We construct a zig–zag from the once delooped space of pseudoisotopies of a closed 2n-disc to the once looped algebraic K-theory space of the integers and show that the maps involved are p-locally (2n-4)\documentclass[12pt]{minimal}
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\begin{document}$$(2n-4)$$\end{document}-connected for n>3\documentclass[12pt]{minimal}
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\begin{document}$$n\,{>}\,3$$\end{document} and large primes p. The proof uses the computation of the stable homology of the moduli space of high-dimensional handlebodies due to Botvinnik–Perlmutter and is independent of the classical approach to pseudoisotopy theory based on Igusa’s stability theorem and work of Waldhausen. Combined with a result of Randal-Williams, one consequence of this identification is a calculation of the rational homotopy groups of BDiff∂(D2n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BDiff}_\partial (D^{2n+1})$$\end{document} in degrees up to 2n-5\documentclass[12pt]{minimal}
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Mathematics Subject Classification57R5219D5057R6555P47publisher-imprint-nameSpringervolume-issue-count3issue-article-count7issue-toc-levels0issue-pricelist-year2022issue-copyright-holderSpringer-Verlag GmbH Germany, part of Springer Natureissue-copyright-year2022article-contains-esmNoarticle-numbering-styleContentOnlyarticle-registration-date-year2021article-registration-date-month9article-registration-date-day24article-toc-levels0toc-levels0volume-typeRegularjournal-productNonStandardArchiveJournalnumbering-styleContentOnlyarticle-grants-typeOpenChoicemetadata-grantOpenAccessabstract-grantOpenAccessbodypdf-grantOpenAccessbodyhtml-grantOpenAccessbibliography-grantOpenAccessesm-grantOpenAccessonline-firstfalsepdf-file-referenceBodyRef/PDF/222_2021_Article_1077.pdfpdf-typeTypesettarget-typeOnlinePDFissue-online-date-year2022issue-online-date-month2issue-online-date-day21issue-print-date-year2022issue-print-date-month2issue-print-date-day21issue-typeRegulararticle-typeOriginalPaperjournal-subject-primaryMathematicsjournal-subject-secondaryMathematics, generaljournal-subject-collectionMathematics and Statisticsopen-accesstrue
The homotopy type of the group of smooth pseudoisotopies, or concordance diffeomorphisms,C(M)={ϕ:M×[0,1]⟶≅M×[0,1]∣ϕ|M×{0}∪∂M×[0,1]=id},\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {C}(M)=\{\phi :M\times [0,1]\overset{\cong }{\longrightarrow } M\times [0,1]\mid \phi |_{M\times \{0\}\cup \partial M\times [0,1]}=\mathrm {id}\}, \end{aligned}$$\end{document}of a smooth compact d-dimensional manifold M in the smooth topology has been an object of interest to geometric topologists for many years, not least because of its intimate connection to algebraic K-theory already visible on the level of path components. Building on Cerf’s proof that C(M)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {C}(M)$$\end{document} is connected if M is simply connected and d≥5\documentclass[12pt]{minimal}
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\begin{document}$$d\ge 5$$\end{document} [11], Hatcher and Wagoner [20] computed the group π0C(M)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {C}(M)$$\end{document} of isotopy classes of concordances in high dimensions by relating it to the lower algebraic K-groups of the integral group ring Z[π1M]\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {Z}[\pi _1M]$$\end{document} of the fundamental group of M.1 Beyond its components, the homotopy type of the space of concordances C(M)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {C}(M)$$\end{document} and its relation to K-theory has so far been studied in two steps: deep work of Igusa [22] shows that the stabilisation mapC(M)⟶C(M×[0,1])\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {C}(M)\longrightarrow \mathrm {C}(M\times [0,1]) \end{aligned}$$\end{document}induced by crossing with an interval is min(d-43,d-72)\documentclass[12pt]{minimal}
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\begin{document}$$\min (\frac{d-4}{3},\frac{d-7}{2})$$\end{document}-connected, so in this range up to about a third of the dimension, one may consider the stable concordance spacecolimkC(M×[0,1]k)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {colim}_k\mathrm {C}(M\times [0,1]^k)$$\end{document} instead, which in turn admits a complete description in terms of Waldhausen’s generalised algebraic K-theory for spaces by Waldhausen, Jahren, and Rognes’ foundational stable parametrised h-cobordism theorem [48].
In this work, we focus on the case M=D2n\documentclass[12pt]{minimal}
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\begin{document}$$M=D^{2n}$$\end{document} of a closed disc of even dimension and study its space of concordances via a new route—independent of the classical approach—which does not involve stabilising the dimension and is, vaguely speaking, homological instead of homotopical; we shall elaborate on this at a later point. Our main result relates the delooped concordance space BC(D2n)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BC}(D^{2n})$$\end{document} to the once looped algebraic K-theory space of the integers Ω∞+1K(Z)\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^{\infty +1}\mathrm {K}(\mathbf {Z})$$\end{document} in a range up to approximately the dimension, p-locally for primes p that are large with respect to the dimension and the degree.
Theorem A
For n>3\documentclass[12pt]{minimal}
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\begin{document}$$n>3$$\end{document}, there exists a zig–zagBC(D2n)⟶·⟵Ω0∞+1K(Z)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {BC}(D^{2n})\longrightarrow \cdot \longleftarrow \Omega ^{\infty +1}_0\mathrm {K}(\mathbf {Z})\end{aligned}$$\end{document}whose maps are p-locally min(2n-4,2p-4-n)\documentclass[12pt]{minimal}
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\begin{document}$$\min (2n-4,2p-4-n)$$\end{document}-connected for primes p.
Remark
The result we prove is slightly stronger than stated here (see Theorem 5.1) and implies for instance that π2n-4BC(D2n)⊗Q\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{2n-4}\mathrm {BC}(D^{2n})\otimes \mathbf {Q}$$\end{document} surjects onto K2n-3(Z)⊗Q\documentclass[12pt]{minimal}
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\begin{document}$$K_{2n-3}(\mathbf {Z})\otimes \mathbf {Q}$$\end{document} as long as n>3\documentclass[12pt]{minimal}
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\begin{document}$$n>3$$\end{document}.
When combined with Borel’s work on the stable rational cohomology of arithmetic groups [9], Theorem A provides an isomorphismπ∗BC(D2n)⊗Q≅K∗+1(Z)⊗Q≅Qif∗≡0(mod4)0otherwisefor0<∗<2n-4,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\pi _*\mathrm {BC}(D^{2n})\otimes \mathbf {Q}\cong & {} K_{*+1}(\mathbf {Z})\otimes \mathbf {Q}\\ {}\cong & {} {\left\{ \begin{array}{ll}\mathbf {Q}&{}{}\text{ if } *\equiv 0\ (\mathrm {mod}\ 4)\\ 0&{}{}\text{ otherwise }\end{array}\right. }\quad \text{ for } 0<*<2n-4,\end{aligned}$$\end{document}and an epimorphism π∗BC(D2n)⊗Q→K∗+1(Z)⊗Q\documentclass[12pt]{minimal}
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\begin{document}$$\pi _*\mathrm {BC}(D^{2n})\otimes \mathbf {Q}\rightarrow K_{*+1}(\mathbf {Z})\otimes \mathbf {Q}$$\end{document} in degree 2n-4\documentclass[12pt]{minimal}
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\begin{document}$$2n-4$$\end{document}, which goes significantly beyond the range that was previously accessible by relying on Igusa’s stability result and shows for instance that BC(D8)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BC}(D^{8})$$\end{document} is nontrivial, even rationally. Given that the K-groups K∗(Z)\documentclass[12pt]{minimal}
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\begin{document}$$K_{*}(\mathbf {Z})$$\end{document} are known to contain p-torsion for comparatively large primes with respect to the degree due to contributions from Bernoulli numbers, Theorem A also exhibits many new torsion elements in π∗BC(D2n)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _*\mathrm {BC}(D^{2n})$$\end{document}, such as one of order 691 in π21BC(D2n)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{21}\mathrm {BC}(D^{2n})$$\end{document} as long as n>12\documentclass[12pt]{minimal}
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\begin{document}$$n>12$$\end{document} resulting from the fact that K22(Z)\documentclass[12pt]{minimal}
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\begin{document}$$K_{22}(\mathbf {Z})$$\end{document} is cyclic of that order.
In the remainder of this introduction, we explain more direct applications of Theorem A and conclude by indicating some ideas that go into its proof.
Diffeomorphisms and concordances of odd discs
Restricting a concordance to the moving part of its boundary induces a homotopy fibre sequenceDiff∂(Dd+1)⟶C(Dd)⟶Diff∂(Dd)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {Diff}_\partial (D^{d+1})\longrightarrow \mathrm {C}(D^{d})\longrightarrow \mathrm {Diff}_\partial (D^{d})\end{aligned}$$\end{document}that compares the group of concordances C(Dd)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {C}(D^{d})$$\end{document} of a d-disc to its group of its diffeomorphisms Diff∂(Dd)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Diff}_\partial (D^d)$$\end{document} fixing the boundary pointwise. By a result of Randal-Williams [37, Thm 4.1] based on Morlet’s lemma of disjunction and work of Berglund and Madsen [8] (a combination which incidentally inspired parts of our strategy to prove Theorem A), the space BDiff∂(D2n)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BDiff}_\partial (D^{2n})$$\end{document} is rationally (2n-5)\documentclass[12pt]{minimal}
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\begin{document}$$(2n-5)$$\end{document}-connected, so the delooped mapsBDiff∂(D2n+1)⟶BC(D2n)andBC(D2n+1)⟶BDiff∂(D2n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {BDiff}_\partial (D^{2n+1})\longrightarrow \mathrm {BC}(D^{2n})\quad \text {and}\quad \mathrm {BC}(D^{2n+1})\longrightarrow \mathrm {BDiff}_\partial (D^{2n+1})\end{aligned}$$\end{document}are rationally (2n-5)\documentclass[12pt]{minimal}
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\begin{document}$$(2n-5)$$\end{document}-connected as well, resulting in the following corollary of Theorem A.
The range in Corollary B is nearly optimal: by work of Watanabe [45] or as consequence of Weiss’ results on topological Pontryagin classes [46], the group π2n-2BDiff∂(D2n+1)⊗Q\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{2n-2}\mathrm {BDiff}_\partial (D^{2n+1})\otimes \mathbf {Q}$$\end{document} is known to be nontrivial for many values of n for which K2n-1(Z)⊗Q\documentclass[12pt]{minimal}
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\begin{document}$$K_{2n-1}(\mathbf {Z})\otimes \mathbf {Q}$$\end{document} vanishes by Borel’s work.
A combination of the strengthening of Theorem A mentioned earlier with recent work of Kupers and Randal-Williams [27] improves the range of Corollary B by one degree from 2n-5\documentclass[12pt]{minimal}
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\begin{document}$$2n-5$$\end{document} to 2n-4\documentclass[12pt]{minimal}
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In the range captured by Igusa’s stability result, i.e. up to approximately degree 2n/3, Corollary B was previously known as a result of a classical computation due to Farrell and Hsiang [16] based on Waldhausen’s approach to pseudoisotopy theory (of which the proof of Corollary B is independent).
Homeomorphisms of Euclidean spaces
By an enhancement of a result due to Morlet (see e.g. [5, Thm 4.4]), there are homotopy equivalencesBDiff∂(Dd)≃Ω0dTop(d)/O(d)andBC(Dd)≃Ωdhofib(Top(d)/O(d)→Top(d+1)/O(d+1))\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {BDiff}_\partial (D^d)&\simeq \Omega ^d_0\mathrm {Top}(d)/\mathrm {O}(d)\quad \text{ and }\\\mathrm {BC}(D^d)&\simeq \Omega ^d\mathrm {hofib}\Big (\mathrm {Top}(d)/\mathrm {O}(d)\rightarrow \mathrm {Top}(d+1)/\mathrm {O}(d+1)\Big )\end{aligned}$$\end{document}for d≥5\documentclass[12pt]{minimal}
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\begin{document}$$d\ge 5$$\end{document} that relate the groups of diffeomorphisms and concordances of a d-disc to the homotopy fibre Top(d)/O(d)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Top}(d)/\mathrm {O}(d)$$\end{document} of the map BO(d)→BTop(d)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BO}(d)\rightarrow \mathrm {BTop}(d)$$\end{document} that classifies the inclusion of the orthogonal group O(d)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {O}(d)$$\end{document} into the topological group Top(d)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Top}(d)$$\end{document} of homeomorphisms of Rd\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {R}^d$$\end{document}, and to its stabilisation map Top(d)/O(d)→Top(d+1)/O(d+1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Top}(d)/\mathrm {O}(d)\rightarrow \mathrm {Top}(d+1)/\mathrm {O}(d+1)$$\end{document} induced by taking products with R\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {R}$$\end{document}. Theorem A and Corollary B can thus be reformulated in terms of these equivalent spaces and result in particular in the following corollary, using the fact that π∗Top(d)/O(d)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _*\mathrm {Top}(d)/\mathrm {O}(d)$$\end{document} is finite for ∗<d+2\documentclass[12pt]{minimal}
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\begin{document}$$*<d+2$$\end{document} and d≠4\documentclass[12pt]{minimal}
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\begin{document}$$d\ne 4$$\end{document} [28, Essay V, 5.0].
As per item (ii) of the previous remark, this range can be improved by one degree.
