110∈Sp2q(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( {\begin{matrix}0 &{} -1 \\ 1 &{} 0 \end{matrix}}\right) \in \mathrm {Sp}_2^q(\mathbf {Z})$$\end{document}. By Lemma A.1, this is an isomorphism on abelianisations for g≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 3$$\end{document} and thus induces a splitting of the universal coefficient theorema∗⊕h:H2(Sp2gq(Z);Z)⟶≅H2(Z/4;Z)⊕Hom(H2(Sp2gq(Z);Z),Z).\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} a^*\oplus h:\mathrm {H}^2(\mathrm {Sp}_{2g}^q(\mathbf {Z});\mathbf {Z})\overset{\cong }{\longrightarrow } \mathrm {H}^2(\mathbf {Z}/4;\mathbf {Z})\oplus \mathrm {Hom}(\mathrm {H}_2(\mathrm {Sp}^q_{2g}(\mathbf {Z});\mathbf {Z}),\mathbf {Z}). \end{aligned}$$\end{document}This splitting is compatible with the inclusion Sp2gq(Z)⊂Sp2g+2q(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}^q(\mathbf {Z})\subset \mathrm {Sp}^q_{2g+2}(\mathbf {Z})$$\end{document}, so we can define a lift of the divided signature sgn/8\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {sgn}/8$$\end{document} to a class H2(Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\mathrm {Sp}_{2g}^q(\mathbf {Z});\mathbf {Z})$$\end{document} as follows.
Definition 3.17
Define the class sgn8∈H2(Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \textstyle {\frac{\mathrm {sgn}}{8}\in \mathrm {H}^2(\mathrm {Sp}_{2g}^q(\mathbf {Z});\mathbf {Z})} \end{aligned}$$\end{document} for g≫0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\gg 0$$\end{document} via the splitting (3.10) by declaring its image in the first summand to be trivial and to be sgn/8\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {sgn}/8$$\end{document} in the second. For small g, the class sgn8∈H2(Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\mathrm {sgn}}{8}\in \mathrm {H}^2(\mathrm {Sp}^q_{2g}(\mathbf {Z});\mathbf {Z})$$\end{document} is defined as the pullback of sgn8∈H2(Sp2g+2hq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\mathrm {sgn}}{8}\in \mathrm {H}^2(\mathrm {Sp}^q_{2g+2h}(\mathbf {Z});\mathbf {Z})$$\end{document} for h≫0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$h\gg 0$$\end{document}.
Define the class sgn8∈H2(Γg,1/2n;Z)forn≠1,3,7odd\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \textstyle {\frac{\mathrm {sgn}}{8}\in \mathrm {H}^2(\Gamma ^n_{g,1/2};\mathbf {Z})}\quad \text {for } n\ne 1,3,7\text { odd} \end{aligned}$$\end{document} as the pullback of the same-named class along the map Γg,1/2n→Gg≅Sp2gq(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1/2}\rightarrow G_g\cong \mathrm {Sp}_{2g}^q(\mathbf {Z})$$\end{document} induced by the action on the middle cohomology.
Framing obstructions
To describe the invariantχ2:H2(BDiff∂/2(Wg,1);Z)⟶Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \chi ^2:\mathrm {H}_2(\mathrm {BDiff}_{\partial /2}(W_{g,1});\mathbf {Z})\longrightarrow \mathbf {Z}\end{aligned}$$\end{document}explained in Sect. 3.1 more algebraically, note that a map f:S→B(Z2g⋊Sp2g(Z))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$f:S\rightarrow \mathrm {B}(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z}))$$\end{document} from an oriented closed connected surface S to B(Z2g⋊Sp2g(Z))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {B}(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z}))$$\end{document} induces a 1-cocycleπ1(S;∗)⟶Z2g⋊Sp2g(Z)⟶Z2g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \pi _1(S;*)\longrightarrow \mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z})\longrightarrow \mathbf {Z}^{2g} \end{aligned}$$\end{document}and hence a class [f]∈H1(S;f∗H(g))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[f]\in \mathrm {H}^1(S; f^*\mathcal {H}(g))$$\end{document}. The composition S→B(Z2g⋊Sp2g(Z))→BSp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S\rightarrow \mathrm {B}(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z}))\rightarrow \mathrm {BSp}_{2g}(\mathbf {Z})$$\end{document} defines a bilinear form ⟨-,-⟩f\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\langle -,-\rangle _f$$\end{document} on H1(S;f∗H(g))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^1(S; f^*\mathcal {H}(g))$$\end{document} as explained in Sect. 3.4, and hence a number ⟨[f],[f]⟩f∈Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\langle [f],[f]\rangle _f\in \mathbf {Z}$$\end{document}. Varying f, this gives a morphismχ2:H2(Z2g⋊Sp2g(Z);Z)⟶Z,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \chi ^2:\mathrm {H}_2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z})\longrightarrow \mathbf {Z}, \end{aligned}$$\end{document}which is compatible with natural inclusion Z2g⋊Sp2g(Z)⊂Z2g+2⋊Sp2g+2(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z})\subset \mathbf {Z}^{2g+2}\rtimes \mathrm {Sp}_{2g+2}(\mathbf {Z})$$\end{document} and takes for n≡3(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 3\ (\mathrm {mod}\ 4)$$\end{document} part in a compositionH2(Γg,1/2n;Z)→(sF,p)H2(Z2g⋊Sp2g(Z);Z)⟶χ2Z,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {H}_2(\Gamma ^n_{g,1/2};\mathbf {Z})\xrightarrow {(s_F,p)}\mathrm {H}_2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z})\overset{\chi ^2}{\longrightarrow }\mathbf {Z}, \end{aligned}$$\end{document}where the first morphism is induced by acting on a stable framing F of Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}$$\end{document} as in Sect. 2. A priori, this requires three choices: a stable framing, a generator πnSO≅Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _n\mathrm {SO}\cong \mathbf {Z}$$\end{document}, and a symplectic basis H(g)≅Z2g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$H(g)\cong \mathbf {Z}^{2g}$$\end{document}. However, the composition turns out to not be affected by these choices and the following proposition shows that it is related to the invariant of (Wg,1,D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(W_{g,1},D^{2n-1})$$\end{document}-bundles explained in Example 3.1.
Proposition 3.18
For n≡3(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 3\ (\mathrm {mod}\ 4)$$\end{document}, the compositionsends the class of a (Wg,1,D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(W_{g,1},D^{2n-1})$$\end{document}-bundle π:E→S\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi :E\rightarrow S$$\end{document} to χ2(TπE)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2(T_\pi E)$$\end{document}.
Proof
The relative Serre spectral sequence of(Wg,1,∂Wg,1)⟶(E,∂E)⟶πS\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} (W_{g,1},\partial W_{g,1})\longrightarrow (E,\partial E)\overset{\pi }{\longrightarrow } S \end{aligned}$$\end{document}induces canonical isomorphismsH2n+2(E,∂E;Z)≅H2(S;Z)andHn+1(E,∂E;Z)≅H1(S;f∗H(g)),\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {H}^{2n+2}(E,\partial E;\mathbf {Z})\cong \mathrm {H}^2(S;\mathbf {Z})\quad \text {and}\quad \mathrm {H}^{n+1}(E,\partial E;\mathbf {Z})\cong \mathrm {H}^1(S;f^*\mathcal {H}(g)), \end{aligned}$$\end{document}where f denotes the compositionS⟶BDiff∂/2(Wg,1)⟶BΓg,1/2n⟶pBSp2g(Z).\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} S\longrightarrow \mathrm {BDiff}_{\partial /2}(W_{g,1})\longrightarrow \mathrm {B}\Gamma _{g,1/2}^n\overset{p}{\longrightarrow }\mathrm {BSp}_{2g}(\mathbf {Z}). \end{aligned}$$\end{document}By the compatibility of the Serre spectral sequence with the cup-product and after identifying H1(S;f∗H(g))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^1(S;f^*\mathcal {H}(g))$$\end{document} with H1(π1(S;∗);H(g)⊗SπnSO(n))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^1(\pi _1(S;*);H(g)\otimes S\pi _n\mathrm {SO}(n))$$\end{document}, it suffices to show that the second isomorphism sends χ(TπE)∈Hn+1(E,∂E;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi (T_\pi E)\in \mathrm {H}^{n+1}(E,\partial E;\mathbf {Z})$$\end{document} up to signs to the class represented by the cocycleπ1(S;∗)⟶Γg,1/2n⟶sF[Wg,1,SO]∗≅H(g)⊗πnSO,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \pi _1(S;*) \longrightarrow \Gamma ^n_{g,1/2}\overset{s_F}{\longrightarrow }[W_{g,1},\mathrm {SO}]_*\cong H(g)\otimes \pi _n\mathrm {SO}, \end{aligned}$$\end{document}involving the choice of stable framing F:TWg,1⊕εk≅ε2n+k\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$F:TW_{g,1}\oplus \varepsilon ^k\cong \varepsilon ^{2n+k}$$\end{document} as in Sect. 2. As a first step, we describe this isomorphism more explicitly:
Note that Hn+1(E,∂E;πnSO)≅Hn+1(E;πnSO)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^{n+1}(E,\partial E;\pi _n\mathrm {SO})\cong \mathrm {H}^{n+1}(E;\pi _n\mathrm {SO})$$\end{document} as H∗(∂E;πnSO)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^{*}(\partial E;\pi _n\mathrm {SO})$$\end{document} is trivial for ∗=n,n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$*=n,n+1$$\end{document}. Unwinding the construction of the Serre spectral sequence using a skeletal filtration of S, one sees that after fixing an identification Wg,1≅π-1(∗)⊂E\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}\cong \pi ^{-1}(*)\subset E$$\end{document}, the image of a class x∈Hn+1(E,πnSO)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$x\in \mathrm {H}^{n+1}(E,\pi _n\mathrm {SO})$$\end{document} under the isomorphism in question is represented by the cocycle π1(S;∗)→H(g)⊗πnSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _1(S;*)\rightarrow H(g)\otimes \pi _n\mathrm {SO}$$\end{document} which maps a loop ω:([0,1],{0,1})→(S,∗)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\omega :([0,1],\{0,1\})\rightarrow (S,*)$$\end{document} to the class obtained from a choice of lift x~∈Hn+1(E,π-1(∗);πnSO)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\widetilde{x}\in \mathrm {H}^{n+1}(E,\pi ^{-1}(*);\pi _n\mathrm {SO})$$\end{document} by pulling it back along(Wg,1×[0,1],Wg,1×{0,1})⟶(ω∗E,Wg,1×{0,1})⟶(E,π-1(∗)),\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} (W_{g,1}\times [0,1],W_{g,1}\times \{0,1\})\longrightarrow (\omega ^*E,W_{g,1}\times \{0,1\})\longrightarrow (E,\pi ^{-1}(*)), \end{aligned}$$\end{document}where the second morphism is induced by pulling back the bundle π:E→S\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi :E\rightarrow S$$\end{document} along ω\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\omega $$\end{document} and the first morphism is the unique (up to homotopy) trivialisation ω∗E≅[0,1]×Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\omega ^*E\cong [0,1]\times W_{g,1}$$\end{document} relative to Wg,1×{0}\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}\times \{0\}$$\end{document} of the pullback bundle over [0, 1]; here we used the canonical isomorphism Hn+1(Wg,1×[0,1],Wg,1×{0,1};πnSO)≅H(g)⊗πnSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^{n+1}(W_{g,1}\times [0,1],W_{g,1}\times \{0,1\};\pi _n\mathrm {SO})\cong H(g)\otimes \pi _n\mathrm {SO}$$\end{document}.