Idea of proof
Instead of sketching the proof of Theorem A, we outline a strategy to achieve a seemingly different task: relating the p-local homology of BC(D2n)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BC}(D^{2n})$$\end{document} to K-theory in a range of degrees. This should, however, convey the main ideas; the actual proof of Theorem A uses a similar strategy to construct a zig–zag between BC(D2n)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BC}(D^{2n})$$\end{document} and Ω0∞+1K(Z)\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^{\infty +1}_0\mathrm {K}(\mathbf {Z})$$\end{document} that consists of maps that are p-local homology isomorphisms in a range and then argues that the maps are actually p-locally highly connected. In the sketch that follows, we allow ourselves to be somewhat vague; full details shall be given in the body of this work.
The root of the proof of Theorem A is to consider the odd-dimensional disc D2n+1\documentclass[12pt]{minimal}
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\begin{document}$$D^{2n+1}$$\end{document} as the 0th member of a whole family of manifolds—the high-dimensional handlebodiesVg:=♮gDn+1×Sn\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}V_g:=\natural ^gD^{n+1}\times S^n\end{aligned}$$\end{document}given as iterated boundary connected sums of Dn+1×Sn\documentclass[12pt]{minimal}
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\begin{document}$$D^{n+1}\times S^n$$\end{document}. Comparing the groups of diffeomorphisms DiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Diff}_{D^{2n}}(V_g)$$\end{document} that pointwise fix a chosen disc D2n⊂∂Vg\documentclass[12pt]{minimal}
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\begin{document}$$D^{2n}\subset \partial V_g$$\end{document} in the boundary to the corresponding block diffeomorphism groups yields homotopy fibre sequencesDiff~D2n(Vg)/DiffD2n(Vg)⟶BDiffD2n(Vg)⟶BDiff~D2n(Vg),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\widetilde{\mathrm {Diff}}_{D^{2n}}(V_g)/\mathrm {Diff}_{D^{2n}}(V_g)\longrightarrow \mathrm {BDiff}_{D^{2n}}(V_g)\longrightarrow \mathrm {B\widetilde{Diff}}_{D^{2n}}(V_g),\end{aligned}$$\end{document}one for each g. Varying g, these fibre sequences are connected by stabilisation maps induced by extending (block) diffeomorphisms along the inclusion Vg⊂Vg+1\documentclass[12pt]{minimal}
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\begin{document}$$V_g\subset V_{g+1}$$\end{document} by the identity, and Morlet’s lemma of disjunction ensures that the map between homotopy fibresDiff~D2n(Vg)/DiffD2n(Vg)⟶Diff~D2n(Vg+1)/DiffD2n(Vg+1)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\widetilde{\mathrm {Diff}}_{D^{2n}}(V_g)/\mathrm {Diff}_{D^{2n}}(V_g)\longrightarrow \widetilde{\mathrm {Diff}}_{D^{2n}}(V_{g+1})/\mathrm {Diff}_{D^{2n}}(V_{g+1})\end{aligned}$$\end{document}is highly connected. It is not hard to see that the space BC(D2n)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BC}(D^{2n})$$\end{document} of interest is equivalent to this fibre for g=0\documentclass[12pt]{minimal}
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\begin{document}$$g=0$$\end{document}, so to access BC(D2n)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BC}(D^{2n})$$\end{document} in a range, we may as well study the homotopy fibre of the sequence obtained from the previous one by taking homotopy colimits,Diff~D2n(V∞)/DiffD2n(V∞)⟶BDiffD2n(V∞)⟶BDiff~D2n(V∞).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \widetilde{\mathrm {Diff}}_{D^{2n}}(V_\infty )/\mathrm {Diff}_{D^{2n}}(V_\infty )\longrightarrow \mathrm {BDiff}_{D^{2n}}(V_\infty )\longrightarrow \mathrm {B\widetilde{Diff}}_{D^{2n}}(V_\infty ).\end{aligned}$$\end{document}By work of Botvinnik and Perlmutter [10], the homology of BDiffD2n(V∞)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BDiff}_{D^{2n}}(V_\infty )$$\end{document} has a surprisingly simple description in homotopy theoretical terms, so to compute the homology of the fibre of (1) and hence that of BC(D2n)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BC}(D^{2n})$$\end{document} in a range, one might try to compute the homology of BDiff~D2n(V∞)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {B\widetilde{Diff}}_{D^{2n}}(V_\infty )$$\end{document} and analyse the Serre spectral sequence of (1). This is essentially what we do, and it involves several steps of which some might be of independent interest:
In Sect. 2, we use surgery theory to express the space of block diffeomorphisms of a general manifold triad satisfying a π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document}-condition p-locally for large primes in terms of its homotopy automorphisms covered by certain bundle data. A similar result in the rational setting which inspired ours but applies to another class of triads was obtained by Berglund and Madsen [8] (see also Remark 2.3).
Sect. 3 serves to compute variants of the mapping class group π0DiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_g)$$\end{document} up to extensions in terms of automorphisms groups of the integral homology of Vg\documentclass[12pt]{minimal}
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\begin{document}$$V_g$$\end{document}.
In Sect. 4, we calculate the p-local homotopy and homology groups of the delooped space of homotopy automorphisms BhAutD2n(Vg,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BhAut}_{D^{2n}}(V_g,W_{g,1})$$\end{document} of Vg\documentclass[12pt]{minimal}
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\begin{document}$$D^{2n}$$\end{document} and restrict to a homotopy automorphism of the complement of the boundary as a module over the group π0hAutD2n(Vg,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {hAut}_{D^{2n}}(V_g,W_{g,1})$$\end{document} in a range of degrees. This uses some pieces of the apparatus of rational homotopy theory, as well as an ad-hoc p-local generalisation we provide along the way.
The action on the nth homology group Hn(Vg;Z)≅Zg\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {H}_n(V_g;\mathbf {Z})\cong \mathbf {Z}^g$$\end{document} induces a map BDiff~D2n(V∞)⟶BGL∞(Z)+≃Ω0∞K(Z),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {B\widetilde{Diff}}_{D^{2n}}(V_\infty )\longrightarrow \mathrm {BGL}_\infty (\mathbf {Z})^+\simeq \Omega ^{\infty }_0 \mathrm {K}(\mathbf {Z}),\end{aligned}$$\end{document} a variant of which we show in Sect. 5 with the help of all previous steps to be a p-local homology isomorphism in a range of degrees.
Outlook
In [24], we will take a different approach and study concordance spaces C(M)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {C}(M)$$\end{document} without restriction on the p-torsion. As a byproduct, the setup of [24] will also make apparent that the zig–zag of Theorem A is compatible with the iterated stabilisation map C(D2n)→C(D2n×[0,1]2)≃C(D2n+2)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {C}(D^{2n})\rightarrow \mathrm {C}(D^{2n}\times [0,1]^2)\simeq \mathrm {C}(D^{2n+2})$$\end{document} and agrees up to equivalence with the zig–zagBC(D2n)→Ω0∞+1WhDiff(∗)=Ω0∞fib(S→K(S))→Ω0∞fib(S→K(Z))←Ω0∞+1K(Z)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\mathrm {BC}(D^{2n})\rightarrow \Omega ^{\infty +1}_0\mathrm {Wh}^{\mathrm {Diff}}(*)=\Omega ^{\infty }_0\mathrm {fib}\big (\mathbf {S}\rightarrow K(\mathbf {S})\big )\\&\quad \rightarrow \Omega _0^{\infty }\mathrm {fib}\big (\mathbf {S}\rightarrow K(\mathbf {Z})\big )\leftarrow \Omega _0^{\infty +1}K(\mathbf {Z}) \end{aligned}$$\end{document}known from Waldhausen’s work [42] (see also Sect. 5.4).
Preliminaries
We start off with a lemma on semi-simplicial actions and a short recollection on nilpotent spaces for later reference, followed by foundational material on various types automorphisms of manifolds with bundle data. Primarily, this serves us to set up a convenient theory of block automorphism spaces with tangential structures.
Semi-simplicial monoids and their actions
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\begin{document}$$X_\bullet /\!/M_\bullet $$\end{document}whose space of p-simplices is defined as the bar-construction B(Xp,Mp,∗)\documentclass[12pt]{minimal}
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\begin{document}$$X_\bullet $$\end{document}. For X∙=∗∙\documentclass[12pt]{minimal}
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\begin{document}$$X_\bullet =*_\bullet $$\end{document} the semi-simplicial point, i.e. ∗p\documentclass[12pt]{minimal}
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\begin{document}$$*_p$$\end{document} a singleton for all p, we abbreviate X∙//M∙\documentclass[12pt]{minimal}
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\begin{document}$$X_\bullet /\!/M_\bullet $$\end{document} by BM∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {B}M_\bullet $$\end{document}. The unique semi-simplicial map X∙→∗∙\documentclass[12pt]{minimal}
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\begin{document}$$X_\bullet \rightarrow *_\bullet $$\end{document} induces a natural map X∙//M∙→BM∙\documentclass[12pt]{minimal}
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\begin{document}$$X_\bullet /\!/M_\bullet \rightarrow \mathrm {B}M_\bullet $$\end{document} which is well-known to geometrically realise to a quasi-fibration with fibre the realisation of X∙\documentclass[12pt]{minimal}
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\begin{document}$$X_\bullet $$\end{document} if M∙\documentclass[12pt]{minimal}
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\begin{document}$$M_\bullet $$\end{document} is a group-like simplicial monoid acting simplicially on a simplicial set X∙\documentclass[12pt]{minimal}
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\begin{document}$$X_\bullet $$\end{document}. To explain a generalisation of this fact for semi-simplicial M∙\documentclass[12pt]{minimal}
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\begin{document}$$|-|$$\end{document} and consider the natural zig–zag|X∙|×|M∙|⟵|X∙×M∙|⟶|X∙|\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} |X_\bullet |\times |M_\bullet |\longleftarrow |X_\bullet \times M_\bullet |\longrightarrow |X_\bullet |\end{aligned}$$\end{document}whose left map is induced by the projections and the right map by the action. If the underlying semi-simplicial sets of M∙\documentclass[12pt]{minimal}
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\begin{document}$$M_\bullet $$\end{document} and X∙\documentclass[12pt]{minimal}
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\begin{document}$$X_\bullet $$\end{document}admit degeneracies, i.e. if they agree (as semi-simplicial sets) with the underlying semi-simplicial sets of simplicial sets, then the left arrow is an equivalence (see e.g. [15, Thm 7.2]), so a contractible choice of a homotopy inverse yields an action map μ:|X∙|×|M∙|→|X∙|\documentclass[12pt]{minimal}
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\begin{document}$$\mu :|X_\bullet |\times |M_\bullet |\rightarrow |X_\bullet |$$\end{document}. In this situation, we say that M∙\documentclass[12pt]{minimal}
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\begin{document}$$M_\bullet $$\end{document}acts onX∙\documentclass[12pt]{minimal}
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\begin{document}$$X_\bullet $$\end{document}by equivalences if μ(-,m):|X∙|→|X∙|\documentclass[12pt]{minimal}
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\begin{document}$$\mu (-,m):|X_\bullet |\rightarrow |X_\bullet |$$\end{document} is an equivalence for all m∈|M∙|\documentclass[12pt]{minimal}
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\begin{document}$$m\in |M_\bullet |$$\end{document}. The following lemma shows that this is sufficient to conclude that the natural map X∙//M∙→BM∙\documentclass[12pt]{minimal}
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\begin{document}$$X_\bullet /\!/M_\bullet \rightarrow \mathrm {B}M_\bullet $$\end{document} realises to a quasi-fibration.
Lemma 1.1
For a semi-simplicial monoid M∙\documentclass[12pt]{minimal}
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\begin{document}$$X_\bullet $$\end{document} is by equivalences, the sequenceX∙⟶X∙//M∙⟶BM∙\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}X_\bullet \longrightarrow X_\bullet //M_\bullet \longrightarrow \mathrm {B}M_\bullet \end{aligned}$$\end{document}induces a quasi-fibration on geometric realisations.
Remark 1.2
In all situations we encounter, the condition that M∙\documentclass[12pt]{minimal}
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\begin{document}$$X_\bullet $$\end{document} admit degeneracies is ensured by them being Kan (every semi-simplicial Kan complex admits degeneracies [23]) and the condition that M∙\documentclass[12pt]{minimal}
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\begin{document}$$X_\bullet $$\end{document} with the action.
The condition that M∙\documentclass[12pt]{minimal}
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\begin{document}$$X_\bullet =G_\bullet ^{\le 0}$$\end{document} which agrees with G0\documentclass[12pt]{minimal}
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\begin{document}$$G_\bullet $$\end{document} via right translations. In this case, the realisation |X∙//G∙|\documentclass[12pt]{minimal}
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\begin{document}$$|G_\bullet ^{\le 0}|\simeq G_0$$\end{document} is rarely equivalent to Ω|BG∙|≃|G∙|\documentclass[12pt]{minimal}
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\begin{document}$$\Omega |\mathrm {B}G_\bullet |\simeq |G_\bullet |$$\end{document}.
Under the assumption of Lemma 1.1, the long exact sequence induced by the quasi-fibration yields a bijection π0|X∙|/π0|M∙|≅π0|X∙//M∙|\documentclass[12pt]{minimal}
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Proof of Lemma 1.1
The sequence in question is the geometric realisation of a sequenceof simplicial semi-simplicial sets, where the simplicial (bar-)direction is indicated by the square and the semi-simplicial direction by the bullet ∙\documentclass[12pt]{minimal}
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\begin{document}$$\bullet $$\end{document}; the fibre X∙\documentclass[12pt]{minimal}
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\begin{document}$$p\ge 0$$\end{document}. For the left square, this follows directly from the definition of the bar-construction together with the above mentioned fact that the realisation of the product of two semi-simplicial sets that admit degeneracies is canonically equivalent to the product of their realisations. Spelling out the definition of the bar-construction, the right hand square is given as
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\begin{document}$$(|\mu _\bullet |, |(\mathrm {pr}_2)_\bullet |):|X_\bullet \times M_\bullet |\rightarrow |X_\bullet |\times |M_\bullet |$$\end{document} is an equivalence. This map fits into a commutative triangle
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\begin{document}$$m\in |M_\bullet |$$\end{document} agrees up to equivalence with the action map μ(-,m):|X∙|→|X∙|\documentclass[12pt]{minimal}
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\begin{document}$$\mu (-,m):|X_\bullet |\rightarrow |X_\bullet |$$\end{document}. The latter is an equivalence by assumption, so the shear map is an equivalence and the claim follows. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Nilpotent spaces
A space is nilpotent if it is path connected and its fundamental group is nilpotent and acts nilpotently on all higher homotopy groups. Such spaces have an unambiguous p-localisation at a prime p, which on homology and homotopy groups (including the fundamental group) has the expected effect of p-localisation in the algebraic sense [31, Thm 6.1.2]. Localisations are defined in terms of a universal property [31, Def. 5.2.3], which ensures that they are unique and functorial up to homotopy. A map between nilpotent spaces is p-locallyk-connected for a prime p and k≥1\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 1$$\end{document} if the induced map on p-localisations is k-connected in the usual sense.
Lemma 1.3
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\begin{document}$$k\ge 1$$\end{document}, the following statements are equivalent:
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\begin{document}$$X\rightarrow Y$$\end{document} is p-locally k-connected.
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\begin{document}$$(\pi _*X)_{(p)}\rightarrow (\pi _*Y)_{(p)}$$\end{document} is an isomorphism for ∗<k\documentclass[12pt]{minimal}
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\begin{document}$$*<k$$\end{document} and surjective for ∗=k\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {H}_*(X;\mathbf {Z}_{(p)})\rightarrow \mathrm {H}_* (Y;\mathbf {Z}_{(p)})$$\end{document} is an isomorphism for ∗<k\documentclass[12pt]{minimal}
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\begin{document}$$*< k$$\end{document} and surjective for ∗=k\documentclass[12pt]{minimal}
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\begin{document}$$*=k$$\end{document}.
Proof
The above mentioned fact that p-localisation of nilpotent spaces commutes with taking homotopy groups shows that the first two items are equivalent. The equivalence between the second two follows for k=1\documentclass[12pt]{minimal}
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\begin{document}$$k=1$$\end{document} from the well-known fact that a morphism G→H\documentclass[12pt]{minimal}
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\begin{document}$$G\rightarrow H$$\end{document} between nilpotent groups is surjective if and only if it is surjective on abelianisations which one sees as follows: writing Γ1(G)=G\documentclass[12pt]{minimal}
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\begin{document}$$\otimes ^n_{\mathbf {Z}}G^{\mathrm {ab}}\rightarrow \Gamma ^{n}(G)/\Gamma ^{n+1}(G)$$\end{document} induced by taking left-normed commutators (see e.g. [44, Thm 3.1]), so one may prove the claim by an induction on the nilpotence degree. For k≥2\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 2$$\end{document}, the equivalence between the second two items is a consequence of the relative Hurewicz theorem for nilpotent spaces (see e.g. [19, Cor. 3.4]) applied to the p-localisation of the map in question. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
At several points in this work, we will make use of the fact that nilpotent spaces behave well with respect to taking homotopy fibres (see e.g. [31, Prop. 4.4.1] for a proof).
Lemma 1.4
Let π:E→B\documentclass[12pt]{minimal}
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Block spaces
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\begin{document}$$M=\partial W$$\end{document}, we write Diff~∂(W,N)\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{\mathrm {Diff}}_\partial (W,N)$$\end{document} instead of Diff~M(W,N)\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{\mathrm {Diff}}_M(W,N)$$\end{document}. Making use of the collaring condition, one shows that Diff~M(W,N)∙\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{\mathrm {Diff}}_M(W,N)_\bullet $$\end{document} satisfies the Kan property (see [18, p. 58-59] for a proof in the case M=N=∅\documentclass[12pt]{minimal}
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\begin{document}$$M=N=\varnothing $$\end{document}; the general case follows in the same way2). The semi-simplicial subgroup of diffeomorphismsDiffM(W,N)∙⊂Diff~M(W,N)∙\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {Diff}_M(W,N)_\bullet \subset \widetilde{\mathrm {Diff}}_M(W,N)_\bullet \end{aligned}$$\end{document}is defined by requiring the diffeomorphisms of Δp×M\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p\times M$$\end{document} to commute with the projection to the simplex Δp\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p$$\end{document} instead of just preserving its faces. This semi-simplicial subgroup agrees with the (collared and smooth) singular set of the topological group DiffM(W,N)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Diff}_M(W,N)$$\end{document} of diffeomorphisms by which we mean the set of 0-simplices DiffM(W,N)0\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Diff}_M(W,N)_0$$\end{document} equipped with the smooth Whitney topology, so there is a canonical weak equivalence |DiffM(W,N)∙|→DiffM(W,N)\documentclass[12pt]{minimal}
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\begin{document}$$|\mathrm {Diff}_M(W,N)_\bullet |\rightarrow \mathrm {Diff}_M(W,N)$$\end{document} and we shall not distinguish between these spaces.
Homotopy automorphisms
The p-simplices of the semi-simplicial monoid of block homotopy automorphismshAut~M(W,N)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {\widetilde{hAut}}_M(W,N)_\bullet $$\end{document} are the block homotopy automorphisms of Δp×W\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p\times W$$\end{document} that fix Δp×M\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p\times N$$\end{document}. Unlike for Diff~M(W,N)∙\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{\mathrm {Diff}}_M(W,N)_\bullet $$\end{document}, it is straight-forward to see that hAut~M(W,N)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {\widetilde{hAut}}_M(W,N)_\bullet $$\end{document} is Kan: a map from the semi-simplicial horn (Λip)∙\documentclass[12pt]{minimal}
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\begin{document}$$(\Lambda _i^p)_\bullet $$\end{document} is represented by a homotopy equivalence ψ:Λip×W→Λjp×W\documentclass[12pt]{minimal}
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\begin{document}$$\psi :\Lambda _i^p\times W\rightarrow \Lambda _j^p\times W$$\end{document} and a lift to a p-simplex Δp×W→Δp×W\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p\times W\rightarrow \Delta ^p\times W$$\end{document} is given by (φi×idW)∘(id[0,1]×ψ)∘(φi-1×idW)\documentclass[12pt]{minimal}
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\begin{document}$$(\varphi _i\times \mathrm {id}_W)\circ (\mathrm {id}_{[0,1]}\times \psi )\circ (\varphi _i^{-1}\times \mathrm {id}_W)$$\end{document}, where φi:[0,1]×Λip→Δp\documentclass[12pt]{minimal}
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\begin{document}$$\varphi _i:[0,1]\times \Lambda _i^p\rightarrow \Delta ^p$$\end{document} is any homeomorphism that extends the inclusion on {0}×Λip⊂Δp\documentclass[12pt]{minimal}
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\begin{document}$$ \{0\}\times \Lambda _i^p\subset \Delta ^p$$\end{document} and restricts to a homeomorphism from [0,1]×∂Λip∪{1}×Λip\documentclass[12pt]{minimal}
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\begin{document}$$[0,1]\times \partial \Lambda _i^p\cup \{1\}\times \Lambda _i^p$$\end{document} onto the ith face Δip⊂Δp\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p_i\subset \Delta ^p$$\end{document}. As for diffeomorphisms, insisting that the homotopy equivalences of Δp×M\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p\times M$$\end{document} be over Δp\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p$$\end{document} defines a sub semi-simplicial monoid of homotopy automorphismshAutM(W,N)∙⊂hAut~M(W,N)∙,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {hAut}_M(W,N)_\bullet \subset \mathrm {\widetilde{hAut}}_M(W,N)_\bullet , \end{aligned}$$\end{document}which agrees with the singular set of the space hAutM(W,N)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hAut}_M(W,N)$$\end{document} obtained by equipping the set of homotopy equivalences hAutM(W,N)0\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hAut}_M(W,N)_0$$\end{document} with the compact open topology, so also |hAutM(W,N)∙|\documentclass[12pt]{minimal}
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\begin{document}$$|\mathrm {hAut}_M(W,N)_\bullet |$$\end{document} and hAutM(W,N)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hAut}_M(W,N)$$\end{document} are canonically equivalent. An aspect which distinguishes the situation for homotopy automorphisms from that for diffeomorphisms is that the inclusion (3) of Kan complexes induces an equivalence on geometric realisation, which one can see from the combinatorial description of their homotopy groups together with the contractibility of hAut∂Δp(Δp)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hAut}_{\partial \Delta ^p}(\Delta ^p)$$\end{document}.
Bundle maps, unstably
A bundle map between two vector bundles ξ→X\documentclass[12pt]{minimal}
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\begin{document}$$\xi \rightarrow X$$\end{document} and ν→Y\documentclass[12pt]{minimal}
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\begin{document}$$\nu \rightarrow Y$$\end{document} over CW complexes X and Y is a commutative square of the form
whose induced maps on vertical fibres are linear isomorphisms. Of course the underlying map of spaces ϕ¯\documentclass[12pt]{minimal}
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\begin{document}$$\bar{\phi }$$\end{document} can be recovered from ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document}, so we often omit it. Given a subcomplex A⊂X\documentclass[12pt]{minimal}
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\begin{document}$$A\subset X$$\end{document} and a bundle map ℓ0:ξ|A→ν\documentclass[12pt]{minimal}
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\begin{document}$$\xi $$\end{document} to A, the semi-simplicial set of block bundle mapsBun~A(ξ,ν;ℓ0)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {\widetilde{Bun}}_A(\xi ,\nu ;\ell _0)_\bullet $$\end{document} has as its p-simplices the bundle maps Δp×ξ→Δp×ν\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {id}_{\Delta ^p}\times \ell _0$$\end{document} on Δp×ξ|A\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p\times \xi |_{A}$$\end{document} and whose underlying map Δp×X→Δp×Y\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p\times X\rightarrow \Delta ^p\times Y$$\end{document} is a block map. As before, the semi-simplicial structure is induced by restriction to subspaces σ×X\documentclass[12pt]{minimal}
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\begin{document}$$\sigma \subset \Delta ^p$$\end{document}. Insisting that the underlying map between base spaces be over Δp\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p$$\end{document} defines the sub semi-simplicial setBunA(ξ,ν;ℓ0)∙⊂Bun~A(ξ,ν;ℓ0)∙\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {Bun}_A(\xi ,\nu ;\ell _0)_\bullet \subset \mathrm {\widetilde{Bun}}_A(\xi ,\nu ;\ell _0)_\bullet \end{aligned}$$\end{document}of bundle maps, which agrees with the singular set of the space BunA(ξ,ν;ℓ0)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Bun}_A(\xi ,\nu ;\ell _0)$$\end{document} obtained by equipping the set BunA(ξ,ν;ℓ0)0\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Bun}_A(\xi ,\nu ;\ell _0)_0$$\end{document} of bundle maps ξ→ν\documentclass[12pt]{minimal}
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\begin{document}$$\ell _0=\mathrm {inc}$$\end{document} then the semi-simplicial sets of (block) bundle maps Bun~A(ξ,ξ;inc)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Bun}_A(\xi ,\xi ;\mathrm {inc})_\bullet $$\end{document} are semi-simplicial monoids under composition, and they act by precomposition on Bun~A(ξ,ν;ℓ0)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {\widetilde{Bun}}_A(\xi ,\nu ;\ell _0)_\bullet $$\end{document} respectively BunA(ξ,ν;ℓ0)∙\documentclass[12pt]{minimal}
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\begin{document}$$\nu $$\end{document} and bundle map ℓ0:ξ|A→ν\documentclass[12pt]{minimal}
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\begin{document}$$\ell _0:\xi |_{A}\rightarrow \nu $$\end{document}. For a subcomplex C⊂X\documentclass[12pt]{minimal}
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\begin{document}$$C\subset X$$\end{document}, we denote byhAut~A(ξ,C)∙⊂Bun~A(ξ,ξ;inc)∙\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {\widetilde{hAut}}_A(\xi ,C)_\bullet \subset \mathrm {\widetilde{Bun}}_A(\xi ,\xi ;\mathrm {inc})_\bullet \end{aligned}$$\end{document}the submonoid of block bundle maps whose underlying selfmap of Δp×X\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p\times C$$\end{document}. The submonoidhAutA(ξ,C)∙⊂BunA(ξ;ξ,inc)∙\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {hAut}_A(\xi ,C)_\bullet \subset \mathrm {Bun}_A(\xi ;\xi ,\mathrm {inc})_\bullet \end{aligned}$$\end{document}is defined analogously.