Recall from Example 3.1 that the class χ(TπE)∈Hn+1(E;πnSO)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi (T_\pi E)\in \mathrm {H}^{n+1}(E;\pi _n\mathrm {SO})$$\end{document} is the primary obstruction to extending the canonical τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}$$\end{document}-structure on TπE\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$T_\pi E$$\end{document} to a τ>n+1BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n+1}\mathrm {BO}$$\end{document}-structure. The choice of framing F induces such an extension on the fibre Wg,1≅π-1(∗)⊂E\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}\cong \pi ^{-1}(*)\subset E$$\end{document} and thus induces a lift of χ(TπE)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi (T_\pi E)$$\end{document} to a relative class in Hn+1(E,π-1(∗);πnSO)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^{n+1}(E,\pi ^{-1}(*);\pi _n\mathrm {SO})$$\end{document}. This uses that the τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}$$\end{document} structure on Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}$$\end{document} induced by the framing agrees with the restriction of the τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}$$\end{document}-structure on TπE\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$T_\pi E$$\end{document} to π-1(∗)≅Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _{-1}(*)\cong W_{g,1}$$\end{document} by obstruction theory. Using the above description, we see that the image of χ(TπE)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi (T_\pi E)$$\end{document} under the isomorphism in question is represented by the cocyle π1(S;∗)→H(g)⊗πnSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _1(S;*)\rightarrow H(g)\otimes \pi _n\mathrm {SO}$$\end{document} that sends a loop ω\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\omega $$\end{document} to the primary obstruction in Hn(Wg,1;πnSO)≅H(g)⊗πnSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^{n}(W_{g,1};\pi _n\mathrm {SO})\cong H(g)\otimes \pi _n\mathrm {SO}$$\end{document} to solving the lifting problem
which agrees with the corresponding obstruction when replacing BSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {BSO}$$\end{document} by τ>nBSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BSO}$$\end{document}. Here the left square in the diagram is given by the trivialisation explained earlier. There is a useful alternative description of this obstruction class: relative to the subspace Wg,1×{0}⊂Wg,1×{0,1}\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}\times \{0\}\subset W_{g,1}\times \{0,1\}$$\end{document} there is a unique (up to homotopy) lift in (3.12), so the obstruction to finding a lift relative to the subspace Wg,1×{0,1}\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}\times \{0,1\}$$\end{document} can be seen as an element in the group of path-components of the fibre of the principal fibration Map∗(Wg,1,τ>n+1BSO)→Map∗(Wg,1,BSO)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Map}_*(W_{g,1},\tau _{>n+1}\mathrm {BSO})\rightarrow \mathrm {Map}_*(W_{g,1},\mathrm {BSO})$$\end{document}, which is exactly [Wg,1,τ≤nSO]∗≅H(g)⊗πnSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[W_{g,1},\tau _{\le n}\mathrm {SO}]_*\cong H(g)\otimes \pi _n\mathrm {SO}$$\end{document}.
To see that the cocyle we just described agrees with (3.11), note that the function sF:Γg,1/2n→H(g)⊗πnSO≅[Wg,1,τ≤nSO]∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$s_F:\Gamma ^n_{g,1/2}\rightarrow H(g)\otimes \pi _n\mathrm {SO}\cong [W_{g,1},\tau _{\le n}\mathrm {SO}]_*$$\end{document} induced by acting on the stable framing F arises as the connecting map π1(BDiff∂/2τ>n+1(Wg,1);[F])→π0Map∗(Wg,1,τ≤nSO)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi _1(\mathrm {BDiff}^{\tau _{>n+1}}_{\partial /2}(W_{g,1});[F])\rightarrow \pi _0\mathrm {Map}_*(W_{g,1},\tau _{\le n}\mathrm {SO})$$\end{document} of the fibrationMap∗(Wg,1,τ≤nSO)⟶BDiff∂/2τ>n+1(Wg,1)⟶BDiff∂/2(Wg,1),\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {Map}_*(W_{g,1},\tau _{\le n}\mathrm {SO})\longrightarrow \mathrm {BDiff}^{\tau _{>n+1}}_{\partial /2}(W_{g,1})\longrightarrow \mathrm {BDiff}_{\partial /2}(W_{g,1}), \end{aligned}$$\end{document}where BDiff∂/2τ>n+1(Wg,1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {BDiff}^{\tau _{>n+1}}_{\partial /2}(W_{g,1})$$\end{document} is the space that classifies (Wg,1,D2n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(W_{g,1},D^{2n-1})$$\end{document}-bundles with a τ>n+1BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n+1}\mathrm {BO}$$\end{document}-structure on the vertical tangent bundle extending the given τ>n+1BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n+1}\mathrm {BO}$$\end{document}-structure on the restriction to the trivial D2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^{2n-1}$$\end{document}-subbundle induced by the standard framing of D2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^{2n-1}$$\end{document}; here we identified the space of τ>n+1BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n+1}\mathrm {BO}$$\end{document}-structures of Wg,1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}$$\end{document} relative to D2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^{2n-1}$$\end{document} with the mapping space Map∗(Wg,1,τ≤nSO)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Map}_*(W_{g,1},\tau _{\le n}\mathrm {SO})$$\end{document} by using the choice of stable framing F, which also induces the basepoint [F]∈BDiff∂/2τ>n+1(Wg,1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[F]\in \mathrm {BDiff}^{\tau _{>n+1}}_{\partial /2}(W_{g,1})$$\end{document}. This shows that the value of (3.11) on a loop [ω]∈π1(S;∗)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[\omega ]\in \pi _1(S;*)$$\end{document} is given by the component in [Wg,1,τ≤nSO]∗≅H(g)⊗πnSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[W_{g,1},\tau _{\le n}\mathrm {SO}]_*\cong H(g)\otimes \pi _n\mathrm {SO}$$\end{document} obtained by evaluating a choice of path-lift
at the end point. Such a path-lift precisely classifies a lift as in (3.12) relative to the subspace Wg,1×{0}⊂Wg,1×{0,1}\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g,1}\times \{0\}\subset W_{g,1}\times \{0,1\}$$\end{document}, so the claim follows from the second description of the obstruction to solving the lifting problem (3.12) mentioned above. □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
By the compatibility of χ2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2$$\end{document} with the stabilisation maps, it suffices to show the first part for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}, which follows from checking that the image of a generator in H2(Z2;Z)≅Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathbf {Z}^{2} ;\mathbf {Z})\cong \mathbf {Z}$$\end{document} is mapped to ±2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pm 2$$\end{document} under χ2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2$$\end{document} by chasing through the definition. As the signature morphism pulls back from Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document}, the second part follows from the first part and Lemma 3.15 by showing that the image of χ2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2$$\end{document} is for Sp2gq(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}^q(\mathbf {Z})$$\end{document} always divisible by 2·Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2\cdot \mathbf {Z}$$\end{document} and for Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} divisible by 2 if and only if g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}. In the case of Sp2gq(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}^q(\mathbf {Z})$$\end{document}, this can be shown “geometrically” as in the proof of Lemma 3.15: choose n≡3(mod4),n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 3\ (\mathrm {mod}\ 4), n\ne 3,7$$\end{document} and consider the compositionThe first morphism is surjective by the second reminder at the beginning of Sect. 3.3 and the second morphism is an isomorphism as a result of Sect. 2, so it suffices to show that the composition is divisible by 2, which in turn follows from a combination of Lemma 3.18, Theorem 3.12 and the fact that in these dimensions, an n-connected almost closed (2n+2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(2n+2)$$\end{document}-manifold E′\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$E'$$\end{document} satisfies χ2(E′)∈2·Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2(E')\in 2\cdot \mathbf {Z}$$\end{document} by a combination of Theorem 3.2 and (3.3). For Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document}, we argue as follows: in the case g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}, we show that the morphism H2(Z2;Z)→H2(Z2⋊Sp2(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathbf {Z}^2;\mathbf {Z})\rightarrow \mathrm {H}_2(\mathbf {Z}^{2}\rtimes \mathrm {Sp}_{2}(\mathbf {Z});\mathbf {Z})$$\end{document} is surjective, which will exhibit the claimed divisibility as a consequence of (i). It follows from Lemma A.3 that the group H1(Sp2(Z);Z2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\mathrm {Sp}_{2}(\mathbf {Z});\mathbf {Z}^2)$$\end{document} vanishes and as Sp2(Z)=SL2(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_2(\mathbf {Z})=\mathrm {SL}_2(\mathbf {Z})$$\end{document}, we also have H2(Sp2(Z);Z)=0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathrm {Sp}_2(\mathbf {Z});\mathbf {Z})=0$$\end{document},3 so an application of the Serre spectral sequence to Z2⋊Sp2(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}^{2}\rtimes \mathrm {Sp}_{2}(\mathbf {Z})$$\end{document} shows the claimed surjectivity. This leaves us with proving that χ2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2$$\end{document} is not divisible by 2 for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document} for which we use that there is class [f:π1S→Sp2g(Z)]∈H2(Sp2g(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[f:\pi _1S\rightarrow \mathrm {Sp}_{2g}(\mathbf {Z})]\in \mathrm {H}_2(\mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z})$$\end{document} of signature 4 by Lemma 3.15, so the form ⟨-,-⟩f\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\langle -,-\rangle _f$$\end{document} cannot be even and hence there is a 1-cocycle g:π1S→Z2g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g:\pi _1S\rightarrow \mathbf {Z}^{2g}$$\end{document} for which ⟨[g],[g]⟩f\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\langle [g],[g]\rangle _f$$\end{document} is odd, which means that the image χ2([g,f])\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2([g,f])$$\end{document} of the class [(g,f)]∈H2(Z2g⋊Sp2g(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[(g,f)]\in \mathrm {H}_2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z})$$\end{document} induced by the morphism (g,f):π1S→Z2⋊Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(g,f):\pi _1S\rightarrow \mathbf {Z}^2\rtimes \mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} is odd. For the last part, note that the argument we gave for the divisibility in the second part for Sp2gq(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}^q(\mathbf {Z})$$\end{document} shows for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document} that the image of the composition in (iii) is contained in 8·Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$8\cdot \mathbf {Z}$$\end{document}, since χ2(E′)-sgn(E′)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2(E')-\mathrm {sgn}(E')$$\end{document} is divisible by 8 if n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}. Hence, to finish the proof, it suffices to establish the existence of a class in H2(Γg,1/2n;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\Gamma ^n_{g,1/2};\mathbf {Z})$$\end{document} for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document} on which the composition evaluates to 8. To this end, we consider the square
induced by the embedding (sF,p)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(s_F,p)$$\end{document} of the extension describing Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g,1/2}^n$$\end{document} into the trivial extension of Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} by Z2g⊗πnSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}^{2g}\otimes \pi _n\mathrm {SO}$$\end{document} (see Sect. 2). By the first part, there is a class [f]∈H2(Z2g⊗πnSO;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[f]\in \mathrm {H}_2(\mathbf {Z}^{2g}\otimes \pi _n\mathrm {SO};\mathbf {Z})$$\end{document} with χ2([f])=2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2([f])=2$$\end{document} and trivial signature, since the signature morphism pulls back from Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document}. As a result of Lemma 2.1, the cokernel of the left vertical map in the square is 4-torsion if n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}, so 4·[f]\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$4\cdot [f]$$\end{document} lifts to H2(Z2g⊗SπnSO(n);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathbf {Z}^{2g}\otimes S\pi _n\mathrm {SO}(n);\mathbf {Z})$$\end{document} and provides a class as desired. □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
Similar to the construction of sgn8∈H2(Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\mathrm {sgn}}{8}\in \mathrm {H}^2(\mathrm {Sp}_{2g}^q(\mathbf {Z}) ;\mathbf {Z})$$\end{document}, we would like to lift the morphism resulting from the second part of Lemma 3.19χ2/2:H2(Z2g⋊Sp2gq(Z);Z)→Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \chi ^2/2:\mathrm {H}_2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}^q(\mathbf {Z}) ;\mathbf {Z})\rightarrow \mathbf {Z}\end{aligned}$$\end{document}to a class χ22∈H2(Z2g⋊Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\chi ^2}{2}\in \mathrm {H}^2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}^q(\mathbf {Z}) ;\mathbf {Z})$$\end{document}. To this end, observe that Sp2gq(Z)⊂Z2g⋊Sp2gq(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}^q(\mathbf {Z})\subset \mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}^q(\mathbf {Z})$$\end{document} induces an isomorphism on abelianisations for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document} since the coinvariants (Z2g)Sp2gq(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\mathbf {Z}^{2g})_{\mathrm {Sp}_{2g}^q(\mathbf {Z})}$$\end{document} vanish in this range by Lemma A.2. The morphisma:Z/4⟶Sp2q(Z)⊂Z2g⋊Sp2gq(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} a:\mathbf {Z}/4\longrightarrow \mathrm {Sp}_2^q(\mathbf {Z})\subset \mathbf {Z}^{2g}\rtimes \mathrm {Sp}^q_{2g}(\mathbf {Z}) \end{aligned}$$\end{document}considered in Sect. 3.4 thus induces a splittinga∗⊕h:H2(Z2g⋊Sp2gq(Z))→≅H2(Z/4)⊕Hom(H2(Z2g⋊Sp2gq(Z)),Z),\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} a^*\oplus h:\mathrm {H}^2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}^q(\mathbf {Z}) )\xrightarrow {\cong } \mathrm {H}^2(\mathbf {Z}/4 )\oplus \mathrm {Hom}(\mathrm {H}_2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}^q(\mathbf {Z}) ),\mathbf {Z}), \end{aligned}$$\end{document}for g≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 3$$\end{document}, analogous to the splitting (3.10). As before, this splitting is compatible with the inclusions Z2g⋊Sp2gq(Z)⊂Z2g+2⋊Sp2g+2q(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}^q(\mathbf {Z})\subset \mathbf {Z}^{2g+2}\rtimes \mathrm {Sp}_{2g+2}^q(\mathbf {Z})$$\end{document}, so the following is valid.