Tangential bundle maps
Introducing yet another variant of bundle maps, we define the semi-simplicial set of tangential block bundle mapsBun~A(ξ,ν;ℓ0)∙τ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {\widetilde{Bun}}_A(\xi ,\nu ;\ell _0)^\tau _\bullet $$\end{document} as follows: writing τM\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {\widetilde{Bun}}_A(\xi ,\nu ;\ell _0)^\tau _\bullet $$\end{document} are the bundle maps φ:τΔp×ξ→τΔp×ν\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p\times X\rightarrow \Delta ^p\times Y$$\end{document} to be over Δp\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p$$\end{document} defines the sub semi-simplicial setBunA(ξ,ν;ℓ0)∙τ⊂Bun~A(ξ,ν;ℓ0)∙τ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {Bun}_A(\xi ,\nu ;\ell _0)^\tau _\bullet \subset \mathrm {\widetilde{Bun}}_A(\xi ,\nu ;\ell _0)^\tau _\bullet \end{aligned}$$\end{document}of tangential bundle maps. As before, we have a chain of semi-simplicial monoidshAutA(ξ,C)∙τ⊂hAut~A(ξ,C)∙τ⊂Bun~A(ξ,ξ;inc)∙τ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {hAut}_A(\xi ,C)^\tau _\bullet \subset \mathrm {\widetilde{hAut}}_A(\xi ,C)^\tau _\bullet \subset \mathrm {\widetilde{Bun}}_A(\xi ,\xi ;\mathrm {inc})^\tau _\bullet \end{aligned}$$\end{document}which are defined in the same way as for non-tangential (block) bundle maps. Note that there is a canonical mapBunA(ξ,ν;ℓ0)∙⟶BunA(ξ,ν;ℓ0)∙τ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {Bun}_A(\xi ,\nu ;\ell _0)_\bullet \longrightarrow \mathrm {Bun}_A(\xi ,\nu ;\ell _0)_\bullet ^\tau \end{aligned}$$\end{document}given by extending a bundle map Δp×ξ→Δp×ν\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p\times \xi \rightarrow \Delta ^p\times \nu $$\end{document} over Δp\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p$$\end{document} to τΔp×ξ→τΔp×ν\documentclass[12pt]{minimal}
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\begin{document}$$\tau _{\Delta ^p}\times \xi \rightarrow \tau _{\Delta ^p}\times \nu $$\end{document} by the identity. This map is not an equivalence, but we shall see in the next paragraph that it becomes one after stabilisation.
Remark 1.5
Note that the map (6) does not extend to a map Bun~A(ξ,ν;ℓ0)∙→Bun~A(ξ,ν;ℓ0)∙τ\documentclass[12pt]{minimal}
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\begin{document}$$ \mathrm {\widetilde{Bun}}_A(\xi ,\nu ;\ell _0)_\bullet \rightarrow \mathrm {\widetilde{Bun}}_A(\xi ,\nu ;\ell _0)_\bullet ^\tau $$\end{document} in an obvious way. This is because, for a bundle map ϕ:Δp×ξ→Δp×ν\documentclass[12pt]{minimal}
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\begin{document}$$\phi :\Delta ^p\times \xi \rightarrow \Delta ^p\times \nu $$\end{document} covering a block map, the map τΔp×ξ→τΔp×ν\documentclass[12pt]{minimal}
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\begin{document}$$(x,\phi (\overline{x},y))$$\end{document} is not a bundle map unless ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document} commutes with the projection on Δp\documentclass[12pt]{minimal}
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\begin{document}$$x\in \tau _{\Delta ^p}$$\end{document}.
Bundle maps, stably
A stable vector bundle is a sequence of vector bundles ψ={ψk→Bk}k≥l\documentclass[12pt]{minimal}
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\begin{document}$$\xi \rightarrow X$$\end{document}, a stable vector bundle ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document}, and a bundle map ℓ0:ξ|A⊕εk→ψd+k\documentclass[12pt]{minimal}
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\begin{document}$$\ell _0:\xi |_{A}\oplus \varepsilon ^k\rightarrow \psi _{d+k}$$\end{document} for some k, the semi-simplicial set of stable bundle maps is the colimitBunA(ξs,ψ;ℓ0)∙:=colimm≥kBunA(ξ⊕εm,ψd+m;ℓ0)∙\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {Bun}_A(\xi ^s,\psi ;\ell _0)_\bullet :=\mathrm {colim}_{m\ge k}\mathrm {Bun}_A(\xi \oplus \varepsilon ^m,\psi _{d+m};\ell _0)_\bullet \end{aligned}$$\end{document}over the stabilisation mapsBunA(ξ⊕εm,ψd+m;ℓ0)∙⟶BunA(ξ⊕εm+1,ψd+m+1;ℓ0)∙\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {Bun}_A(\xi \oplus \varepsilon ^m,\psi _{d+m};\ell _0)_\bullet \longrightarrow \mathrm {Bun}_A(\xi \oplus \varepsilon ^{m+1},\psi _{d+m+1};\ell _0)_\bullet \end{aligned}$$\end{document}given by adding a trivial line bundle followed by postcomposition with the structure map ψd+m⊕ε→ψd+m+1\documentclass[12pt]{minimal}
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\begin{document}$$\psi _{d+m}\oplus \varepsilon \rightarrow \psi _{d+m+1}$$\end{document}. Analogously, we define stable tangential bundle maps as the colimitBunA(ξs,ψ;ℓ0)∙τ:=colimm≥kBunA(ξ⊕εm,ψd+m;ℓ0)∙τ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {Bun}_A(\xi ^s,\psi ;\ell _0)^\tau _\bullet :=\mathrm {colim}_{m\ge k}\mathrm {Bun}_A(\xi \oplus \varepsilon ^m,\psi _{d+m};\ell _0)^\tau _\bullet .\end{aligned}$$\end{document}As in Section 1.6, there are semi-simplicial sub-monoidshAutA(ξs;C)∙⊂BunA(ξs,ξs;inc)∙andhAutA(ξs;C)∙τ⊂BunA(ξs,ξs;inc)∙τ,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\mathrm {hAut}_A(\xi ^s;C)_\bullet \subset \mathrm {Bun}_A(\xi ^s,\xi ^s;\mathrm {inc})_\bullet \quad \text {and}\\&\mathrm {hAut}_A(\xi ^s;C)_\bullet ^\tau \subset \mathrm {Bun}_A(\xi ^s,\xi ^s;\mathrm {inc})^\tau _\bullet ,\end{aligned}$$\end{document}and also block variants of these semi-simplicial sets, defined by adding appropriate tildes. As the extension map (6) is compatible with the stabilisation maps, it gives rise to mapswhich we show in Lemma A.4 to be equivalences if X is a finite CW complex.
Tangential structures
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\begin{document}$$\mathrm {Bun}_A(\xi ,\theta ^*\gamma _d;\ell _0)$$\end{document} from ξ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document}, relative to a fixed bundle map ℓ0:ξ|A→θd∗γd\documentclass[12pt]{minimal}
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\begin{document}$$\ell _0:\xi |_A\rightarrow \theta _d^*\gamma _d$$\end{document}. We denote the homotopy quotient (in the sense of Sect. 1.1) of the action of hAutA(ξ,C)∙⊂BunA(ξ;ξ,inc)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hAut}_A(\xi ,C)_\bullet \subset \mathrm {Bun}_A(\xi ;\xi ,\mathrm {inc})_\bullet $$\end{document} on the semi-simplicial set of θ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document}-structures byBhAutAθ(ξ,C;ℓ0)∙:=BunA(ξ,θ∗γd;ℓ0)∙//hAutA(ξ,C)∙.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {BhAut}^{\theta }_A(\xi ,C;\ell _0)_\bullet :=\mathrm {Bun}_A(\xi ,\theta ^*\gamma _d;\ell _0)_\bullet //\mathrm {hAut}_A(\xi ,C)_\bullet .\end{aligned}$$\end{document}
Remark 1.6
Note that BhAutAθ(ξ,C;ℓ0)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BhAut}^{\theta }_A(\xi ,C;\ell _0)_\bullet $$\end{document} is in many cases empty or disconnected, so despite the suggestive notation, it is in general not the classifying space of any kind of group or monoid, (semi-)simplicial or topological.
Stable tangential structures
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\begin{document}$$\mathrm {BO}(d)\rightarrow \mathrm {BO}$$\end{document}. Note, however, that not all d-dimensional tangential structures arise this way, for instance the tangential structure EO(d)→BO(d)\documentclass[12pt]{minimal}
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\begin{document}$$\Xi ^*\gamma =\{\Xi _{d}^*\gamma _d\}_{d\ge 0}$$\end{document} whose structure maps are induced by the canonical bundle map γd⊕ε→γd+1\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BO}(d)\rightarrow \mathrm {BO}(d+1)$$\end{document}. Given a stable bundle map ℓ0:ξs|A→Ξ∗γ\documentclass[12pt]{minimal}
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\begin{document}$$\ell _0:\xi ^s|_{A}\rightarrow \Xi ^*\gamma $$\end{document}, we call BunA(ξs,Ξ∗γ;ℓ0)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Bun}_A(\xi ^s,\Xi ^*\gamma ;\ell _0)$$\end{document} the semi-simplicial set of stableθ\documentclass[12pt]{minimal}
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\begin{document}$$\xi $$\end{document}. As in the unstable case, we abbreviateBhAutAΞ(ξs,C;ℓ0)∙:=BunA(ξs,Ξ∗γ;ℓ0)∙//hAutA(ξs,C)∙.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {BhAut}^{\Xi }_A(\xi ^s,C;\ell _0)_\bullet :=\mathrm {Bun}_A(\xi ^s,\Xi ^*\gamma ;\ell _0)_\bullet //\mathrm {hAut}_A(\xi ^s,C)_\bullet .\end{aligned}$$\end{document}Tangential and or block variants of the previous definitions are defined by using the respective variants of bundle maps and adding appropriate tildes and or τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}-superscripts.
Forgetting tangential structures
It follows from Lemma 1.1 that the sequence of semi-simplicial spacesBunA(ξ,θ∗γd;ℓ0)∙⟶BhAutAθ(ξ,C;ℓ0)∙⟶BhAutA(ξ,C)∙\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {Bun}_A(\xi ,\theta ^*\gamma _d;\ell _0)_\bullet \longrightarrow \mathrm {BhAut}^{\theta }_A(\xi ,C;\ell _0)_\bullet \longrightarrow \mathrm {BhAut}_A(\xi ,C)_\bullet \end{aligned}$$\end{document}realises to a quasi-fibration, because BunA(ξ,θ∗γd;ℓ0)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Bun}_A(\xi ,\theta ^*\gamma _d;\ell _0)_\bullet $$\end{document} and hAutA(ξ,C)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hAut}_A(\xi ,C)_\bullet $$\end{document} admit degeneracies since they satisfy the Kan property (they agree with the singular complex of a space, see Sect. 1.6) and the action is by equivalences as hAutA(ξ,C)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hAut}_A(\xi ,C)_\bullet $$\end{document} is group-like. The same argument applies to the stable analogue of this sequence involving (8) and, using Corollary A.2 (assuming that X is a finite complex), also to its variants involving tangential and or block bundle maps. By definition of the universal bundle, the semi-simplicial set BunA(ξ,θ∗γd;ℓ0)∙\documentclass[12pt]{minimal}
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\begin{document}$$\theta =\mathrm {id}$$\end{document}, so the second map in (9) is an equivalence for this particular choice of θ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document}. The analogous statement holds in the stable case as well and, as a result of Lemmas A.3 and A.4, also for the tangential and or block variants.
The derivative maps
Taking fibrewise derivatives of diffeomorphisms φ:Δp×W→Δp×W\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p$$\end{document} induces a canonical semi-simplicial mapDiffM(W,N)∙⟶hAutM(τW,N)∙,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {Diff}_M(W,N)_\bullet \longrightarrow \mathrm {hAut}_M(\tau _W,N)_\bullet ,\end{aligned}$$\end{document}which we call the derivative map. Furthermore, the notion of a tangential bundle map is designed exactly so that there is a block derivative mapDiff~M(W,N)∙⟶hAut~M(τWs,N)∙τ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \widetilde{\mathrm {Diff}}_M(W,N)_\bullet \longrightarrow \mathrm {\widetilde{hAut}}_M(\tau ^s_W,N)^\tau _\bullet .\end{aligned}$$\end{document}given by assigning a block diffeomorphism Δp×W→Δp×W\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p\times W\rightarrow \Delta ^p\times W$$\end{document} its derivative τΔp×τW→τΔp×τW,\documentclass[12pt]{minimal}
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\begin{document}$$\tau _{\Delta ^p}\times \tau _W\rightarrow \tau _{\Delta ^p}\times \tau _W,$$\end{document} which indeed makes the square (4) commute as φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document} is assumed to be collared in the sense of Section 1.3.
Remark 1.7
A different model of the block derivative map (11) was considered by Berglund and Madsen in their prominent study of the rational homotopy type of spaces of block diffeomorphisms of manifolds with certain boundary conditions [8] (see Section 4.3 loc. cit. and Remark 2.3 below).