Definition 3.20
Define the class χ22∈H2(Z2g⋊Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \textstyle {\frac{\chi ^2}{2}\in \mathrm {H}^2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}^q(\mathbf {Z});\mathbf {Z})} \end{aligned}$$\end{document} for g≫0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\gg 0$$\end{document} via the splitting (3.13) by declaring its image in the first summand to be trivial and to be χ2/2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2/2$$\end{document} in the second. For small g, the class χ22\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\chi ^2}{2}$$\end{document} is defined as the pullback of the class for g≫0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\gg 0$$\end{document}.
Define the class χ22∈H2(Γg,1n;Z)forn≠1,3,7andn≡3(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \textstyle {\frac{\chi ^2}{2}\in \mathrm {H}^2(\Gamma ^n_{g,1};\mathbf {Z})\quad \text {for }n\ne 1,3,7\text { and }n\equiv 3\ (\mathrm {mod}\ 4)}\end{aligned}$$\end{document} as the pullback of the same-named class along the map Γg,1/2n⟶(sF,p)(H(g)⊗SπnSO(n))⋊Gg≅Z2g⋊Sp2gq(Z).\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \Gamma ^n_{g,1/2}\overset{(s_F,p)}{\longrightarrow } \big (H(g)\otimes S\pi _n\mathrm {SO}(n)\big )\rtimes G_g\cong \mathbf {Z}^{2g}\rtimes \mathrm {Sp}_{2g}^q(\mathbf {Z}). \end{aligned}$$\end{document}
Define the class χ2-sgn8∈H2(Γg,1/2n;Z)≅Hom(H2(Γg,1/2n;Z),Z)forn=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \textstyle {\frac{\chi ^2-\mathrm {sgn}}{8}\in \mathrm {H}^2(\Gamma ^n_{g,1/2};\mathbf {Z})\cong \mathrm {Hom}(\mathrm {H}_2(\Gamma ^n_{g,1/2} ;\mathbf {Z}),\mathbf {Z})\quad \text {for }n=3,7} \end{aligned}$$\end{document} for g≫0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\gg 0$$\end{document} as image of (χ2-sgn)/8\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\chi ^2-\mathrm {sgn})/8$$\end{document} ensured by Lemma 3.19, and for small g as the pullback of χ2-sgn8∈H2(Γg+h,1/2;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\chi ^2-\mathrm {sgn}}{8}\in \mathrm {H}^2(\Gamma _{g+h,1/2};\mathbf {Z})$$\end{document} for h≫0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$h\gg 0$$\end{document}.
The isomorphism H2(Γg,1/2n;Z)≅Hom(H2(Γg,1/2n;Z),Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\Gamma ^n_{g,1/2};\mathbf {Z})\cong \mathrm {Hom}(\mathrm {H}_2(\Gamma ^n_{g,1/2};\mathbf {Z}),\mathbf {Z})$$\end{document} for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document} in the previous definition is assured by Corollary 2.4 and Lemma A.1, as the abelianisation H1(Γg,1/2n;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\Gamma ^n_{g,1/2};\mathbf {Z})$$\end{document} vanishes for g≫0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\gg 0$$\end{document}.
We finish this subsection with an auxiliary lemma convenient for later purposes.
Lemma 3.21
For g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}, the class sgn8∈H2(Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\mathrm {sgn}}{8}\in \mathrm {H}^2(\mathrm {Sp}^q_{2g}(\mathbf {Z});\mathbf {Z})$$\end{document} and the pullback of the class χ22∈H2(Z2g⋊Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\chi ^2}{2}\in \mathrm {H}^2(\mathbf {Z}^{2g}\rtimes \mathrm {Sp}^q_{2g}(\mathbf {Z});\mathbf {Z})$$\end{document} to H2(Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\mathrm {Sp}^q_{2g}(\mathbf {Z});\mathbf {Z})$$\end{document} are trivial.
Proof
Both classes evaluate trivially on H2(Sp2gq(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathrm {Sp}_{2g}^{q}(\mathbf {Z});\mathbf {Z})$$\end{document}. For the first class, this is a consequence of Lemma 3.15, for the second this holds by construction. Moreover, both classes pull back trivially to H2(Z/4;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\mathbf {Z}/4;\mathbf {Z})$$\end{document} along the morphism a:Z/4→Sp2gq(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$a:\mathbf {Z}/4\rightarrow \mathrm {Sp}^q_{2g}(\mathbf {Z})$$\end{document} by definition. Although this morphism does not induce an isomorphism on abelianisations for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}, it still induces one on their torsion subgroups by Lemma A.1 and this is sufficient to deduce the assertion from the universal coefficient theorem. □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
The proof of Theorem B
We are ready to prove our main result Theorem B, which we state equivalently in terms of the central extension0⟶Θ2n+1⟶Γgn⟶Γg,1/2n⟶0,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\longrightarrow \Theta _{2n+1}\longrightarrow \Gamma ^n_{g}\longrightarrow \Gamma ^n_{g,1/2}\longrightarrow 0, \end{aligned}$$\end{document}explained in Sect. 1.3. Our description of its extension class in H2(Γg,1/2n;Θ2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\Gamma _{g,1/2}^n;\Theta _{2n+1})$$\end{document} involves the cohomology classes in H2(Γg,1/2n;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\Gamma _{g,1/2}^n;\mathbf {Z})$$\end{document} of Definitions 3.17 and 3.20, and the two homotopy spheres ΣP\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _P$$\end{document} and ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document} in the subgroup bA2n+2⊂Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}\subset \Theta _{2n+1}$$\end{document} examined in Sect. 3.2.1. We write (-)·Σ∈H2(Γg,1/2n;Z)→H2(Γg,1/2n;Θ2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(-)\cdot \Sigma \in \mathrm {H}^2(\Gamma _{g,1/2}^n;\mathbf {Z})\rightarrow \mathrm {H}^2(\Gamma _{g,1/2}^n;\Theta _{2n+1})$$\end{document} for the change of coefficients induced by Σ∈Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma \in \Theta _{2n+1}$$\end{document}.
As all cohomology classes involved are compatible with the stabilisation map Γg,1/2n→Γg+1,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1/2}\rightarrow \Gamma ^n_{g+1,1/2}$$\end{document}, it is sufficient to show the first part of the claim for g≫0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\gg 0$$\end{document} (see Sect. 1.4). We assume n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document} first. Identifying Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1/2}$$\end{document} with (H(g)⊗πnSO)⋊Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(H(g)\otimes \pi _n\mathrm {SO})\rtimes G_g$$\end{document} via the isomorphism (sF,p)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(s_F,p)$$\end{document} of Sect. 2, the morphism a:Z/4→(H(g)⊗πnSO)⋊Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$a:\mathbf {Z}/4\rightarrow (H(g)\otimes \pi _n\mathrm {SO})\rtimes G_g$$\end{document} of the previous section induces a morphism between the sequences of the universal coefficient theorem
By the exactness of the rows and the vanishing of H2(Z/4;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathbf {Z}/4;\mathbf {Z})$$\end{document}, it is sufficient to show that the extension class in consideration agrees with the classes in the statement when mapped to Hom(H2(Γg,1/2n;Z),Θ2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Hom}(\mathrm {H}_2(\Gamma ^n_{g,1/2};\mathbf {Z}),\Theta _{2n+1})$$\end{document} and H2(Z/4,Θ2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\mathbf {Z}/4,\Theta _{2n+1})$$\end{document}. Regarding the images in the Hom-term, it is enough to identify them after precomposition with the epimorphismH2(BDiff∂/2(Wg,1);Z)⟶H2(Γg,1/2n;Z),\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {H}_2(\mathrm {BDiff}_{\partial /2}(W_{g,1});\mathbf {Z})\longrightarrow \mathrm {H}_2(\Gamma ^n_{g,1/2};\mathbf {Z}), \end{aligned}$$\end{document}so from the construction of sgn8\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\mathrm {sgn}}{8}$$\end{document} and χ22\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\chi ^2}{2}$$\end{document} together with Lemmas 3.15 and 3.19, we see that it suffices to show that H2(BDiff∂/2(Wg,1);Z)→Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_2(\mathrm {BDiff}_{\partial /2}(W_{g,1});\mathbf {Z})\rightarrow \Theta _{2n+1}$$\end{document} induced by the extension class maps the class of a bundle π:E→S\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi :E\rightarrow S$$\end{document} to sgn(E)/8·ΣP\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {sgn}(E)/8\cdot \Sigma _P$$\end{document} if n≡1(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 1\ (\mathrm {mod}\ 4)$$\end{document} and to sgn(E)/8·ΣP+χ2(E)/2·ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {sgn}(E)/8\cdot \Sigma _P+\chi ^2(E)/2\cdot \Sigma _Q$$\end{document} otherwise, which is a consequence of Theorem 3.12 combined with Proposition 3.9. By construction, the classes sgn8·ΣP\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\mathrm {sgn}}{8}\cdot \Sigma _P$$\end{document} and χ22·ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\chi ^2}{2}\cdot \Sigma _Q$$\end{document} vanish in H2(Z/4;Θ2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\mathbf {Z}/4;\Theta _{2n+1})$$\end{document}, so the claim for n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document} follows from showing that the extension class is trivial in H2(Z/4;Θ2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\mathbf {Z}/4;\Theta _{2n+1})$$\end{document}, i.e. that the pullback of the extension to Z/4\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}/4$$\end{document} splits, which is in turn equivalent to the existence of a lift
Using the standard embedding W1=Sn×Sn⊂Rn+1×Rn+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_1=S^n\times S^n\subset \mathbf {R}^{n+1}\times \mathbf {R}^{n+1}$$\end{document}, we consider the diffeomorphismSn×Sn⟶Sn×Sn(x1,…,xn+1,y1,…,yn+1)⟼(-y1,…,yn+1,x1,…,xn+1),\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \begin{array}{rcl} S^n\times S^n &{} \longrightarrow &{} S^n\times S^n \\ (x_1,\ldots ,x_{n+1},y_1,\ldots ,y_{n+1}) &{} \longmapsto &{} (-y_1,\ldots ,y_{n+1},x_1,\ldots ,x_{n+1}), \end{array} \end{aligned}$$\end{document}which is of order 4, maps to 0-110∈Sp2q(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( {\begin{matrix} 0 &{} -1 \\ 1 &{} 0 \end{matrix}}\right) \in \mathrm {Sp}_2^q(\mathbf {Z})$$\end{document}, and has constant differential, so it vanishes in H(g)⊗πnSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$H(g)\otimes \pi _n\mathrm {SO}$$\end{document}. As the natural map Γ1,1n→Γ1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{1,1}^n\rightarrow \Gamma ^n_1$$\end{document} is an isomorphism by Lemma 1.1, this diffeomorphism induces a lift as required for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}, which in turn provides a lift for all g≥1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 1$$\end{document} via the stabilisation map Γ1,1n→Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{1,1}\rightarrow \Gamma ^n_{g,1}$$\end{document}. For n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}, the abelianisation of Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1/2}$$\end{document} vanishes for g≫0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\gg 0$$\end{document} due to Corollary 2.4, so it suffices to identify the extension class with the classes in the statement in Hom(H2(Γg,1/2n;Z),Θ2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Hom}(\mathrm {H}_2(\Gamma ^n_{g,1/2};\mathbf {Z}),\Theta _{2n+1})$$\end{document} which follows as in the case n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document}.
Lemma 3.19 (iii) implies that the image of d2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_2$$\end{document} for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document} is generated by ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document} for all g≥1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 1$$\end{document}. For n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document}, the map Γg,1/2n→(H(g)⊗πnSO)⋊Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g,1/2}^n\rightarrow (H(g)\otimes \pi _n\mathrm {SO})\rtimes G_g$$\end{document} is an isomorphism (see Sect. 2), so Lemma 3.19 tells us that the image of d2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_2$$\end{document} for n≡3(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 3\ (\mathrm {mod}\ 4)$$\end{document} is generated by ΣP\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _P$$\end{document} and ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document} if g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document} and by ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document} if g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}. For n≡1(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 1\ (\mathrm {mod}\ 4)$$\end{document}, it follows from Lemma 3.15 that the image of d2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_2$$\end{document} is generated by ΣP\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _P$$\end{document} if g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document} and that it is trivial for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}. In sum, this implies the second part of the claim by Corollary 3.5, and also that the differential d2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_2$$\end{document} does not vanish for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document}, so the extension is nontrivial in these cases. For n≡3(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 3\ (\mathrm {mod}\ 4)$$\end{document}, the homotopy sphere ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document} is nontrivial by Theorem 3.3, so d2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_2$$\end{document} does not vanish for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} either. Finally, in the case n≡1(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 1\ (\mathrm {mod}\ 4)$$\end{document}, the extension is classified by sgn8·ΣP\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\mathrm {sgn}}{8}\cdot \Sigma _P$$\end{document}, which is trivial for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} by Lemma 3.21, so the extension splits and the proof is finished. □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
It is time to make good for the missing part of the proof of Theorem 2.2.
Proof of Theorem2.2forn=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}. We have Gg=Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g=\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document}, so the case g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} follows from the fact that H2(Sp2g(Z);Z2g⊗SπnSO(n))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z}^{2g}\otimes S\pi _n\mathrm {SO}(n))$$\end{document} vanishes by Lemma A.3. To prove the case g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document}, note that a hypothetical splitting s:Sp2g(Z)→Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$s:\mathrm {Sp}_{2g}(\mathbf {Z})\rightarrow \Gamma ^n_{g,1/2}$$\end{document} of the upper row of the commutative diagram
induces a splitting (sF,p)∘s\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(s_F,p)\circ s$$\end{document} of the lower row, which agrees with the canonical splitting of the lower row up to conjugation with Z2g⊗πnSO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}^{2g}\otimes \pi _n\mathrm {SO}$$\end{document}, because such splittings up to conjugation are a torsor for H1(Sp2g(Z);Z2g⊗πnSO)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^1(\mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z}^{2g}\otimes \pi _n\mathrm {SO})$$\end{document} which vanishes by Lemma A.3. Lemma 3.15 on the other hand ensures that there is a class [f]∈H2(Sp2g(Z);Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[f]\in \mathrm {H}_2(\mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z})$$\end{document} with signature 4, so s∗[f]∈H2(Γg,1/2n;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$s_*[f]\in \mathrm {H}_2(\Gamma ^n_{g,1/2};\mathbf {Z})$$\end{document} satisfies sgn(s∗[f])=4\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {sgn}(s_*[f])=4$$\end{document} and χ2(s∗[f])=0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\chi ^2(s_*[f])=0$$\end{document}, which contradicts Lemma 3.19 (iii).□\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
Kreck’s extensions and their abelian quotients
The inclusion Tg,1n⊂Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}^n_{g,1}\subset \Gamma _{g,1}^n$$\end{document} of the Torelli group extends to a pullback diagram
of extensions whose bottom row we identified in Theorem 3.22. We now apply this to obtain information about the top row, and moreover to compute the abelianisations of Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g,1}^n$$\end{document} and Tg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_{g,1}^n$$\end{document} in terms of the homotopy sphere ΣQ∈Θ2n+2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q\in \Theta _{2n+2}$$\end{document}, the subgroup bA2n+2⊂Θ2n+2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}\subset \Theta _{2n+2}$$\end{document} of Sect. 3.2, and the abelianisation of Γg,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g,1/2}^n$$\end{document} as computed in Corollary 2.4.
Theorem 4.1
Let n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} be odd and g≥1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 1$$\end{document}.
The kernel Kg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$K_g$$\end{document} of the morphism Θ2n+1→H1(Γg,1n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}\rightarrow \mathrm {H}_1(\Gamma ^n_{g,1})$$\end{document} is generated by ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document} for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} and agrees with the subgroup bA2n+2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}$$\end{document} for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document}. The induced extension 0⟶Θ2n+1/Kg⟶H1(Γg,1n)⟶H1(Γg,1/2n)⟶0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\longrightarrow \Theta _{2n+1}/K_g\longrightarrow \mathrm {H}_1(\Gamma _{g,1}^n)\longrightarrow \mathrm {H}_1(\Gamma ^n_{g,1/2})\longrightarrow 0 \end{aligned}$$\end{document} splits.
By the naturality of the Serre spectral sequence, the morphism of extension (4.1) induces a ladder of exact sequences
from which we see that the kernel Kg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$K_g$$\end{document} in question agrees with the image of the differential d2Γ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_2^\Gamma $$\end{document}, which we described in Theorem 3.22. By the universal coefficient theorem, the pushforward of the extension class of the bottom row of (4.1) along the quotient map Θ2n+1→Θ2n+1/im(d2Γ)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}\rightarrow \Theta _{2n+1}/\mathrm {im}(d_2^\Gamma )$$\end{document} is classified by a class in Ext(H1(Γg,1/2n),Θ2n+1/im(d2Γ))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Ext}(\mathrm {H}_1(\Gamma ^n_{g,1/2}),\Theta _{2n+1}/\mathrm {im}(d_2^\Gamma ))$$\end{document}, which also describes the exact sequence in (i). By a combination of Theorem 3.22 and Corollary 3.5, this class is trivial for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}, and for n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document} as long as g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document}. For g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} and n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document}, this class agrees with the image of sgn8·ΣP\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\mathrm {sgn}}{8}\cdot \Sigma _P$$\end{document} in H2(Γg,1n;Θ2n+1/im(d2Γ))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(\Gamma ^n_{g,1};\Theta _{2n+1}/\mathrm {im}(d_2^\Gamma ))$$\end{document} and therefore vanishes as sgn8∈H2(Gg;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\mathrm {sgn}}{8}\in \mathrm {H}^{2}(G_g;\mathbf {Z})$$\end{document} is trivial in view of Lemma 3.21.
The extension class of the bottom row of (4.1), determined in Theorem 3.22, pulls back to the extension class of the top one. Since sgn8∈H2(Γg,1/2n;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\mathrm {sgn}}{8}\in \mathrm {H}^2(\Gamma ^n_{g,1/2};\mathbf {Z})$$\end{document} is pulled back from Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document} by construction, it is trivial in H(g)⊗SπnSO(n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$H(g)\otimes S\pi _n\mathrm {SO}(n)$$\end{document}, so the extension in (ii) is trivial for n≡1(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 1\ (\mathrm {mod}\ 4)$$\end{document}, classified by χ22·ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\chi ^2}{2}\cdot \Sigma _Q$$\end{document} for n≡3(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 3\ (\mathrm {mod}\ 4)$$\end{document} if n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document}, and by χ2-sgn8·ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\chi ^2-\mathrm {sgn}}{8}\cdot \Sigma _Q$$\end{document} if n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}. From this, the claimed image of d2T\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_2^T$$\end{document} follows from Lemma 3.19 and its proof. For n≡3(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 3\ (\mathrm {mod}\ 4)$$\end{document}, the homotopy sphere ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document} is nontrivial, so the extension does not split. This shows the second part of the statement, except for the claim regarding the equivariance, which will follow from the third part, since ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document} is trivial for n≡1(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 1\ (\mathrm {mod}\ 4)$$\end{document} by Theorem 3.3.
The diagram above shows that the commutator subgroup of Tg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_{g,1}^n$$\end{document} agrees with the image of d2T\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_2^{\mathrm {T}}$$\end{document}, which we already showed to be generated by ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document}. To construct a Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document}-equivariant splitting as claimed, note that the quotient of the lower row of (4.1) by ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document} pulls back from Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document} by Theorem 3.22 because sgn8∈H2(Γg,1n;Θ2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\frac{\mathrm {sgn}}{8}\in \mathrm {H}^{2}(\Gamma ^n_{g,1};\Theta _{2n+1})$$\end{document} has this property. Consequently, there is a central extension 0→Θ2n+1/ΣQ→E→Gg→0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$0\rightarrow \Theta _{2n+1}/\Sigma _Q\rightarrow E\rightarrow G_g\rightarrow 0$$\end{document} fitting into a commutative diagram
whose middle vertical composition induces a splitting as claimed, using that the right column in the diagram is exact. □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
The previous theorem, together with Lemma 1.1, Corollary 2.4, and Theorem 3.2 has Theorem C (ii) and Theorem D as a consequence. It also implies the following.
Corollary 4.2
For n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} odd and g≥1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 1$$\end{document}, Tg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_{g,1}^n$$\end{document} is abelian if and only if n≡1(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 1\ (\mathrm {mod}\ 4)$$\end{document}.
A geometric splitting
The splittings of the abelianisations of Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g,1}^n$$\end{document} and Tg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_{g,1}^n$$\end{document} provided by Theorem 4.1 are of a rather abstract nature. Aiming towards splitting these sequences more geometrically, we consider the following construction.
A diffeomorphism ϕ∈Diff∂(Wg,1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\phi \in \mathrm {Diff}_\partial (W_{g,1})$$\end{document} fixes a neighbourhood of the boundary pointwise, so its mapping torus Tϕ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$T_\phi $$\end{document} comes equipped with a canonical germ of a collar of its boundary S1×∂Wg,1⊂Tϕ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^1\times \partial W_{g,1}\subset T_\phi $$\end{document} using which we obtain a closed oriented (n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(n-1)$$\end{document}-connected (2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(2n+1)$$\end{document}-manifold T~ϕ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tilde{T}_\phi $$\end{document} by gluing in D2×S2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^2\times S^{2n-1}$$\end{document}. By obstruction theory and the fact that Wg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$W_{g}$$\end{document} is n-parallelisable, the stable normal bundle T~ϕ→BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tilde{T}_\phi \rightarrow \mathrm {BO}$$\end{document} has a unique lift to τ>nBO→BO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}\rightarrow \mathrm {BO}$$\end{document} compatible with the lift on D2×S2n-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^2\times S^{2n-1}$$\end{document} induced by its standard stable framing. This gives rise to a morphismt:Γg,1n⟶Ω2n+1τ>n,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} t:\Gamma ^n_{g,1}\longrightarrow \Omega ^{\tau _{>n}}_{2n+1}, \end{aligned}$$\end{document}which is compatible with the stabilisation map s:Γg,1n→Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$s:\Gamma _{g,1}^n\rightarrow \Gamma _{g,1}^n$$\end{document} sincets(ϕ)=[Tϕ♯(S1×Sn×Sn)∪S1×S2n-1D2×S2n-1]=t(ϕ)+[S1×Sn×Sn]=t(ϕ),\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} ts(\phi ){=}[T_\phi \sharp (S^1\times S^n\times S^n)\cup _{S^1\times S^{2n-1}} D^2\times S^{2n-1}]{=}t(\phi )+[S^1 \times S^n\times S^n]{=}t(\phi ), \end{aligned}$$\end{document}where S1×Sn×Sn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^1\times S^n\times S^n$$\end{document} carries the τ>nBO\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tau _{>n}\mathrm {BO}$$\end{document}-structure induced by the standard stable framing, which bounds. Using this, it is straight-forward to see that the compositionΘ2n+1=Γ0,1n⟶Γg,1n⟶tΩ2n+1τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \Theta _{2n+1}=\Gamma _{0,1}^{n}\longrightarrow \Gamma _{g,1}^n\overset{t}{\longrightarrow }\Omega _{2n+1}^{\tau _{>n}} \end{aligned}$$\end{document}of the iterated stabilisation map with t agrees with the canonical epimorphism appearing in Wall’s exact sequence (3.1), so its kernel agrees with the subgroup bA2n+2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}$$\end{document}. Together with the first part of Theorem 4.1, we conclude that the dashed arrow in the commutative diagram
is an isomorphism if and only if Kg=bA2n+2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$K_g=\mathrm {bA}_{2n+2}$$\end{document} which is the case for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document}, and for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} as long as n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document} since ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document} generates bA2n+2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}$$\end{document} for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document} by Corollary 3.5. Consequently, in these cases, the morphismis an isomorphism, whereas its kernel for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} coincides with the quotient bA2n+2/ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}/\Sigma _Q$$\end{document}, which is nontrivial as long as n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document}, generated by ΣP\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _P$$\end{document}, and whose order can be interpreted in terms of signatures (see Lemma 3.10). Since p∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$p_*$$\end{document} splits by Theorem 4.1 and the natural map coker(J)2n+1/[ΣQ]→Ω2n+1τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {coker}(J)_{2n+1}/[\Sigma _Q]\rightarrow \Omega _{2n+1}^{\tau _{>n}}$$\end{document} is an isomorphism by Corollary 3.6, the morphism (4.2) splits for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} if and only if the natural mapdoes. Brumfiel [10, Thm 1.3] has shown that this morphism always splits before taking quotients, so the map (4.3) (and hence also (4.2)) in particular splits whenever [ΣQ]∈coker(J)2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[\Sigma _Q]\in \mathrm {coker}(J)_{2n+1}$$\end{document} is trivial, which is conjecturally always the case (see Conjecture 3.7) and known in many cases as a result of Theorem 3.3 (see also Remark 3.8).