Lemmas A.3 and A.4 provide an equivalence hAut~M(τWs,N)∙τ≃hAutM(τWs,N)∙,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {\widetilde{hAut}}_M(\tau ^s_W,N)^\tau _\bullet \simeq \mathrm {hAut}_M(\tau ^s_W,N)_\bullet ,\end{aligned}$$\end{document} so the delooped space BhAut~M(τWs,N)∙τ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {B\widetilde{hAut}}_M(\tau ^s_W,N)^\tau _\bullet $$\end{document} classifies fibrations πW:E→B\documentclass[12pt]{minimal}
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\begin{document}$$\pi _W: E\rightarrow B$$\end{document} with fibre W together with the following data:
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\begin{document}$$\pi _M\rightarrow \pi _W\leftarrow \pi _N$$\end{document} over the identity whose induced maps on fibres is equivalent to the system of inclusions M⊂W⊃N\documentclass[12pt]{minimal}
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a stable vector bundle E→BO\documentclass[12pt]{minimal}
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\begin{document}$$E\rightarrow \mathrm {BO}$$\end{document} over the total space of πW\documentclass[12pt]{minimal}
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\begin{document}$$\pi _W$$\end{document} whose restriction to each fibre agrees with the stable tangent bundle of W.
From this point of view, the block derivative (11) comes as no surprise: it is reminiscent of the fact that a block bundle has an underlying fibration and a stable vertical tangent bundle by [14]. However, somewhat curiously, the block derivative map (11) obviously factors over the variant hAut~M(τW,N)∙τ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {\widetilde{hAut}}_M(\tau _W,N)^\tau _\bullet $$\end{document} involving the unstable tangent bundle of W, giving rise to an unstable block derivative mapBDiff~M(W,N)∙⟶BhAut~M(τW,N)∙τ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {B\widetilde{Diff}}_M(W,N)_\bullet \longrightarrow \mathrm {B\widetilde{hAut}}_M(\tau _W,N)^\tau _\bullet .\end{aligned}$$\end{document} The target of this map is neither equivalent to BhAutM(τWs,N)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BhAut}_M(\tau ^s_W,N)_\bullet $$\end{document} nor BhAutM(τW,N)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BhAut}_M(\tau _W,N)_\bullet $$\end{document}, and it would be interesting to have a good description of what it classifies.
We denote the submonoids of the components hit by the derivative maps byhAutM≅(τW,N)∙⊂hAutM(τW,N)∙andhAut~M≅(τWs,N)∙τ⊂hAut~M(τWs,N)∙τ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \mathrm {hAut}^{\cong }_{M}(\tau _W,N)_\bullet&\subset \mathrm {hAut}_M(\tau _W,N)_\bullet \quad \text {and}\\ \mathrm {\widetilde{hAut}}^{\cong }_M(\tau ^s_W,N)^\tau _\bullet&\subset \mathrm {\widetilde{hAut}}_M(\tau ^s_W,N)^\tau _\bullet \end{aligned}\end{aligned}$$\end{document}and add the same ≅\documentclass[12pt]{minimal}
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\begin{document}$$\cong $$\end{document}-superscript to (8) and its tangential block variant to indicate when we take homotopy quotients by the submonoids (12) instead of the full monoids. DefiningBDiff~MΞ(W,N;ℓ0)∙:=Bun~M(τWs,Ξ∗γ;ℓ0)∙τ//Diff~M(W,N)∙\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {B\widetilde{Diff}}_M^{\Xi }(W,N;\ell _0)_\bullet :=\mathrm {\widetilde{Bun}}_M(\tau ^s_W,{\Xi }^*\gamma ;\ell _0)^\tau _\bullet //\widetilde{\mathrm {Diff}}_M(W,N)_\bullet \end{aligned}$$\end{document}for a stable tangential structure Ξ\documentclass[12pt]{minimal}
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\begin{document}$$\ell _0$$\end{document} on τWs|M\documentclass[12pt]{minimal}
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\begin{document}$$\tau ^s_W|_M$$\end{document} andBDiffMθ(W,N;ℓ0)∙:=BunM(τW,θ∗γd;ℓ0)∙//DiffM(W,N)∙\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {BDiff}_M^{\theta }(W,N;\ell _0)_\bullet :=\mathrm {Bun}_M(\tau _W,\theta ^*\gamma _d;\ell _0)_\bullet //\mathrm {Diff}_M(W,N)_\bullet \end{aligned}$$\end{document}in the unstable case, the two derivative maps fit into a commutative squarewhose vertical maps are induced by the compositionBunM(τW,Ξd∗γd;ℓ0)∙→BunM(τWs,Ξ∗γ;ℓ0)∙→BunM(τWs,Ξ∗γ;ℓ0)∙τ⊂Bun~M(τWs,Ξ∗γ;ℓ0)∙τ.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {Bun}_M(\tau _W,\Xi _d^*\gamma _d;\ell _0)_\bullet\rightarrow & {} \mathrm {Bun}_M(\tau ^s_W,\Xi ^*\gamma ;\ell _0)_\bullet \\\rightarrow & {} \mathrm {Bun}_M(\tau ^s_W,\Xi ^*\gamma ;\ell _0)^\tau _\bullet \subset \mathrm {\widetilde{Bun}}_M(\tau ^s_W,{\Xi }^*\gamma ;\ell _0)^\tau _\bullet . \end{aligned}$$\end{document}All maps in this composition are equivalences, the first one by an exercise in obstruction theory and the second map as well as the final inclusion as a result of Lemmas A.3 and A.4. The composition of equivalences just discussed also induces the vertical maps in the commutative diagramwhose horizontal maps are induced by forgetting tangential structures.
Lemma 1.8
The geometric realisation of (15) is homotopy cartesian.
Proof
By construction the induced map on horizontal strict fibres agrees with the composition of equivalences discussed above (15), so it suffices to show that the horizontal maps of the square realise to quasi-fibrations. As DiffM(W,N)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Diff}_M(W,N)_\bullet $$\end{document} and Diff~M(W,N)∙\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{\mathrm {Diff}}_M(W,N)_\bullet $$\end{document} are Kan and group-like (see Section 1.4), this follows from Lemma 1.1 and Corollary A.2. □\documentclass[12pt]{minimal}
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Surgery theory and spaces of block diffeomorphisms
We use surgery theory to give a partial p-local description of the space BDiff~∂0WΞ(W,∂1W)\documentclass[12pt]{minimal}
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Notation
As a point of notation, we refer to the geometric realisation of any of the semi-simplicial sets or spaces of the previous sections by omitting their ∙\documentclass[12pt]{minimal}
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\begin{document}$$\bullet $$\end{document}-subscripts.
A reminder of surgery theory
A d-dimensional manifold triad is a triple W=(W;∂0W,∂1W)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {S}(W)$$\end{document} of a triad W (see e.g. [43, Ch. 10]) is the collection of equivalence classes of simple homotopy equivalences of triads N→W\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {S}(W)$$\end{document} is the surgery exact sequencewhich relates the structure sets S(W×Dk)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {S}(W\times D^k)$$\end{document} of the triadsW×Dk=(W×Dk;∂0W×Dk∪W×∂Dk;∂1W×Dk)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}W\times D^k=(W\times D^k;\partial _0W\times D^k\cup W\times \partial D^k;\partial _1W\times D^k)\end{aligned}$$\end{document}to the sets of normal invariantsN(W×Dk)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {N}(W\times D^k)$$\end{document} and the L-groupsL(W×Dk)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}(W\times D^k)$$\end{document}. Assuming d≥6\documentclass[12pt]{minimal}
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\begin{document}$$d\ge 6$$\end{document}, this sequence is an exact sequence of abelian groups until L(W×D1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}(W\times D^1)$$\end{document} where it continues as an exact sequence of based sets (see e.g. [43, Ch. 10]). The similarity with the long exact sequence of homotopy groups induced by a fibration is no coincidence: Quinn’s surgery fibration [34, 35] is a homotopy fibration of based spacesS~(W)⟶N(W)⟶L(W)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \tilde{\mathbf {S}}(W)\longrightarrow \mathbf {N}(W)\longrightarrow \mathbf {L}(W)\end{aligned}$$\end{document}that induces (16) on homotopy groups (see also [43, Ch. 17A], or [32] for a detailed account in the topological category). We refrain from describing (16) or (17) in detail; all we shall need to know are a few basic properties, which we explain in the following.
The block structure space
Assuming d≥6\documentclass[12pt]{minimal}
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\begin{document}$$d\ge 6$$\end{document}, an application of the s-cobordism theorem results in a preferred homotopy equivalencehAut~∂0W≅(W,∂1W)/Diff~∂0W(W)≃S~(W)id\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {\widetilde{hAut}}^{\cong }_{\partial _0W}(W,\partial _1W)/\widetilde{\mathrm {Diff}}_{\partial _0W}(W)\simeq \tilde{\mathbf {S}}(W)_{\mathrm {id}}\end{aligned}$$\end{document}between the homotopy fibre of the map BDiff~∂0W(W)→BhAut~∂0W≅(W,∂1W)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {B\widetilde{Diff}}_{\partial _0W}(W)\rightarrow \mathrm {B\widetilde{hAut}}^{\cong }_{\partial _0W}(W,\partial _1W)$$\end{document} induced by inclusion and the basepoint component S~(W)id⊂S~(W)\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{\mathbf {S}}(W)_{\mathrm {id}}\subset \tilde{\mathbf {S}}(W)$$\end{document} of the block structure space (cf. [43, Ch. 17A] or [7, p. 33-34]). HerehAut~∂0W≅(W,∂1W)⊂hAut~∂0W(W,∂1W)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {\widetilde{hAut}}^{\cong }_{\partial _0W}(W,\partial _1W)\subset \mathrm {\widetilde{hAut}}_{\partial _0W}(W,\partial _1W)\end{aligned}$$\end{document}are the components in the image of the map Diff~∂0W(W)→hAut~∂0W(W,∂1W)\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{\mathrm {Diff}}_{\partial _0W}(W)\rightarrow \!\!\mathrm {\widetilde{hAut}}_{\partial _0W}(W,\partial _1W)$$\end{document}. Note that a diffeomorphism of W that fixes ∂0W\documentclass[12pt]{minimal}
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\begin{document}$$\partial _0W$$\end{document} pointwise automatically preserves ∂1W\documentclass[12pt]{minimal}
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\begin{document}$$\partial _1W$$\end{document} setwise, since ∂1W\documentclass[12pt]{minimal}
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\begin{document}$$\partial _1W$$\end{document} is the complement of the interior of ∂0W⊂∂W\documentclass[12pt]{minimal}
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\begin{document}$$\partial _0W\subset \partial W$$\end{document}. On homotopy groups, the equivalence (18) can be described as follows: using the combinatorial description of the relative homotopy groups of a semi-simplicial Kan pair, a class inπk(hAut~∂0W≅(W,∂1W)/Diff~∂0W(W);id)≅πk(hAut~∂0W≅(W,∂1W),Diff~∂0W(W);id)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\pi _k(\mathrm {\widetilde{hAut}}^{\cong }_{\partial _0W}(W,\partial _1W)/\widetilde{\mathrm {Diff}}_{\partial _0W}(W);\mathrm {id})\\&\cong \pi _k(\mathrm {\widetilde{hAut}}^{\cong }_{\partial _0W}(W,\partial _1W),\widetilde{\mathrm {Diff}}_{\partial _0W}(W);\mathrm {id})\end{aligned}$$\end{document}is represented by a simple homotopy equivalence of triads W×Dk→W×Dk\documentclass[12pt]{minimal}
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\begin{document}$$W\times D^k\rightarrow W\times D^k$$\end{document} which is the identity on ∂0W×Dk\documentclass[12pt]{minimal}
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\begin{document}$$\partial _0W\times D^k$$\end{document} and restricts to a diffeomorphism on W×∂Dk\documentclass[12pt]{minimal}
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\begin{document}$$W\times \partial D^k$$\end{document}, so it defines a class in the structure set S(W×Dk)≅πk(S~(W);id)\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {S}(W\times D^k)\cong \pi _k(\tilde{\mathbf {S}}(W); \mathrm {id})$$\end{document}.