The situation for H1(Tg,1n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\mathrm {T}_{g,1}^n)$$\end{document} is similar. By Theorem 4.1, the morphism ρ∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\rho _*$$\end{document} in
splits Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document}-equivariantly and since the morphism t∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_*$$\end{document} is defined on H1(Γg,1n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\Gamma ^n_{g,1})$$\end{document}, its restriction to H1(Tg,1n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\mathrm {T}^n_{g,1})$$\end{document} is Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document}-equivariant when equipping Ω2n+1τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Omega _{2n+1}^{\tau _{>n}}$$\end{document} with the trivial action. By an analogous discussion to the one above, the kernel of the resulting morphism of Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document}-modulesis trivial for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document} and given by the quotient bA2n+2/ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}/\Sigma _Q$$\end{document} for n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document}. Moreover, this morphism splits if and only if it splits Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document}-equivariantly,4 which is precisely the case if the natural map (4.3) admits a splitting. We summarise this discussion in the following corollary, which implies Theorem E when combined with Lemma 3.10.
The morphism is an isomorphism for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document}. For g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}, it is an epimorphism and its kernel is given by bA2n+2/ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}/\Sigma _Q$$\end{document}.
The morphism is an epimorphism and has kernel bA2n+2/ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}/\Sigma _Q$$\end{document}.
The morphism t∗⊕p∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_*\oplus p_*$$\end{document} splits for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} if and only if splits, which is the case for n≡1(mod4)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\equiv 1\ (\mathrm {mod}\ 4)$$\end{document}. The same holds for t∗⊕ρ∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_*\oplus \rho _*$$\end{document} for all g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document}.
Abelianising Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g,1}^n$$\end{document} and Tg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_{g,1}^n$$\end{document} for n even
For n≥4\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 4$$\end{document} even, the abelianisations of Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g,1}^n$$\end{document} and Tg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_{g,1}^n$$\end{document} can be computed without fully determining the extensions0⟶Θ2n+1⟶Γg,1n⟶Γg,1/2n⟶00⟶H(g)⊗SπnSO(n)⟶Γg,1/2n⟶Gg⟶0.\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned}&\displaystyle 0\longrightarrow \Theta _{2n+1}\longrightarrow \Gamma _{g,1}^n \longrightarrow \Gamma _{g,1/2}^n\longrightarrow 0\\&\displaystyle 0\longrightarrow H(g)\otimes S\pi _n\mathrm {SO}(n)\longrightarrow \Gamma _{g,1/2}^n\longrightarrow G_g\longrightarrow 0. \end{aligned}$$\end{document}Indeed, arguing similarly as in the proof of Corollary 2.4, the second extension provides an isomorphism H1(Γg,1/2n)≅H1(Gg)⊕(H(g)⊗SπnSO(n))Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\Gamma _{g,1/2}^n)\cong \mathrm {H}_1(G_g)\oplus ( H(g)\otimes S\pi _n\mathrm {SO}(n))_{G_g}$$\end{document}, using that the coinvariants vanish also for Gg=Og,g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g=\mathrm {O}_{g,g}(\mathbf {Z})$$\end{document} as long as g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document} by Lemma A.2 and that the extension splits for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}, which is straightforward to check by noting that it is easy to lift elements ofG1=⟨-100-1,0110⟩≅Z/2⊕Z/2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} G_1=\langle \left( {\begin{matrix}-1&{}\quad 0\\ 0&{}\quad -1\end{matrix}}\right) , \left( {\begin{matrix}0&{}\quad 1\\ 1&{}\quad 0\end{matrix}}\right) \rangle \cong \mathbf {Z}/2\oplus \mathbf {Z}/2 \end{aligned}$$\end{document}to (so in particular to Γ1,1/2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{1,1/2}^n$$\end{document}) using the flip of the factors and a diffeomorphism of Sn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^n$$\end{document} of degree -1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$-1$$\end{document}.5 In contrast to the case n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document}, the resulting analogues of the morphisms (4.2) and (4.4) for n≥4\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 4$$\end{document} even are isomorphisms for allg≥1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 1$$\end{document}; this is Theorem G.
Proof of Theorem G
Wall’s exact sequence (3.1) is also valid for n≥4\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 4$$\end{document} even, so implies similarly to the case n odd that both morphisms in question are surjective and that their kernels agree with the quotients of bA2n+2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}$$\end{document} by the images of the differentialsH2(Γg,1/2n;Z)⟶d2Θ2n+1andH2(H(g)⊗SπnSO(n);Z)⟶d2Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {H}_2(\Gamma _{g,1/2}^n;\mathbf {Z})\overset{d_2}{\longrightarrow }\Theta _{2n+1}\quad \text {and}\quad \mathrm {H}_2(H(g)\otimes S\pi _n\mathrm {SO}(n);\mathbf {Z})\overset{d_2}{\longrightarrow }\Theta _{2n+1} \end{aligned}$$\end{document}induced by the extensions (1.5) and (1.6), so we have to show that these images agree with bA2n+2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}$$\end{document}. By comparing the extensions for different g via the stabilisation map (see Sect. 1.4) and noting that the second differential factors through the first, we see that it suffices to show this for the second differential in the case g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}. Plumbing disc bundles over Sn+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^{n+1}$$\end{document} defines a pairing of the formSπnSO(n)⊗SπnSO(n)→A2n+2τ>n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} S\pi _n\mathrm {SO}(n)\otimes S\pi _n\mathrm {SO}(n)\rightarrow A_{2n+2}^{\tau _{>n}} \end{aligned}$$\end{document}which can be seen to be surjective for n≥4\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 4$$\end{document} even, unlike in the case n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} odd [57, p. 295]. The composition of the pairing with the boundary map ∂:A2n+2τ>n→Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\partial :A_{2n+2}^{\tau _{>n}}\rightarrow \Theta _{2n+1}$$\end{document} with image bA2n+2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}$$\end{document} is usually called the Milnor pairing. Using the surjectivity, it is enough to show that the image of the second differential contains the image of the Milnor pairing. To do so, we rewrite the extension inducing the second differential via the canonical isomorphism resulting from Lemma 1.1 as0⟶Θ2n+1⟶ιT1n⟶ρSπnSO(n)⊕2⟶0.\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\longrightarrow \Theta _{2n+1}\overset{\iota }{\longrightarrow } \mathrm {T}_1^n \overset{\rho }{\longrightarrow } S\pi _n\mathrm {SO}(n)^{\oplus 2}\longrightarrow 0. \end{aligned}$$\end{document}and constructsin an explicit cocycle f:SπnSO(n)⊕2×SπnSO(n)⊕2→Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$f:S\pi _n\mathrm {SO}(n)^{\oplus 2}\times S\pi _n\mathrm {SO}(n)^{\oplus 2}\rightarrow \Theta _{2n+1}$$\end{document} classifying this extension as follows: define morphismst1:SπnSO(n)→T1nandt2:SπnSO(n)→T1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} t_1:S\pi _n\mathrm {SO}(n)\rightarrow \mathrm {T}_1^n\quad \text {and}\quad t_2:S\pi _n\mathrm {SO}(n)\rightarrow \mathrm {T}_1^n \end{aligned}$$\end{document}by assigning to a class η∈SπnSO(n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\eta \in S\pi _n\mathrm {SO}(n)$$\end{document} represented by η:(Sn,D+n)→(SO(n),id)⊂(SO(n+1),id)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\eta :(S^n,D^n_+)\rightarrow (\mathrm {SO}(n),\mathrm {id})\subset (\mathrm {SO}(n+1),\mathrm {id})$$\end{document} the diffeomorphism t1(η)(x,y)=(x,η(x)·y)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_1(\eta )(x,y)=(x,\eta (x)\cdot y)$$\end{document} and t2(η)(x,y)=(η(y)·x,y)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_2(\eta )(x,y)=(\eta (y)\cdot x,y)$$\end{document}, where D+n⊂Sn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$D^n_+\subset S^n$$\end{document} is the upper half-disc. Choosing the centre ∗∈D+n⊂Sn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$*\in D^n_+\subset S^n$$\end{document} as the base point, the diffeomorphisms t1(η)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_1(\eta )$$\end{document} and t2(η)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_2(\eta )$$\end{document} fix both spheres ∗×Sn,Sn×∗⊂Sn×Sn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$*\times S^n,S^n\times *\subset S^n\times S^n$$\end{document}, one because η\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\eta $$\end{document} is a based map and the other because it factors through the stabilisation map. From the description of ρ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\rho $$\end{document} in Sect. 1.3, we see that ρt1(η)=(η,0)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\rho t_1(\eta )=(\eta ,0)$$\end{document} and ρt2(η)=(0,η)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\rho t_2(\eta )=(0,\eta )$$\end{document}, so the function SπnSO(n)⊕2→T1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S\pi _n\mathrm {SO}(n)^{\oplus 2}\rightarrow \mathrm {T}_1^n$$\end{document} mapping (η,ξ)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\eta ,\xi )$$\end{document} to t1(η)∘t2(ξ)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_1(\eta )\circ t_2(\xi )$$\end{document} is a set-theoretical section to ρ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\rho $$\end{document}, which implies that the function defined byf:SπnSO(n)⊕2×SπnSO(n)⊕2⟶Θ2n+1((η1,ξ1),(η2,ξ2))⟼t1(η1)t2(ξ1)t1(η2)t2(ξ2)(t1(η1+η2)t2(ξ1+ξ2))-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} {f:}{S\pi _n\mathrm {SO}(n)^{\oplus 2}\times S\pi _n\mathrm {SO}(n)^{\oplus 2}}\longrightarrow & {} {\Theta _{2n+1}}\\ {\big ((\eta _1,\xi _1),(\eta _2,\xi _2)\big )}\longmapsto & {} {t_1(\eta _1) t_2(\xi _1) t_1(\eta _2) t_2(\xi _2) \big (t_1(\eta _1+\eta _2) t_2(\xi _1+\xi _2)\big )^{-1}} \end{aligned}$$\end{document}defines a 2-cocycle that represents the extension class of (4.5) in H2(SπnSO(n)⊕2;Θ2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(S\pi _n\mathrm {SO}(n)^{\oplus 2};\Theta _{2n+1})$$\end{document} (see e.g. [9, Ch. IV.3]); here we identified Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}$$\end{document} with its image in T1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {T}_1^n$$\end{document}. The differential is the image of this class under the map H2(SπnSO(n)⊕2;Θ2n+1)→Hom(H2(SπnSO(n)⊕2);Θ2n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(S\pi _n\mathrm {SO}(n)^{\oplus 2};\Theta _{2n+1})\rightarrow \mathrm {Hom}(\mathrm {H}_2(S\pi _n\mathrm {SO}(n)^{\oplus 2});\Theta _{2n+1})$$\end{document} participating in the universal coefficient theorem (see Sect. 3.3), so with respect to the canonical isomorphism Λ2(SπnSO(n)⊕2)≅H2(SπnSO(n)⊕2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Lambda ^2(S\pi _n\mathrm {SO}(n)^{\oplus 2})\cong \mathrm {H}_2(S\pi _n\mathrm {SO}(n)^{\oplus 2})$$\end{document} (see e.g. [9, Thm V.6.4 (iii)], the differential takes the form (see e.g. [9, Ex. IV.4.8 (c), Ex. V.6.5])Λ2(SπnSO(n)⊕2)⟶Θ2n+1((η1,ξ1)∧(η2,ξ2))⟼f((η1,ξ1),(η2,ξ2))-f((η2,ξ2),(η1,ξ1)).\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \begin{array}{rcl} \Lambda ^2(S\pi _n\mathrm {SO}(n)^{\oplus 2}) &{} \longrightarrow &{} \Theta _{2n+1} \\ \big ((\eta _1,\xi _1)\wedge (\eta _2,\xi _2)\big ) &{} \longmapsto &{} f\big ((\eta _1,\xi _1),(\eta _2,\xi _2)\big )-f\big ((\eta _2,\xi _2),(\eta _1,\xi _1)\big ). \end{array} \end{aligned}$$\end{document}The precomposition of this differential with the map SπnSO(n)⊗SπnSO(n)→Λ2(SπnSO(n)⊕2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S\pi _n\mathrm {SO}(n)\otimes S\pi _n\mathrm {SO}(n)\rightarrow \Lambda ^2(S\pi _n\mathrm {SO}(n)^{\oplus 2})$$\end{document} mapping η⊗ξ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\eta \otimes \xi $$\end{document} to (η,0)∧(0,ξ)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\eta ,0)\wedge (0,\xi )$$\end{document} thus agrees with the pairingSπnSO(n)⊗SπnSO(n)→Θ2n+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} S\pi _n\mathrm {SO}(n)\otimes S\pi _n\mathrm {SO}(n)\rightarrow \Theta _{2n+1} \end{aligned}$$\end{document}that sends η⊗ξ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\eta \otimes \xi $$\end{document} to the commutator t1(η)t2(ξ)t1(η)-1t2(ξ)-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_1(\eta )t_2(\xi )t_1(\eta )^{-1}t_2(\xi )^{-1}$$\end{document}. By the discussion in [40, p. 834], this coincides with the Milnor pairing, so its image is bA2n+2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}$$\end{document} by the discussion above, which concludes the claim. □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
Splitting the homology action
We conclude our study of Kreck’s extensions and its abelianisations with the following result, which proves the remaining first part of Theorem C. The reader shall be reminded once more of Lemma 1.1, saying that the natural map Γg,1n→Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{g,1}^n\rightarrow \Gamma ^n_g$$\end{document} is an isomorphism for n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document}.
Theorem 4.4
For n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document} odd, the extension0⟶Tg,1n⟶Γg,1n⟶Gg⟶0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\longrightarrow \mathrm {T}_{g,1}^n\longrightarrow \Gamma _{g,1}^n\longrightarrow G_g\longrightarrow 0 \end{aligned}$$\end{document}does not split for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document}, but admits a splitting for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} and n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document}.
Proof
The case n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document} and g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document} is clear, since Theorem A shows that under this assumption even the quotient of the extension by the subgroup Θ2n+1⊂Tg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}\subset \mathrm {T}_{g,1}^n$$\end{document} does not split. To deal with the other cases, we consider the morphism of extensions
By the naturality of the Serre spectral sequence, we have a commutative square
whose right vertical map has kernel generated by ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Sigma _Q$$\end{document} by the computation of H1(Tg,1n;Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\mathrm {T}_{g,1}^n;\mathbf {Z})$$\end{document} as a Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document}-module in Theorem 4.1. Therefore, to finish the proof of the first claim, it suffices to show that the differential d2:H2(Γg,1/2n;Z)→Θ2n+1/ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_2:\mathrm {H}_2(\Gamma ^n_{g,1/2};\mathbf {Z})\rightarrow \Theta _{2n+1}/\Sigma _Q$$\end{document} is nontrivial for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document} and n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document}, which follows from Theorem 3.22 together with the fact that bA2n+2/ΣQ\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {bA}_{2n+2}/\Sigma _Q$$\end{document} is nontrivial in these cases by Lemma 3.10 and Remark 3.11. Turning towards the second claim, we assume n≠3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 3,7$$\end{document} and recall that the isomorphism (sF,p):Γg,1/2n→(H(g)⊗πnSO)⋊Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(s_F,p):\Gamma ^n_{g,1/2}\rightarrow (H(g)\otimes \pi _n\mathrm {SO})\rtimes G_g$$\end{document} induces a splitting of the right vertical map Γg,1/2n→Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1/2}\rightarrow G_g$$\end{document} in the above diagram (see Sect. 2), so the claim follows from showing that the pullback of Γg,1n→Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1}\rightarrow \Gamma _{g,1}^n$$\end{document} along Gg⊂(H(g)⊗πnSO)⋊Gg≅Γg,1n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g\subset (H(g)\otimes \pi _n\mathrm {SO})\rtimes G_g\cong \Gamma _{g,1}^n$$\end{document} splits for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}, which is a consequence of Theorem B and Lemma 3.21. □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
Remark 4.5
Theorem 4.4 leaves open whether Γg,1n→Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma ^n_{g,1}\rightarrow G_g$$\end{document} admits a splitting for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} in dimensions n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}. Krylov [37, Thm 2.1] and Fried [19, Sect. 2] showed that this can not be the case for n=3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3$$\end{document} and we expect the same to hold for n=7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=7$$\end{document}.
Homotopy equivalences
Our final result Corollary F is concerned with the morphism of extensions
underlying work of Baues [1, Thm 10.3], relating the mapping class group Γgn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _g^n$$\end{document} to the group of homotopy classes of orientation preserving homotopy equivalences. The left vertical morphism is induced by the restriction of the unstable J homomorphism J:πnSO(n+1)→π2n+1Sn+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$J:\pi _n\mathrm {SO}(n+1)\rightarrow \pi _{2n+1}S^{n+1}$$\end{document} to the image of the stabilisation S:πnSO(n)→πnSO(n+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S:\pi _n\mathrm {SO}(n)\rightarrow \pi _n\mathrm {SO}(n+1)$$\end{document} in the source and to the image of the suspension map S:π2nSn→π2n+1Sn+1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S:\pi _{2n}S^{n}\rightarrow \pi _{2n+1}S^{n+1}$$\end{document} in the target, justified by the fact that the square
commutes up to sign (see [53, Cor. 11.2]).
Proof of Corollary F
By Theorem A, the upper row of (5.1) splits for n≠1,3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ne 1,3,7$$\end{document} odd, so the first part of (i) is immediate. The second part follows from Theorem A as well if we show that the existence of a splitting of the lower row for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document} is equivalent to one of the upper row. To this end, note that for n=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3,7$$\end{document}, we have isomorphisms SπnSO(n)≅Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S\pi _n\mathrm {SO}(n)\cong \mathbf {Z}$$\end{document} and Sπ2nSn=Tor(π2n+1Sn+1)≅Z/dn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S\pi _{2n}S^n=\mathrm {Tor}(\pi _{2n+1}S^{n+1})\cong \mathbf {Z}/d_n$$\end{document} for d3=12\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_3=12$$\end{document} and d7=120\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_7=120$$\end{document} with respect to which the J-homomorphism J:SπnSO(n)→SπnSn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$J:S\pi _n\mathrm {SO}(n)\rightarrow S\pi _{n}S^n$$\end{document} is given by reduction by dn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_n$$\end{document} (see e.g. [53, Ch. XIV]). By Lemma A.3, the group H2(Gg;H(g))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^2(G_g;H(g))$$\end{document} is annihilated by 2, so in particular by dn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d_n$$\end{document}. Using this, the first claim follows from the long exact sequence on cohomology induced by the exact sequence0→H(g)⊗SπnSO(n)→dn·(-)H(g)⊗SπnSO(n)→J∗H(g)⊗Sπ2nSn→0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\rightarrow H(g)\otimes S\pi _n\mathrm {SO}(n)\xrightarrow {d_n\cdot (-)} H(g)\otimes S\pi _n\mathrm {SO}(n)\xrightarrow {J_*} H(g)\otimes S\pi _{2n}S^n\rightarrow 0 \end{aligned}$$\end{document}of Gg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_g$$\end{document}-modules, since the extension class of the lower row is obtained from that of the upper one by the change of coefficients J∗:H(g)⊗SπnSO(n)→H(g)⊗Sπ2nSn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$J_*:H(g)\otimes S\pi _n\mathrm {SO}(n)\rightarrow H(g)\otimes S\pi _{2n}S^n$$\end{document}.
To prove the second part, we consider the exact sequenceinduced by the Serre spectral sequence of the lower row of (5.1). The left morphism is split injective as long as the extension splits, which, together with the first part and a consultation of Lemma A.2, exhibits to be as asserted. □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
Acknowledgements
I would like to thank Oscar Randal-Williams for several valuable discussions, Aurélien Djament for an explanation of an application of a result of his, and Fabian Hebestreit for many useful comments on an earlier version of this work. I was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 756444).
ReferencesBauesHJOn the group of homotopy equivalences of a manifold1996348124737477313401680866.55007BasterraMBobkovaIPontoKTillmannUYeakelSInfinite loop spaces from operads with homological stability201732139143037157151391.18012BensonDCampagnoloCRanickiARoviCSignature cocycles on the mapping class group and symplectic groups202022411106400410448707206683BensonDTheta functions and a presentation of 21+(2g+1)Sp(2g,2)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2^{1+(2g+1)}\text{Sp}(2g,2)$$\end{document}20195272042403924432BotvinnikBEbertJWraithDJOn the topology of the space of Ricci-positive metrics20201483997400641278431444.53030BerglundAMadsenIHomological stability of diffeomorphism groups20139114831264991295.57038BerglundAMadsenIRational homotopy theory of automorphisms of manifolds202022416718540867151441.57033Browder, W.: Surgery on Simply-Connected Manifolds. Springer, New York (1972) Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 65BrownKS1982New YorkSpringerBrumfielGOn the homotopy groups of BPL\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\rm BPL}$$\end{document} and PL/O\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${\rm PL/O}$$\end{document}1968882913112344580162.27302Burklund, R., Hahn, J., Senger, A.: On the boundaries of highly connected, almost closed manifolds. arXiv e-prints (2019)Burklund, R., Senger, A.: On the high-dimensional geography problem. arXiv e-prints (2020)CerfJLa stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie19703951732920890213.25202ChernSSHirzebruchFSerreJ-POn the index of a fibered manifold19578587596879430083.17801CrowleyDJOn the mapping class groups of #r(Sp×Sp)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\#_r(S^p\times S^p)$$\end{document} for p=3,7\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$p=3,7$$\end{document}20112693–41189119928602831236.57038DjamentASur l’homologie des groupes unitaires à coefficients polynomiaux20121018713929905631281.19004EndresRMultiplikatorsysteme der symplektischen Thetagruppe19829442812976853750497.10021EbertJRandal-WilliamsOTorelli spaces of high-dimensional manifolds201581386433352481387.55020FriedDWord maps, isotopy and entropy198629628518598466090647.58037GritsenkoVHulekKCommutator coverings of Siegel threefolds199894350954216395311037.11503GalatiusSRandal-WilliamsOAbelian quotients of mapping class groups of highly connected manifolds20163651–285787934989291343.55005GalatiusSRandal-WilliamsOMillerHModuli spaces of manifolds: a user’s guide2020New YorkChapman and Hall/CRC1205.55007GreyMOn rational homological stability for block automorphisms of connected sums of products of spheres20191973359340740453561432.55024HaefligerAPlongements différentiables de variétés dans variétés19613647821455380102.38603HillMAHopkinsMJRavenelDCOn the nonexistence of elements of Kervaire invariant one20161841126235051791366.55007HochschildGSerreJ-PCohomology of group extensions1953741110134524380050.02104JohnsonDMillsonJJModular Lagrangians and the theta multiplier1990100114316510371450699.10042KervaireMASome nonstable homotopy groups of Lie groups196041611691132370105.35302KauffmanLHKrylovNAKernel of the variation operator and periodicity of open books20051481–318320021189641066.57025KervaireMAMilnorJWGroups of homotopy spheres. I1963775045371480750115.40505KosińskiAAOn the inertia group of π\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\pi $$\end{document}-manifolds1967892272482140850172.25303Krannich, M., Kupers, A.: Some Hermitian K-groups via geometric topology. Proc. Am. Math. Soc. (2020). https://doi.org/10.17863/CAM.50639KrannichMRandal-WilliamsOMapping class groups of simply connected high-dimensional manifolds need not be arithmetic20203584469473413425607226693KrannichMReinholdJCharacteristic numbers of manifold bundles over surfaces with highly connected fibers20201022879904Kreck, M.: Isotopy classes of diffeomorphisms of (k-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(k-1)$$\end{document}-connected almost-parallelizable 2k\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2k$$\end{document}-manifolds. Proceedings of Symposium University of Aarhus, Aarhus, 1978, Lecture Notes in Mathematics, vol. 763. Springer, Berlin, pp. 643–663 (1979)Krylov, N.A.: Mapping class groups of (k-1)-connected almost-parallelizable 2k-manifolds, ProQuest LLC, Ann Arbor, MI, 2002, Thesis (Ph.D.)—University of Illinois at ChicagoKrylovNAOn the Jacobi group and the mapping class group of S3×S3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^3\times S^3$$\end{document}20033551991171015.57020KrylovNAPseudo-isotopy classes of diffeomorphisms of the unknotted pairs (Sn+2,Sn)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(S^{n+2}, S^n)$$\end{document} and (S2p+2,Sp×Sp)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(S^{2p+2}, S^p\times S^p)$$\end{document}20075139379501153.57020KupersASome finiteness results for groups of automorphisms of manifolds20192352277233340198941437.57035LawsonTCRemarks on the pairings of Bredon, Milnor, and Milnor–Munkres–Novikov1972228338433125060238.57017Levine, J.P.: Lectures on Groups of Homotopy Spheres. Algebraic and Geometric Topology (New Brunswick, N.J., 1983). Lecture Notes in Mathematics, vol. 1126. Springer, Berlin, pp. 62–95 (1985)Meyer, W.: Die Signatur von lokalen Koeffizientensystemen und Faserbündeln, Bonn. Math. Schr. no. 53, viii+59 (1972)MeyerWDie Signatur von Flächenbündeln19732012392643313820241.55019Milnor, J.W., Kervaire, M.A.: Bernoulli numbers, homotopy groups, and a theorem of Rohlin. Proceedings of International Congress of Mathematicians 1958. Cambridge University Press, New York, pp. 454–458 (1960)NewmanMSmartJRSymplectic modulary groups1964983891628620135.06502O’Meara, O.T.: Symplectic Groups. Mathematical Surveys, vol. 16. American Mathematical Society, Providence, RI (1978)PutmanAThe Picard group of the moduli space of curves with level structures2012161462367428915311241.30015SatoHDiffeomorphism group of Sp×Sq\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^{p}\times S^{q}$$\end{document} and exotic spheres196920255276SatoMThe abelianization of the level d\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d$$\end{document} mapping class group20103484788227463401209.57012SchultzRComposition constructions on diffeomorphisms of Sp×Sq\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$S^{p}\times S^{q}$$\end{document}1972427397540242.57013SpanierEH1966NewYorkMcGraw-Hill Book Co.0145.43303ThurstonWFoliations and groups of diffeomorphisms1974803043073392670295.57014TodaH1962PrincetonPrinceton University Press0101.40703WallCTCThe action of Γ2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _{2n}$$\end{document} on (n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(n-1)$$\end{document}-connected 2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2n$$\end{document}-manifolds1962139439441432230132.19901WallCTCClassification of (n-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(n-1)$$\end{document}-connected 2n\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2n$$\end{document}-manifolds1962751631891455400218.57022WallCTCClassification problems in differential topology. II. Diffeomorphisms of handlebodies196322632721563540123.16204WallCTCClassification problems in differential topology. VI. Classification of (s-1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(s-1)$$\end{document}-connected (2s+1)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(2s+1)$$\end{document}-manifolds196762732962165100173.26102Weintraub, S.H.: The Abelianization of the Theta Group in Low Genus. Algebraic Topology Poznań 1989. Lecture Notes in Mathematics, vol. 1474. Springer, Berlin, pp. 382–388 (1991)Appendix A: Low-degree cohomology of symplectic groups
This appendix contains various results on the low-degree (co)homology of the integral symplectic group Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} and its theta subgroup Sp2gq(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}^q(\mathbf {Z})$$\end{document} (see Sect. 1.2).