The space of normal invariants
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\begin{document}$$\mathbf {N}(W)$$\end{document} admits a preferred homotopy equivalence to the pointed mapping space Maps∗(W/∂0W,G/O)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Maps}_*(W/\partial _0W,\mathrm {G}/\mathrm {O})$$\end{document} based at the constant map, where G/O\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {G}/\mathrm {O}$$\end{document} is the homotopy fibre of the canonical map BO→BG\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BO}\rightarrow \mathrm {BG}$$\end{document} witnessing the fact that a stable vector bundle has an underlying stable spherical fibration (see e.g. [34] or [43, Ch. 10, 17A]). This map is one of infinite loop spaces, so its homotopy fibre G/O\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {G}/\mathrm {O}$$\end{document} is an infinite loop space and hence so is the mapping space Maps∗(W/∂0W,G/O)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Maps}_*(W/\partial _0W,\mathrm {G}/\mathrm {O})$$\end{document}. On homotopy groups, the compositionS~(W)→N(W)≃Maps∗(W/∂0W,G/O)→Maps∗(W/∂0W,BO)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\tilde{\mathbf {S}}(W)\rightarrow \mathbf {N}(W)\simeq \mathrm {Maps}_*(W/\partial _0W,\mathrm {G}/\mathrm {O})\rightarrow \mathrm {Maps}_*(W/\partial _0W,\mathrm {BO})\end{aligned}$$\end{document}has the following geometric description (see e.g. [43, p. 113-114]): given a class in the structure set πk(S~(W);∗)≅S(W×Dk)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _k(\tilde{\mathbf {S}}(W);*)\cong \mathcal {S}(W\times D^k)$$\end{document} represented by a simple homotopy equivalence φ:N→W×Dk\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{\varphi }:W\times D^k\rightarrow N$$\end{document} of triads that agrees with (φ|∂0N)-1\documentclass[12pt]{minimal}
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\begin{document}$$(\varphi |_{\partial _0N})^{-1}$$\end{document} on ∂0(W×Dk)\documentclass[12pt]{minimal}
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\begin{document}$$W\times D^k$$\end{document} comes with a trivialisation on the subspace∂0(W×Dk)=∂0W×Dk∪W×∂Dk\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\partial _0(W\times D^k)=\partial _0W\times D^k\cup W\times \partial D^{k}\end{aligned}$$\end{document}by making use of the diffeomorphism φ~|∂0(W×Dk)\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{\varphi }|_{\partial _0(W\times D^k)}$$\end{document}, and hence gives a class∈πk(Maps∗(W/∂0W,BO);∗).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\in \pi _k(\mathrm {Maps}_*(W/\partial _0W,\mathrm {BO});*).\end{aligned}$$\end{document}
The L-theory space
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\begin{document}$$\mathbf {L}(W)$$\end{document} is an infinite loop space as well (see e.g. [32, Prop. 2.2.2]), and its homotopy groups are canonically isomorphic to Wall’s quadratic L-groups (see e.g. [32, Prop. 2.2.4]). We shall not need to know much about these groups, except that πk(L(W);∗)≅L(W×Dk)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _k(\mathbf {L}(W);*)\cong \mathcal {L}(W\times D^k)$$\end{document} vanishes if W satisfies the π\documentclass[12pt]{minimal}
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\begin{document}$$\partial _1W\subset W$$\end{document} induces an equivalence on fundamental groupoids. This is a consequence of the exact sequence of L-groups of a triad (or more generally, n-ad) described for instance in [43, Thm 3.1]. In other words, under these assumptions, the L-theory space L(W)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {L}(W)$$\end{document} is weakly contractible, so (17) induces a preferred equivalence S~(W)≃N(W)\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{\mathbf {S}}(W)\simeq \mathbf {N}(W)$$\end{document}—an instance of the so-called π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document}-theorem.
The block derivative map of a triad
With these basics of space-level surgery theory in mind, we now turn towards studying connectivity properties of the block derivative map resulting from (11) and (12)BDiff~∂0W(W)⟶BhAut~∂0W≅(τWs,∂1W)τ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {B\widetilde{Diff}}_{\partial _0W}(W)\longrightarrow \mathrm {B\widetilde{hAut}}^{\cong }_{\partial _0W}(\tau _W^s,\partial _1W)^\tau \end{aligned}$$\end{document}and its enhancement involving tangential structures (i.e. the realisation of the bottom row of (14)), beginning with a technical but useful lemma on the homotopy fibre featuring in (18). We refer Sect. 1.2 for a recollection on nilpotent spaces.
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\begin{document}$$\mathrm {B\widetilde{Diff}}_{\partial _0W}(W)\rightarrow \mathrm {BhAut}^{\cong }_{\partial _0W}(W,\partial _1W)$$\end{document} is surjective on fundamental groups, so its homotopy fibre is connected. To see that it is nilpotent, we use the equivalence (18) to the identity component S~(W)id\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{\mathbf {S}}(W)_{\mathrm {id}}$$\end{document} of the block structure space, which is itself equivalent to a component of the homotopy fibre of the surgery obstruction map N(W)→L(W)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {N}(W)\rightarrow \mathbf {L}(W)$$\end{document} of (17). Given this description of the space in consideration, the claim follows from an application of Lemma 1.4, using the fact that, being an infinite loop space, the space of normal invariants N(W)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {N}(W)$$\end{document} is nilpotent (see Sect. 2.1.2). □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
To state the main result of this section, some notation is in order. For a stable tangential structure Ξ:B→BO\documentclass[12pt]{minimal}
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\begin{document}$$\ell :\tau ^s_W\rightarrow \Xi ^*\gamma $$\end{document} (see Sect. 1.8.1), we write ℓ0\documentclass[12pt]{minimal}
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\begin{document}$$\tau ^s_{W}|_{\partial _0W}$$\end{document} and denote byBDiff~∂0WΞ(W;ℓ0)ℓandBhAut~∂0WΞ,≅(τWs,∂1W;ℓ0)ℓτ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {B\widetilde{Diff}}^\Xi _{\partial _0W}(W;\ell _0)_\ell \quad \text {and}\quad \mathrm {B\widetilde{hAut}}^{\Xi ,\cong }_{\partial _0W}(\tau _W^s,\partial _1W;\ell _0)^\tau _\ell \end{aligned}$$\end{document}the components of the spaces (see Sects.1.8.1 and 1.9 for the notation)BDiff~∂0WΞ(W;ℓ0)=Bun~∂0W(τWs,Ξ∗γ;ℓ0)τ//Diff~∂0W(W)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {B\widetilde{Diff}}^\Xi _{\partial _0W}(W;\ell _0)=\mathrm {\widetilde{Bun}}_{\partial _0W}(\tau ^s_W,{\Xi }^*\gamma ;\ell _0)^\tau //\widetilde{\mathrm {Diff}}_{\partial _0W}(W) \end{aligned}$$\end{document}andBhAut~∂0WΞ,≅(τWs,∂1W;ℓ0)τ=Bun~∂0W(τWs,Ξ∗γ;ℓ0)τ//hAut~∂0W≅(τWs,∂1W)τ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {B\widetilde{hAut}}^{\Xi ,\cong }_{\partial _0W}(\tau _W^s,\partial _1W;\ell _0)^\tau =\mathrm {\widetilde{Bun}}_{\partial _0W}(\tau ^s_W,{\Xi }^*\gamma ;\ell _0)^\tau //\mathrm {\widetilde{hAut}}^{\cong }_{\partial _0W}(\tau _W^s,\partial _1W)^\tau \end{aligned}$$\end{document}that correspond to ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document} under the bijectionπ0BhAut~∂0WΞ,≅(τWs,∂1W;ℓ0)τ≅π0BDiff~∂0WΞ(W;ℓ0)≅π0Bun~∂0W(τWs,Ξ∗γ;ℓ0)τ/π0Diff~∂0W(W)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _0\mathrm {B\widetilde{hAut}}^{\Xi ,\cong }_{\partial _0W}(\tau _W^s,\partial _1W;\ell _0)^\tau\cong & {} \pi _0\mathrm {B\widetilde{Diff}}^\Xi _{\partial _0W}(W;\ell _0)\\\cong & {} \pi _0\mathrm {\widetilde{Bun}}_{\partial _0W}(\tau ^s_W,{\Xi }^*\gamma ;\ell _0)^\tau /\pi _0\widetilde{\mathrm {Diff}}_{\partial _0W}(W)\end{aligned}$$\end{document}resulting from Remark 1.21.2 and the discussion in Sects.1.8.1 and 1.9, using that the map induced by the block derivativeπ0Diff~∂0W(W)⟶π0hAut~∂0W≅(τWs,∂1W)τ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\pi _0\widetilde{\mathrm {Diff}}_{\partial _0W}(W)\longrightarrow \pi _0\mathrm {\widetilde{hAut}}^{\cong }_{\partial _0W}(\tau _W^s,\partial _1W)^\tau \end{aligned}$$\end{document}is surjective by the definition of the target in (12). Recall from Sects.2.1.3 that a manifold triad W=(W;∂0W,∂1W)\documentclass[12pt]{minimal}
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\begin{document}$$W=(W;\partial _0W,\partial _1W)$$\end{document} satisfies the π\documentclass[12pt]{minimal}
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\begin{document}$$\partial _1W\subset W$$\end{document} induces an equivalence on fundamental groupoids.
Theorem 2.2
Let d≥6\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document} on τWs\documentclass[12pt]{minimal}
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\begin{document}$$\tau _W^s$$\end{document}, the homotopy fibre of the mapBDiff~∂0WΞ(W;ℓ0)ℓ⟶BhAut~∂0WΞ,≅(τWs,∂1W;ℓ0)ℓτ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {B\widetilde{Diff}}^\Xi _{\partial _0W}(W;\ell _0)_\ell \longrightarrow \mathrm {B\widetilde{hAut}}^{\Xi ,\cong }_{\partial _0W}(\tau _W^s,\partial _1W;\ell _0)^\tau _\ell \end{aligned}$$\end{document}is nilpotent and has finite homotopy groups. Moreover, this fibre is p-locally (2p-4-k)\documentclass[12pt]{minimal}
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\begin{document}$$(2p-4-k)$$\end{document}-connected for primes p, where k is the relative handle dimension of the inclusion ∂0W⊂W\documentclass[12pt]{minimal}
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\begin{document}$$\partial _0W\subset W$$\end{document}.
Remark 2.3
Theorem 2.2 is inspired by a similar result of Berglund and Madsen [8, Thm 1.1], which applies to a different class of triads, namely those satisfying ∂0W=∂W≅Sd-1\documentclass[12pt]{minimal}
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\begin{document}$$\partial _0W=\partial W\cong S^{d-1}$$\end{document}. Another point in which their result differs from ours is that it is purely rational, and does in fact not seem to admit a p-local refinement analogous to Theorem 2.2. This is because p-torsion occurs for primes p that can be rather large with respect to the degree, originating from contributions of numerators of divided Bernoulli numbers to the homotopy groups of the homotopy fibre Top/O\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Top}/\mathrm {O}$$\end{document} of the canonical map BO→BTop\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BO}\rightarrow \mathrm {BTop}$$\end{document}.
Proof of Theorem 2.2
Using Corollary A.2, an application of Lemma 1.1 to the horizontal arrows of the canonical squareidentifies the horizontal homotopy fibres with the union of components of the space Bun~∂0W(τWs,Ξ∗γ;ℓ0)τ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {\widetilde{Bun}}_{\partial _0W}(\tau ^s_W,{\Xi }^*\gamma ;\ell _0)^\tau $$\end{document} given by the π0Diff~∂0W(W)\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}, so (20) is homotopy cartesian, which reduces the proof to the case Ξ=id\documentclass[12pt]{minimal}
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\begin{document}$$\Xi =\mathrm {id}$$\end{document} in which both rows of the square are equivalences (see Sect. 1.8.1). To settle the case Ξ=id\documentclass[12pt]{minimal}
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\begin{document}$$\Xi =\mathrm {id}$$\end{document}, we consult the map of fibre sequences induced by the block derivative mapin order to see that the homotopy fibre in question agrees with the homotopy fibre of the left vertical map and is therefore nilpotent by Lemmas 1.4 and 2.1. The bottom right horizontal map of (21) forms the rightmost column of a commutative diagramwhose right horizontal maps are induced by inclusion and are equivalences; the upper by Lemma A.3 and the lower by the discussion in Sect. 1.5. The left upper horizontal map is the equivalence from (7) and the leftmost vertical arrow fits into a commutative square induced by inclusionwhere hAut∂0W(W,∂1W)τWs\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hAut}_{\partial _0W}(W,\partial _1W)_{\tau _W^s}$$\end{document} are the components in the image of the maphAut∂0W(τWs,∂1W)⟶hAut∂0W(W,∂1W)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {hAut}_{\partial _0W}(\tau _W^s,\partial _1W)\longrightarrow \mathrm {hAut}_{\partial _0W}(W,\partial _1W)\end{aligned}$$\end{document}that forgets bundle data. Before taking geometric realisation, the map (24) is easily seen to be a Kan fibration, so the homotopy fibre of the right vertical map in (23) is equivalent to the classifying space of the gauge group of τWs\documentclass[12pt]{minimal}
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\begin{document}$$\partial _0W$$\end{document}. The horizontal arrows in (23) are 1-coconnected by construction, so the same holds for the induced map on vertical homotopy fibres. Combining this with Lemma 4.8 (i), we see that the left vertical map in (21) agrees, up to canonical equivalence and postcomposition with a 1-coconnected map, with a maphAut~∂0W≅(W,∂1W)/Diff~∂0W(W)⟶Maps∂0W(W,BO)τWs,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {\widetilde{hAut}}^{\cong }_{\partial _0W}(W,\partial _1W)/\widetilde{\mathrm {Diff}}_{\partial _0W}(W)\longrightarrow \mathrm {Maps}_{\partial _0W}(W,\mathrm {BO})_{\tau _W^s},\end{aligned}$$\end{document}so it suffices to show that this map between nilpotent spaces is p-locally (2p-3-k)\documentclass[12pt]{minimal}
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\begin{document}$$(2p-3-k)$$\end{document}-connected and that its homotopy fibre has finite homotopy groups. On homotopy groups, this map has the following description: a class in πk(hAut~∂0W≅(W,∂1W),Diff~∂0W(W);id)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _k(\mathrm {\widetilde{hAut}}^{\cong }_{\partial _0W}(W,\partial _1W),\widetilde{\mathrm {Diff}}_{\partial _0W}(W);\mathrm {id})$$\end{document} is represented by a homotopy equivalence of triads φ:W×Dk→W×Dk\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {\widetilde{hAut}}^{\cong }_{\partial _0W}(W,\partial _1W)/\widetilde{\mathrm {Diff}}_{\partial _0W}(W)\simeq \tilde{\mathbf {S}}(W)_{\mathrm {id}}\end{aligned}$$\end{document}and the map S~(W)id→Maps∗(W/∂0W,BO)0\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{\mathbf {S}}(W)_{\mathrm {id}}\rightarrow \mathrm {Maps}_*(W/\partial _0W,\mathrm {BO})_0$$\end{document} on homotopy groups explained in Sections 2.1.1 and 2.1.2, one sees that the diagram of nilpotent spaces
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\begin{document}$$*<2p-3$$\end{document} by a result of Serre [38, p. 498, Prop. 5]. □\documentclass[12pt]{minimal}
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The stable tangential structure we shall be primarily interested in is the one encoding stable framings, which we denote by sfr:EO→BO\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {sfr}:\mathrm {EO}\rightarrow \mathrm {BO}$$\end{document}. In this case, the p-local approximation of the space of block diffeomorphisms with tangential structures provided by Theorem 2.2 can be further simplified in terms of the union of componentshAut~∂0W≅(W,∂1W)ℓ⊂hAut~∂0W≅(W,∂1W)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {\widetilde{hAut}}^{\cong }_{\partial _0W}(W,\partial _1W)_\ell \subset \mathrm {\widetilde{hAut}}^{\cong }_{\partial _0W}(W,\partial _1W)\end{aligned}$$\end{document}given by the image of the canonical map that forgets tangential structuresBhAut~∂0Wsfr,≅(τWs,∂1W;ℓ0)ℓτ⟶BhAut~∂0W≅(W,∂1W)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {B\widetilde{hAut}}^{\mathrm {sfr},\cong }_{\partial _0W}(\tau _W^s,\partial _1W;\ell _0)^\tau _\ell \longrightarrow \mathrm {B\widetilde{hAut}}^{\cong }_{\partial _0W}(W,\partial _1W)\end{aligned}$$\end{document}on fundamental groups based at a fixed stable framing ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document} of τW\documentclass[12pt]{minimal}
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\begin{document}$$\tau _{W}$$\end{document} (see (12) and (19) for the notation). Loosely speaking, these are the components of homotopy equivalences of triads that are homotopic to a diffeomorphism preserving the component of the stable framing ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}.
Corollary 2.4
Let d≥6\documentclass[12pt]{minimal}
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\begin{document}$$d\ge 6$$\end{document} and W be a d-dimensional manifold triad satisfying the π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document}-condition. For a stable framing ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document} of W, the homotopy fibre ofBDiff~∂0Wsfr(W;ℓ0)ℓ⟶BhAut~∂0W≅(W,∂1W)ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {B\widetilde{Diff}}^{\mathrm {sfr}}_{\partial _0W}(W;\ell _0)_\ell \longrightarrow \mathrm {B\widetilde{hAut}}^{\cong }_{\partial _0W}(W,\partial _1W)_\ell \end{aligned}$$\end{document}is nilpotent and has finite homotopy groups. Moreover, this fibre is p-locally (2p-4-k)\documentclass[12pt]{minimal}
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\begin{document}$$(2p-4-k)$$\end{document}-connected for primes p, where k is the relative handle dimension of the inclusion ∂0W⊂W\documentclass[12pt]{minimal}
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\begin{document}$$\partial _0W\subset W$$\end{document}.
Proof
Once we show that the mapBhAut~∂0Wsfr,≅(τWs,∂1W;ℓ0)ℓτ⟶BhAut~∂0W≅(W,∂1W)ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {B\widetilde{hAut}}^{\mathrm {sfr},\cong }_{\partial _0W}(\tau _W^s,\partial _1W;\ell _0)^\tau _\ell \longrightarrow \mathrm {B\widetilde{hAut}}^{\cong }_{\partial _0W}(W,\partial _1W)_\ell \end{aligned}$$\end{document}is an equivalence, the statement is a consequence of Theorem 2.2. By construction, the homotopy fibre of this map is connected. Using (22) and (23), one sees that it is thus sufficient to show that the mapBhAut∂0Wsfr(τWs,∂1W;ℓ0)ℓ⟶BhAut∂0W(W,∂1W)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {BhAut}^{\mathrm {sfr}}_{\partial _0W}(\tau _W^s,\partial _1W;\ell _0)_\ell \longrightarrow \mathrm {BhAut}_{\partial _0W}(W,\partial _1W)\end{aligned}$$\end{document}is an equivalence. Taking vertical homotopy fibres in the map of fibre sequences
where Bun∂0W(τWs,sfr∗γ;ℓ0)ℓ⊂Bun∂0W(τWs,sfr∗γ;ℓ0)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Bun}_{\partial _0W}(\tau _W^s,\mathrm {sfr}^*\gamma ;\ell _0)_\ell \subset \mathrm {Bun}_{\partial _0W}(\tau _W^s,\mathrm {sfr}^*\gamma ;\ell _0)$$\end{document} are the components of the ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {hAut}_{\partial _0W}(\tau _W^s,\partial _1W)$$\end{document}-action, we see that it suffices to show that the induced action of the loop space of the homotopy fibre of the right vertical map on Bun∂0W(τWs,sfr∗γ;ℓ0)ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Bun}_{\partial _0W}(\tau _W^s,\mathrm {sfr}^*\gamma ;\ell _0)_\ell $$\end{document} is a torsor in the homotopical sense, i.e. that the map given by acting on ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document} is an equivalence. Since hAut∂0W(τWs,∂1W)∙→hAut∂0W(W,∂1W)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hAut}_{\partial _0W}(\tau _W^s,\partial _1W)_\bullet \rightarrow \mathrm {hAut}_{\partial _0W}(W,\partial _1W)_\bullet $$\end{document} is a Kan fibration, this loop space is canonically equivalent to the space of bundle self-maps of τWs\documentclass[12pt]{minimal}
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\begin{document}$$\tau _W^s$$\end{document} that cover the identity and agree with the identity on τWs|∂0W\documentclass[12pt]{minimal}
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\begin{document}$$\tau _W^s|_{\partial _0W}$$\end{document}. Moreover, by construction of the top fibration, the induced action on Bun∂0W(τWs,sfr∗γ;ℓ0)ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Bun}_{\partial _0W}(\tau _W^s,\mathrm {sfr}^*\gamma ;\ell _0)_\ell $$\end{document} is given by precomposition. Suitably modeled, this action is simply transitive, so the assertion follows. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
High-dimensional handlebodies and their mapping classes
This section serves to compute variants of the mapping class group of a high-dimensional handlebody up to extensions in terms of automorphisms of the integral homology.
Automorphisms of handlebodies
The proof of Theorem A relies on considering a more general family of manifold than discs, the boundary connected sumsVg:=♮g(Dn+1×Sn),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}V_g:=\natural ^g (D^{n+1}\times S^n),\end{aligned}$$\end{document}and their boundaries as well as the manifolds obtained by cutting out a fixed embedded disc D2n⊂∂Vg\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}W_g:=\partial V_g\cong \sharp ^g(S^n\times S^n)\quad \text {and}\quad W_{g,1}:=W_g\backslash \mathrm {int}(D^{2n}).\end{aligned}$$\end{document}This includes the case g=0\documentclass[12pt]{minimal}
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\begin{document}$$W_{0,1}=D^{2n}$$\end{document}. Using the notation introduced in Sections 1.4 and 1.5, the restriction of diffeomorphisms and relative homotopy automorphisms of Vg\documentclass[12pt]{minimal}
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\begin{document}$$V_g$$\end{document} to its boundary induces a commutative diagramwhere the right horizontal maps are fibrations and the left maps the inclusions of the fibres over the identity. These fibrations need not be surjective; we denote their images byDiff∂ext(Wg,1)⊂Diff∂(Wg,1)andhAut∂ext(Wg,1)⊂hAut∂(Wg,1).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {Diff}^{\mathrm {ext}}_{\partial }(W_{g,1})\subset \mathrm {Diff}_\partial (W_{g,1})\quad \text { and }\quad \mathrm {hAut}^{\mathrm {ext}}_{\partial }(W_{g,1})\subset \mathrm {hAut}_{\partial }(W_{g,1}).\end{aligned}$$\end{document}Furthermore, in agreement with (19) we writehAut∂≅(Vg),hAutD2n≅(Vg,Wg,1),andhAut∂≅(Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {hAut}^{\cong }_\partial (V_g),\quad \mathrm {hAut}^{\cong }_{D^{2n}}(V_g,W_{g,1}),\quad \text {and}\quad \mathrm {hAut}^{\cong }_{\partial }(W_{g,1})\end{aligned}$$\end{document}for the components hit by the vertical maps. Block variants of all of the above automorphism spaces are defined in the same way.
The mapping class group
As a first step in our analysis of the mapping class group π0DiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_g)$$\end{document}, we observe that we may equally study orientation-preserving diffeomorphisms of Vg\documentclass[12pt]{minimal}
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\begin{document}$$V_g$$\end{document} that do not necessarily fix the embedded disc in the boundary D2n⊂∂Vg\documentclass[12pt]{minimal}
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\begin{document}$$D^{2n}\subset \partial V_{g}$$\end{document}, or diffeomorphisms that preserve a disc D2n+1⊂int(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$D^{2n+1}\subset \mathrm {int}(V_g)$$\end{document} in the interior set- or pointwise.
Lemma 3.1
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\begin{document}$$D^{2n+1}\subset \mathrm {int}(V_g)$$\end{document}, the compositions induced by inclusionconsist of isomorphisms.
Proof
Up to isotopy, preserving a disc in the interior setwise is equivalent to preserving a point, so the group agrees with the group of path components of the subgroup of diffeomorphisms that fix the centre ∗∈D2n+1⊂int(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$*\in D^{2n+1}\subset \mathrm {int}(V_g)$$\end{document}. As Vg\documentclass[12pt]{minimal}
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\begin{document}$$*$$\end{document} implies that agrees with , so the second map in the first composition in the statement is an isomorphism. Taking derivatives at ∗\documentclass[12pt]{minimal}
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\begin{document}$$*$$\end{document} yields a homotopy fibre sequence of the formand hence an exact sequenceOn the subgroup SO(n)⊂SO(2n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {SO}(n)$$\end{document}-action with fixed point ∗∈Vg\documentclass[12pt]{minimal}
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\begin{document}$$*\le n-1$$\end{document} into short exact sequences and shows in particular that t is trivial for n≥2\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 2$$\end{document}, so holds as claimed. Replacing (26) by the fibration sequence induced by taking the derivative at the centre ∗∈D2n⊂∂Vg\documentclass[12pt]{minimal}
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\begin{document}$$*\in D^{2n}\subset \partial V_g$$\end{document} of the disc in the boundary, the proof of the claim regarding the maps in the second composition of the statement proceeds analogous to the first part of the proof. □\documentclass[12pt]{minimal}
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The homology action
To obtain further information on the mapping class group π0DiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}H_{W_{g,1}}:=\mathrm {H}_n(W_{g,1};\mathbf {Z})\quad \text {and}\quad H_{V_{g}}:=\mathrm {H}_n(V_{g};\mathbf {Z}).\end{aligned}$$\end{document}This action preserves further structure, such as the intersection pairingλ:HWg,1⊗HWg,1⟶Z,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\lambda :H_{W_{g,1}}\otimes H_{W_{g,1}}\longrightarrow \mathbf {Z},\end{aligned}$$\end{document}which equips HWg,1\documentclass[12pt]{minimal}
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\begin{document}$$H_{W_{g,1}}$$\end{document} with a nondegenerate (-1)n\documentclass[12pt]{minimal}
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\begin{document}$$(-1)^n$$\end{document}-symmetric form by Poincaré duality. In addition, any automorphism of HWg,1\documentclass[12pt]{minimal}
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\begin{document}$$H_{W_{g,1}}$$\end{document} induced by an orientation-preserving diffeomorphism of Wg,1\documentclass[12pt]{minimal}
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\begin{document}$$W_{g,1}$$\end{document} has to preserve the functionα:HWg,1⟶πnBSO(n)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\alpha :H_{W_{g,1}}\longrightarrow \pi _{n}\mathrm {BSO}(n)\end{aligned}$$\end{document}given by representing a homology class by an embedded n-sphere and taking its normal bundle. Wall [39, Thm 2] has shown that λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document} and α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} satisfy the relations
as long as n≥3\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 3$$\end{document}, where∂n:πnBSO(n)⟶πn-1Sn-1≅Z\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\partial _n:\pi _n\mathrm {BSO}(n)\longrightarrow \pi _{n-1}S^{n-1}\cong \mathbf {Z}\end{aligned}$$\end{document}is induced by the fibration Sn-1→BSO(n-1)→BSO(n)\documentclass[12pt]{minimal}
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\begin{document}$$S^{n-1}\rightarrow \mathrm {BSO}(n-1)\rightarrow \mathrm {BSO}(n)$$\end{document} and τSn∈πnBSO(n)\documentclass[12pt]{minimal}
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\begin{document}$$\tau _{S^n}\in \pi _n\mathrm {BSO}(n)$$\end{document} is the class representing the tangent bundle of the n-sphere. In sum, we arrive at a morphismπ0Diff(Wg,1)⟶Gg:=Aut(HWg,1,λ,α)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\pi _0\mathrm {Diff}(W_{g,1})\longrightarrow G_g:=\mathrm {Aut}(H_{W_{g,1}},\lambda ,\alpha )\end{aligned}$$\end{document}to the subgroup Gg⊂GL(HWg)\documentclass[12pt]{minimal}
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\begin{document}$$G_g\subset \mathrm {GL}(H_{W_g})$$\end{document} of automorphisms preserving λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document} and α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}, which is surjective by [40, Lem. 10]. However, we are interested in the mapping class group π0DiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_g)$$\end{document} and not every automorphism in Gg\documentclass[12pt]{minimal}
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\begin{document}$$G_g$$\end{document} is realised by a diffeomorphism of Wg,1\documentclass[12pt]{minimal}
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\begin{document}$$W_{g,1}$$\end{document} that extends to one of Vg\documentclass[12pt]{minimal}
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\begin{document}$$V_g$$\end{document}; it would at least have to preserve the Lagrangian subspaceKg:=ker(HWg,1→HVg),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}K_g:=\ker \big (H_{W_{g,1}}\rightarrow H_{V_{g}}\big ),\end{aligned}$$\end{document}so there is a canonical map π0DiffD2n(Vg)→Ggext\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_g)\rightarrow G_g^{\mathrm {ext}}$$\end{document} to the subgroupGgext:={Φ∈Gg∣Φ(Kg)⊂Kg}\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}G_g^{\mathrm {ext}}:=\{\Phi \in G_g\mid \Phi (K_g)\subset K_g\}\end{aligned}$$\end{document}of automorphisms that preserve this Lagrangian, given by acting on the homology of the boundary. Using the canonical isomorphism HWg,1/Kg≅HVg\documentclass[12pt]{minimal}
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\begin{document}$$H_{W_{g,1}}/K_g\cong H_{V_g}$$\end{document}, the subgroup Ggext\documentclass[12pt]{minimal}
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\begin{document}$$G^{\mathrm {ext}}_g$$\end{document} maps further to GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {GL}(H_{V_g})$$\end{document}. The resulting compositionπ0DiffD2n(Vg)⟶Ggext⟶GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _0\mathrm {Diff}_{D^{2n}}(V_g)\longrightarrow G_g^{\mathrm {ext}}\longrightarrow \mathrm {GL}(H_{V_g})\end{aligned}$$\end{document}agrees with the action on the homology of Vg\documentclass[12pt]{minimal}
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\begin{document}$$V_g$$\end{document} and one may ask for the (co)kernel of the three maps involved. Extending work of Wall [40, 41], we express the answer in Theorem 3.3 below in terms of an exact sequence0⟶HVg∨⊗SπnSO(n)⟶Ng⟶Mg⟶0,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} 0\longrightarrow H_{V_g}^{\vee }\otimes S\pi _n\mathrm {SO}(n)\longrightarrow N_g\longrightarrow M_g\longrightarrow 0,\end{aligned}$$\end{document}of GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {GL}(H_{V_g})$$\end{document}-modules where
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\begin{document}$$S\pi _n\mathrm {SO}(n)\subset \pi _n\mathrm {SO}(n+1)$$\end{document} is the image of the stabilisation map πnSO(n)→πnSO(n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _n\mathrm {SO}(n)\rightarrow \pi _n\mathrm {SO}(n+1)$$\end{document},
Mg⊂(HVg⊗HVg)∨\documentclass[12pt]{minimal}
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\begin{document}$$M_g\subset (H_{V_g}\otimes H_{V_g})^{\vee }$$\end{document} is the submodule of bilinear forms μ∈(HVg⊗HVg)∨\documentclass[12pt]{minimal}
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\begin{document}$$\mu \in (H_{V_g}\otimes H_{V_g})^{\vee }$$\end{document} that are
all equipped with the evident GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {GL}(H_{V_g})$$\end{document}-action through HVg\documentclass[12pt]{minimal}
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\begin{document}$$H_{V_g}$$\end{document}. Here we denoted the integral dual of a G-module M by M∨:=Hom(M,Z)\documentclass[12pt]{minimal}
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\begin{document}$$M^\vee :=\mathrm {Hom}(M,\mathbf {Z})$$\end{document}.