The stabilisation map H1(Sp2g(Z))⟶H1(Sp2g+2(Z))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {H}_1(\mathrm {Sp}_{2g}(\mathbf {Z}))\longrightarrow \mathrm {H}_1(\mathrm {Sp}_{2g+2}(\mathbf {Z})) \end{aligned}$$\end{document} is surjective for all g≥1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 1$$\end{document} and the stabilisation map H1(Sp2gq(Z))⟶H1(Sp2g+2q(Z))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {H}_1(\mathrm {Sp}^q_{2g}(\mathbf {Z}))\longrightarrow \mathrm {H}_1(\mathrm {Sp}^q_{2g+2}(\mathbf {Z})) \end{aligned}$$\end{document} is surjective for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document}, but has cokernel Z/2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}/2$$\end{document} for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}.
Proof
The fact that the abelianisation of Sp2(Z)=SL2(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2}(\mathbf {Z})=\mathrm {SL}_2(\mathbf {Z})$$\end{document} is generated by 1101\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( {\begin{matrix} 1 &{} 1 \\ 0 &{} 1 \end{matrix}}\right) $$\end{document} and of order 12 is well-known, and so is the isomorphism type of H1(Sp2g(Z))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\mathrm {Sp}_{2g}(\mathbf {Z}))$$\end{document} for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document} (see e.g. [3, Lem. A.1 (ii)]). The remaining claims regarding Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} follows from showing that H1(Sp2(Z))→H1(Sp4(Z))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\mathrm {Sp}_{2}(\mathbf {Z}))\rightarrow \mathrm {H}_1(\mathrm {Sp}_{4}(\mathbf {Z}))$$\end{document} is nontrivial, which can for instance be extracted from the proof of [20, Thm 2.1]: in their notation Γ1=Sp4(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Gamma _1=\mathrm {Sp}_4(\mathbf {Z})$$\end{document} and the map i∞:SL2(Z)→Γ1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$i_\infty :\mathrm {SL}_2(\mathbf {Z})\rightarrow \Gamma _1$$\end{document} identifies with the stabilisation Sp2(Z)→Sp4(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_2(\mathbf {Z})\rightarrow \mathrm {Sp}_4(\mathbf {Z})$$\end{document}. The isomorphism type of H1(Sp2gq(Z))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\mathrm {Sp}^q_{2g}(\mathbf {Z}))$$\end{document} for g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document} is determined in [17, Thm 2] and for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document} in [58, Thm 1]. The first claim of (ii) follows from the main formula of [27] for g≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 3$$\end{document} and from [58, Cor. 2] for g=1,2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1,2$$\end{document}, which also gives the claimed generator of the Z\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}$$\end{document}-summand in H1(Sp2q(Z))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\mathrm {Sp}^q_{2}(\mathbf {Z}))$$\end{document}. The proof of [17, Thm 2] provides the asserted generator of the Z/2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}/2$$\end{document}-summand in H1(Sp4q(Z))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\mathrm {Sp}^q_{4}(\mathbf {Z}))$$\end{document}. The final claim follows from the first four items once we show that the image of 1201\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( {\begin{matrix} 1 &{} 2 \\ 0 &{} 1 \end{matrix}}\right) $$\end{document} in H1(Sp4q(Z))\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}_1(\mathrm {Sp}^q_4(\mathbf {Z}))$$\end{document} vanishes, which is another consequence of the formulas in [58, Cor. 2]. □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
Lemma A.2
The (co)invariants of the standard actions of Sp2gq(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}^q_{2g}(\mathbf {Z})$$\end{document} and Og,g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {O}_{g,g}(\mathbf {Z})$$\end{document} on Z2g⊗A\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}^{2g}\otimes A$$\end{document} for an abelian group A satisfy(Z2g⊗A)Sp2gq(Z)≅(Z2g⊗A)Og,g(Z)≅A/2ifg=10ifg≥2and(Z2g⊗A)Sp2gq(Z)≅(Z2g⊗A)Og,g(Z)≅Hom(Z/2,A)ifg=10ifg≥2.\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} (\mathbf {Z}^{2g}\otimes A)_{\mathrm {Sp}^q_{2g}(\mathbf {Z})}\cong & {} (\mathbf {Z}^{2g}\otimes A)_{\mathrm {O}_{g,g}(\mathbf {Z})}\cong {\left\{ \begin{array}{ll}A/2&{}\quad \text{ if } g=1\\ 0&{}\quad \text{ if } g\ge 2\end{array}\right. }\quad \text {and}\\ (\mathbf {Z}^{2g}\otimes A)^{\mathrm {Sp}^q_{2g}(\mathbf {Z})}\cong & {} (\mathbf {Z}^{2g}\otimes A)^{\mathrm {O}_{g,g}(\mathbf {Z})}\cong {\left\{ \begin{array}{ll}\mathrm {Hom}(\mathbf {Z}/2,A)&{}\quad \text{ if } g=1\\ 0&{}\quad \text{ if } g\ge 2\end{array}\right. }. \end{aligned}$$\end{document}The same applies to the action of Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document}, except that the (co)invariants also vanish for g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}.
Proof
We prove the claim for Sp2g(q)(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}^{(q)}(\mathbf {Z})$$\end{document} first. The self-duality of Z2g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}^{2g}$$\end{document} induced by the symplectic form implies(Z2g⊗A)Sp2g(q)(Z)≅Hom(Z2g,A)Sp2g(q)(Z)≅Hom((Z2g)Sp2g(q)(Z),A),\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} (\mathbf {Z}^{2g}\otimes A)^{\mathrm {Sp}^{(q)}_{2g}(\mathbf {Z})}\cong \mathrm {Hom}(\mathbf {Z}^{2g},A)^{\mathrm {Sp}^{(q)}_{2g}(\mathbf {Z})}\cong \mathrm {Hom}((\mathbf {Z}^{2g})_{\mathrm {Sp}^{(q)}_{2g}(\mathbf {Z})},A), \end{aligned}$$\end{document}so the computation of the invariants is a consequence of that of the coinvariants. To settle the case g≥2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\ge 2$$\end{document} for both Sp2gq(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}^q_{2g}(\mathbf {Z})$$\end{document} and Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document}, it thus suffices to prove that the coinvariants (Z2g⊗A)Sp2gq(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\mathbf {Z}^{2g}\otimes A)_{\mathrm {Sp}_{2g}^q(\mathbf {Z})}$$\end{document} with respect to the smaller group vanish. We consider the matricesPσ00Pσ∈Sp2gq(Z)forσ∈Σg,0-IgIg0∈Sp2gq(Z),and1011Ig-21-101Ig-2∈Sp2gq(Z),\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned}&\left( \begin{array}{cc} P_\sigma &{}\quad 0\\ 0&{}\quad P_\sigma \end{array}\right) \in \mathrm {Sp}_{2g}^q(\mathbf {Z})\quad \text {for }\sigma \in \Sigma _g,\quad \left( \begin{array}{cc}0&{}\quad -I_g\\ I_g&{}\quad 0\end{array}\right) \in \mathrm {Sp}_{2g}^q(\mathbf {Z}),\quad \text {and}\\&\quad \left( \begin{array}{cccccc}1&{}\quad 0&{}\quad &{}\quad &{}\quad &{}\quad \\ 1&{}\quad 1&{}\quad &{}\quad &{}\quad &{}\quad \\ {} &{}\quad &{}\quad I_{g-2}&{}\quad &{}\quad &{}\quad \\ {} &{}\quad &{}\quad &{}\quad 1&{}\quad -1&{}\quad \\ {} &{}\quad &{}\quad &{}\quad 0&{}\quad 1&{}\quad \\ {} &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad I_{g-2}\end{array}\right) \in \mathrm {Sp}_{2g}^q(\mathbf {Z}), \end{aligned}$$\end{document}which can be seen to be contained in the subgroup Sp2gq(Z)⊂Sp2g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}^q(\mathbf {Z})\subset \mathrm {Sp}_{2g}(\mathbf {Z})$$\end{document} using the description in Sect. 1.2; here Ig∈GLg(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$I_g\in \mathrm {GL}_g(\mathbf {Z})$$\end{document} is the unit matrix and Pσ∈GLg(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$P_\sigma \in \mathrm {GL}_g(\mathbf {Z})$$\end{document} the permutation matrix associated to σ∈Σg\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\sigma \in \Sigma _g$$\end{document}. It suffices to show that ei⊗a\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$e_i\otimes a$$\end{document} and fi⊗a\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$ f_i\otimes a$$\end{document} are trivial in the coinvariants for 1≤i≤g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$1\le i\le g$$\end{document} and a∈A\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$a\in A$$\end{document}, where (e1,…,eg,f1,…,fg)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(e_1,\ldots ,e_g,f_1,\ldots ,f_g)$$\end{document} is the standard symplectic basis of Z2g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}^{2g}$$\end{document}. Writing [-]\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[-]$$\end{document} for the residue class of an element in the coinvaraints, acting with the permutation matrices shows [ei⊗a]=[ej⊗a]\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[e_i\otimes a]=[ e_j\otimes a]$$\end{document} and [fi⊗a]=[fj⊗a]\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[f_i\otimes a]=[f_j\otimes a]$$\end{document} for all 1≤i,j≤g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${1\le i,j\le g}$$\end{document}, so it suffices to prove [e1⊗a]=[f1⊗a]=0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[e_1\otimes a]=[ f_1\otimes a]=0$$\end{document}. Using the second matrix, we see that [e1⊗a]=[f1⊗a]\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[ e_1\otimes a]=[f_1\otimes a]$$\end{document} and 2[e1⊗a]=0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$2[ e_1\otimes a]=0$$\end{document}, and finally the third matrix shows [e1⊗a]=[e1⊗a]+[e2⊗a]=2[e1⊗a]=0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[ e_1\otimes a]=[e_1\otimes a]+[ e_2\otimes a]=2[ e_1\otimes a]=0$$\end{document}, so the coinvariants are trivial. In the case g=1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g=1$$\end{document}, the first part of the proof is still valid and shows that (Z2⊗A)Sp2q(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\mathbf {Z}^{2}\otimes A)_{\mathrm {Sp}_{2}^q(\mathbf {Z})}$$\end{document} and (Z2⊗A)Sp2q(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\mathbf {Z}^{2}\otimes A)_{\mathrm {Sp}_{2}^q(\mathbf {Z})}$$\end{document} are quotients of A/2 generated by [e1⊗a]=[f1⊗a]\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[ e_1\otimes a]=[ f_1\otimes a]$$\end{document} for a∈A\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$a\in A$$\end{document}. For Sp2(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2}(\mathbf {Z})$$\end{document}, we may use 1101∈Sp2(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( {\begin{matrix} 1&{}1\\ 0&{}1 \end{matrix}}\right) \in \mathrm {Sp}_{2}(\mathbf {Z})$$\end{document} to conclude [f1⊗a]=[e1⊗a]+[f1⊗a]=2[e1⊗a]=0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$[ f_1\otimes a]=[ e_1\otimes a]+[f_1\otimes a]=2[e_1\otimes a]=0$$\end{document}, so the coinvariants vanish. For Sp2q(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}^q_{2}(\mathbf {Z})$$\end{document}, one uses that Sp2q(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}^q_{2}(\mathbf {Z})$$\end{document} is generated by 1201\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( {\begin{matrix} 1&{}2\\ 0&{}1\end{matrix}}\right) $$\end{document} and 01-10\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( {\begin{matrix} 0&{}1\\ -1&{}0\end{matrix}}\right) $$\end{document} (see e.g. [58, p. 385]) to see that the surjection Z2⊗A→A/2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}^{2}\otimes A\rightarrow A/2$$\end{document} induced by adding coordinates is invariant and surjective, and thus induces an isomorphism(Z2g⊗A)Sp2gq(Z)≅A/2,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} (\mathbf {Z}^{2g}\otimes A)_{\mathrm {Sp}_{2g}^q(\mathbf {Z})}\cong A/2, \end{aligned}$$\end{document}as claimed. The proof for Og,g(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {O}_{g,g}(\mathbf {Z})$$\end{document} is almost identical to that for Sp2gq(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Sp}_{2g}^q(\mathbf {Z})$$\end{document}, except that one has to replace the second displayed matrix by 0II0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( {\begin{matrix}0&{}\quad I\\ I&{}\quad 0\end{matrix}}\right) $$\end{document}, also act with -I2g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$-I_{2g}$$\end{document}, and use that O1,1(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {O}_{1,1}(\mathbf {Z})$$\end{document} is generated by -10-10\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( {\begin{matrix}-1&{}\quad 0\\ -1&{}\quad 0\end{matrix}}\right) $$\end{document} and 0110\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( {\begin{matrix}0&{}\quad 1\\ 1&{}\quad 0\end{matrix}}\right) $$\end{document}.□\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
Lemma A.3
The cohomology groups H∗(Sp2g(Z),Z2g)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^*(\mathrm {Sp}_{2g}(\mathbf {Z}),\mathbf {Z}^{2g})$$\end{document} and H∗(Sp2gq(Z),Z2g)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^*(\mathrm {Sp}^{q}_{2g}(\mathbf {Z}),\mathbf {Z}^{2g})$$\end{document} with coefficients in the standard module are annihilated by 2 and in low degrees given byH0(Sp2g(Z);Z2g)=0H0(Sp2gq(Z);Z2g)=0H1(Sp2g(Z);Z2g)≅0H1(Sp2gq(Z);Z2g)≅Z/2g=10g≥2H2(Sp2g(Z);Z2g)≅0forg=1H2(Sp2gq(Z);Z2g)≅Z/2⊕Z/2forg=1.\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \begin{array}{lll} \mathrm {H}^0(\mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z}^{2g})=0 &{}\quad \mathrm {H}^0(\mathrm {Sp}^q_{2g}(\mathbf {Z});\mathbf {Z}^{2g})=0 \\ \mathrm {H}^1(\mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z}^{2g})\cong 0 &{}\quad \mathrm {H}^1(\mathrm {Sp}^{q}_{2g}(\mathbf {Z});\mathbf {Z}^{2g})\cong {\left\{ \begin{array}{ll}\mathbf {Z}/2&{}\quad g=1\\ 0&{}\quad g\ge 2\end{array}\right. }\\ \mathrm {H}^2(\mathrm {Sp}_{2g}(\mathbf {Z});\mathbf {Z}^{2g})\cong 0\ \ \text {for }g=1&{}\quad \mathrm {H}^2(\mathrm {Sp}^{q}_{2g}(\mathbf {Z});\mathbf {Z}^{2g})\cong \mathbf {Z}/2\oplus \mathbf {Z}/2\ \ \text {for }g=1. \end{array} \end{aligned}$$\end{document}
Proof
The negative of the identity matrix -I∈Sp2g(q)(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$-I\in \mathrm {Sp}^{(q)}_{2g}(\mathbf {Z})$$\end{document} is central and acts by -1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$-1$$\end{document} on Z2g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}^{2g}$$\end{document}, so the first claim follows from the “centre kills trick” which is worth recalling: multiplication g·(-):M→M\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\cdot (-):M\rightarrow M$$\end{document} by an element g∈G\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\in G$$\end{document} of a discrete group G on a G-module M is equivariant with respect to the conjugation cg:G→G\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$c_g:G\rightarrow G$$\end{document} by g and induces the identity on H∗(G;M)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^*(G;M)$$\end{document} by the usual argument. If g∈G\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$g\in G$$\end{document} is central, conjugation by g is trivial, so H∗(G;M)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^*(G;M)$$\end{document} is annihilated by the action of (1-g)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(1-g)$$\end{document} on M.
Since H∗(Sp2g(q)(Z);Z2g)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^*(\mathrm {Sp}^{(q)}_{2g}(\mathbf {Z});\mathbf {Z}^{2g})$$\end{document} is 2-torsion, the self-duality of Z2g\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}^{2g}$$\end{document} combined with the universal coefficient theorem for nontrivial coefficients (see e.g. [51, p. 283]) impliesExtZ1(Hi-1(Sp2g(q)(Z);Z2g),Z)≅Hi(Sp2g(q)(Z);Z2g),\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {Ext}^1_\mathbf {Z}(\mathrm {H}_{i-1}(\mathrm {Sp}^{(q)}_{2g}(\mathbf {Z});\mathbf {Z}^{2g}),\mathbf {Z})\cong \mathrm {H}^i(\mathrm {Sp}^{(q)}_{2g}(\mathbf {Z});\mathbf {Z}^{2g}), \end{aligned}$$\end{document}so the computation of H∗(Sp2g(q)(Z);Z2g)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {H}^*(\mathrm {Sp}^{(q)}_{2g}(\mathbf {Z});\mathbf {Z}^{2g})$$\end{document} for ∗≤1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$*\le 1$$\end{document} is a consequence of Lemma A.2.
From the Lyndon–Hochschild–Serre spectral sequence for the extension0⟶{±1}⟶Sp2(q)(Z)⟶PSp2(q)(Z)⟶0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} 0\longrightarrow \{\pm 1\}\longrightarrow \mathrm {Sp}_2^{(q)}(\mathbf {Z})\longrightarrow \mathrm {PSp}_2^{(q)}(\mathbf {Z})\longrightarrow 0 \end{aligned}$$\end{document}with coefficients in Z2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}^2$$\end{document}, we see thatH2(Sp2(q)(Z);Z2)≅H1(PSp2(q)(Z);F22)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {H}^{2}(\mathrm {Sp}^{(q)}_{2}(\mathbf {Z});\mathbf {Z}^{2})\cong {\mathrm {H}^1(\mathrm {PSp}^{(q)}_2(\mathbf {Z});\mathbf {F}_2^2)} \end{aligned}$$\end{document}and the right hand side can easily be computed to be as claimed by applying the Mayer–Vietoris sequence to the presentationPSp2(Z)=⟨S,T∣S2=1,T3=1⟩andPSp2q(Z)=⟨S,R∣S2=1⟩\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \mathrm {PSp}_2(\mathbf {Z})=\langle S,T\mid S^2=1,T^3=1\rangle \quad \text {and}\quad \mathrm {PSp}^q_2(\mathbf {Z})=\langle S,R\mid S^2=1\rangle \end{aligned}$$\end{document}forS=0-110,T=0-111,andR=1201,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} S=\left( {\begin{matrix} 0 &{}\quad -1 \\ 1 &{}\quad 0 \end{matrix}}\right) ,\quad T=\left( {\begin{matrix} 0 &{}\quad -1 \\ 1 &{}\quad 1 \end{matrix}}\right) ,\quad \text {and}\quad R=\left( {\begin{matrix} 1 &{}\quad 2 \\ 0 &{}\quad 1 \end{matrix}}\right) , \end{aligned}$$\end{document}which is well-known for PSp2(Z)≅PSL2(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {PSp}_2(\mathbf {Z})\cong \mathrm {PSL}_2(\mathbf {Z})$$\end{document} and appears for instance in [58, p. 385] for PSp2q(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {PSp}^q_2(\mathbf {Z})$$\end{document}. □\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\square $$\end{document}
Note that Z/Λn\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}/\Lambda _n$$\end{document} is the trivial in the case n=3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=3$$\end{document} where Haefliger’s result does not apply.
Kreck phrases his results in terms of pseudo-isotopy instead of isotopy. By Cerf’s “pseudo-isotopy implies isotopy” [13], this does not make a difference as long as n≥3\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 3$$\end{document}.
This can be seen for instance from the Meyer–Vietoris sequence of the well-known decomposition SL2(Z)≅Z/4∗Z/2Z/6\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {SL}_2(\mathbf {Z})\cong \mathbf {Z}/4*_{\mathbf {Z}/2}\mathbf {Z}/6$$\end{document}, where Z/4\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}/4$$\end{document} is generated by 01-10\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( {\begin{matrix}0&{}1\\ -1&{}0\end{matrix}}\right) $$\end{document}, Z/6\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}/6$$\end{document} by 11-10\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( {\begin{matrix}1&{}1\\ -1&{}0\end{matrix}}\right) $$\end{document}, and Z/2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}/2$$\end{document} by -100-1\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( {\begin{matrix}-1&{}0\\ 0&{} -1\end{matrix}}\right) $$\end{document}.
If (st,sρ)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(s_{t},s_{\rho })$$\end{document} is a non-equivariant splitting of t∗⊕ρ∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_*\oplus \rho _*$$\end{document}, then (st,(id-stt∗)s′)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(s_t,(\mathrm {id}-s_tt_*)s')$$\end{document} is an equivariant splitting of t∗⊕ρ∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$t_*\oplus \rho _*$$\end{document}, where s′\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$s'$$\end{document} an equivariant splitting of ρ∗\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\rho _*$$\end{document} ensured by Theorem 4.1. This uses that st\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$s_t$$\end{document} is already equivariant since ρ∗st=0\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\rho _*s_t=0$$\end{document}, so the image of st\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$s_t$$\end{document} is contained in the image of Θ2n+1/ΣQ→H1(Tg,1n)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\Theta _{2n+1}/\Sigma _Q\rightarrow \mathrm {H}_1(\mathrm {T}^n_{g,1})$$\end{document} which is fixed by the action.
There are several mistakes in the literature related to the fact that G1=O1,1(Z)\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$G_1=\mathrm {O}_{1,1}(\mathbf {Z})$$\end{document} is isomorphic to (Z/2)2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(\mathbf {Z}/2)^2$$\end{document} and not Z/4\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbf {Z}/4$$\end{document}: in [35, p. 645], it should be π~0Diff(S2×S2)≅(Z/2)2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tilde{\pi }_0\mathrm {Diff}(S^2\times S^2)\cong (\mathbf {Z}/2)^2$$\end{document}, in [49, Thm 1] it should be π~0Diff(Sp×Sq)/π~0SDiff(Sp×Sq)≅(Z/2)2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\tilde{\pi }_0 \mathrm {Diff}(S^p\times S^q)/\tilde{\pi }_0S \mathrm {Diff}(S^p\times S^q)\cong (\mathbf {Z}/2)^2$$\end{document} for p=q\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$p=q$$\end{document} even, and finally in the proof of [36, Thm 2.6] it should be Aut(Hk(Sk×Sk))≅(Z/2)2\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathrm {Aut}(\mathrm {H}_k(S^k\times S^k))\cong (\mathbf {Z}/2)^2$$\end{document} for k even.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.