Remark 3.2
The image of the map ∂n+1:πnSO(n+1)→πnSn≅Z\documentclass[12pt]{minimal}
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\begin{document}$$\partial _{n+1}:\pi _{n}\mathrm {SO}(n+1)\rightarrow \pi _nS^n\cong \mathbf {Z}$$\end{document} is generated by the order of the tangent bundle τSn∈πn-1SO(n)\documentclass[12pt]{minimal}
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\begin{document}$$\tau _{S^n}\in \pi _{n-1}\mathrm {SO}(n)$$\end{document}, so we have (see [29, §1B)])im(∂n+1)=0fornevenZforn=1,3,72·Zotherwise,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {im}(\partial _{n+1})= {\left\{ \begin{array}{ll} 0&{}\text{ for } n\text { even}\\ \mathbf {Z}&{}\text{ for } n=1,3,7\\ 2\cdot \mathbf {Z}&{}\text{ otherwise }, \end{array}\right. }\end{aligned}$$\end{document}which exhibits the condition μ(x,x)∈im(∂n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\mu (x,x)\in \mathrm {im}(\partial _{n+1})$$\end{document} in (2) as vacuous unless n≠1,3,7\documentclass[12pt]{minimal}
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\begin{document}$$n\ne 1,3,7$$\end{document} is odd. Moreover, this shows that, after choosing a basis HVg≅Zg\documentclass[12pt]{minimal}
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\begin{document}$$H_{V_g}\cong \mathbf {Z}^{g}$$\end{document}, the module Mg\documentclass[12pt]{minimal}
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\begin{document}$$M_g$$\end{document} can be described equivalently in terms of (-1)n+1\documentclass[12pt]{minimal}
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\begin{document}$$(-1)^{n+1}$$\end{document}-symmetric integral (g×g)\documentclass[12pt]{minimal}
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\begin{document}$$(g\times g)$$\end{document}-matrices, with even diagonal entries if n≠1,3,7\documentclass[12pt]{minimal}
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\begin{document}$$n\ne 1,3,7$$\end{document} is odd.
Moreover, the induced outer actions of these extensions is as specified above and the second extension admits a preferred splitting.
Remark 3.4
For a complete description of π0DiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_g)$$\end{document}, one still needs to determine the extension problems of the first or third part of the theorem, which we do not pursue at this point. Similar extensions by Kreck [26] describing the closely related mapping class group π0Diff∂(Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_\partial (W_{g,1})$$\end{document} for n≥3\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 3$$\end{document} have been resolved in [25] for n odd.
Proof of Theorem 3.3
We begin with three preparatory remarks.
Results of Wall we shall use rely on are phrased in terms of pseudoisotopy instead of isotopy, but these notions agree in our situation by [11].
Justified by Lemma 3.1, we do not distinguish between seemingly different variants of π0DiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_g)$$\end{document} fixing various discs point- or setwise.
We identify Kg\documentclass[12pt]{minimal}
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\begin{document}$$K_g$$\end{document} canonically with the dual HVg∨\documentclass[12pt]{minimal}
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\begin{document}$${H_{V_g}}^\vee $$\end{document} as a Ggext\documentclass[12pt]{minimal}
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\begin{document}$$G^{\mathrm {ext}}_g$$\end{document}-module via the isomorphism induced by the form λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}, and dually HVg\documentclass[12pt]{minimal}
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\begin{document}$$H_{V_g}$$\end{document} with Kg∨\documentclass[12pt]{minimal}
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\begin{document}$$K_g^\vee $$\end{document}.
Wall [40, Lem. 10] showed that the action π0DiffD2n(Vg)→GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_g)\rightarrow \mathrm {GL}(H_{V_g})$$\end{document} is surjective and identified its kernel with those isotopy classes of diffeomorphisms φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document} that are homotopic to the identity. Moreover, in [41, p. 298], he defined a complete obstruction (μφ,βφ)∈Ng\documentclass[12pt]{minimal}
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\begin{document}$$(\mu _\varphi ,\beta _\varphi )\in N_g$$\end{document} for such a homotopically trivial diffeomorphism to be isotopic to the identity. By [40, Lem. 12–13], these obstructions are additive and exhaust Ng\documentclass[12pt]{minimal}
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\begin{document}$$N_g$$\end{document}, so the resulting functionker(π0Diff(Vg)→GL(HVg))⟶Ng\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \ker \big (\pi _0\mathrm {Diff}(V_g)\rightarrow \mathrm {GL}(H_{V_g})\big )\longrightarrow N_g\end{aligned}$$\end{document}is an isomorphism of groups, which establishes the first of the three claims. To demonstrate the second, note that, as automorphisms in Ggext\documentclass[12pt]{minimal}
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\begin{document}$$G_g^{\mathrm {ext}}$$\end{document} preserve the form λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}, the compositionGgext⟶GL(HVg)⟶((-)-1)∨GL(Kg)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}G^{\mathrm {ext}}_g\longrightarrow \mathrm {GL}(H_{V_g})\overset{((-)^{-1})^{\vee }}{\longrightarrow }\mathrm {GL}(K_g)\end{aligned}$$\end{document}agrees with the restriction to Kg\documentclass[12pt]{minimal}
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\begin{document}$$K_g$$\end{document}. In particular, this shows that the kernel of the first map in this composition acts trivially on Kg\documentclass[12pt]{minimal}
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\begin{document}$$K_g$$\end{document}, so there is a canonical monomorphismdefined by sending ϕ∈Ggext\documentclass[12pt]{minimal}
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\begin{document}$$\phi \in G_g^{\mathrm {ext}}$$\end{document} to the linear map Ψ(ϕ):HVg→Kg\documentclass[12pt]{minimal}
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\begin{document}$$\Psi (\phi ):H_{V_g}\rightarrow K_g$$\end{document} induced by the difference (ϕ-id):HWg,1→Kg\documentclass[12pt]{minimal}
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\begin{document}$$(\phi -\mathrm {id}):H_{W_{g,1}}\rightarrow K_g$$\end{document}. This leaves us with identifying the image of Ψ\documentclass[12pt]{minimal}
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\begin{document}$$\Psi $$\end{document} with Mg\documentclass[12pt]{minimal}
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\begin{document}$$M_g$$\end{document} for which it is helpful to note that a morphism f∈Hom(HVg,Kg)\documentclass[12pt]{minimal}
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\begin{document}$$f\in \mathrm {Hom}(H_{V_g},K_g)$$\end{document} lies in the subspaceMg⊂(HVg⊗HVg)∨≅Hom(HVg,Kg)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}M_g\subset (H_{V_g}\otimes H_{V_g})^\vee \cong \mathrm {Hom}(H_{V_g},K_g)\end{aligned}$$\end{document}if and only if f∨=(-1)n+1f\documentclass[12pt]{minimal}
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\begin{document}$$f^{\vee }=(-1)^{n+1}f$$\end{document} and λ(f(x),x~)∈im(∂n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda (f(x),\tilde{x})\in \mathrm {im}(\partial _{n+1})$$\end{document} holds for all x∈HVg\documentclass[12pt]{minimal}
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\begin{document}$$x\in H_{V_g}$$\end{document}. Here x~∈HWg,1\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{x}\in H_{W_{g,1}}$$\end{document} is a choice of preimage of x under the projection HWg,1→HVg\documentclass[12pt]{minimal}
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\begin{document}$$H_{W_{g,1}}\rightarrow H_{V_g}$$\end{document}, but the value λ(f(x),x~)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda (f(x),\tilde{x})$$\end{document} is independent of this choice x~\documentclass[12pt]{minimal}
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\begin{document}$$\tilde{x}$$\end{document} since Kg\documentclass[12pt]{minimal}
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\begin{document}$$K_g$$\end{document} is Lagrangian. For elements of the form Ψ(ϕ)\documentclass[12pt]{minimal}
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\begin{document}$$\Psi (\phi )$$\end{document}, the first property Ψ(ϕ)∨=(-1)n+1Ψ(ϕ)\documentclass[12pt]{minimal}
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\begin{document}$$\Psi (\phi )^{\vee }=(-1)^{n+1}\Psi (\phi )$$\end{document} follows from the fact that ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document} preserves the form λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}. To see the second, we note that, since Kg\documentclass[12pt]{minimal}
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\begin{document}$$K_g$$\end{document} is Lagrangian, the function α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} is additive on Kg\documentclass[12pt]{minimal}
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\begin{document}$$K_g$$\end{document}, so it vanishes on it since the images of the second Sn\documentclass[12pt]{minimal}
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\begin{document}$$S^n$$\end{document}-factors of the connected sum Wg,1≅♯g(Sn×Sn)\Dn\documentclass[12pt]{minimal}
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\begin{document}$$W_{g,1}\cong \sharp ^g(S^n\times S^n)\backslash D^{n}$$\end{document} in Vg=♮g(Dn+1×Sn)\documentclass[12pt]{minimal}
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\begin{document}$$V_g=\natural ^g (D^{n+1}\times S^n)$$\end{document} induce a basis of Kg\documentclass[12pt]{minimal}
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\begin{document}$$K_g$$\end{document} and have trivial normal bundle. Using the fact that ϕ∈Ggext\documentclass[12pt]{minimal}
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\begin{document}$$\phi \in G_g^{\mathrm {ext}}$$\end{document} preserves α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} and property (ii) of α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document}, we computeα(x~)=α(Ψ(ϕ)(x)+x~)=α(Ψ(ϕ)(x))+α(x~)+λ(Ψ(ϕ)(x),x~)·τSn=α(x~)+λ(Ψ(ϕ)(x),x~)·τSn\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \alpha (\tilde{x})=\alpha \big (\Psi (\phi )(x)+\tilde{x}\big )= & {} \alpha \big (\Psi (\phi )(x)\big )+\alpha (\tilde{x})+\lambda \big (\Psi (\phi )(x),\tilde{x}\big )\cdot \tau _{S^n}\\= & {} \alpha (\tilde{x})+\lambda \big (\Psi (\phi )(x),\tilde{x}\big )\cdot \tau _{S^n}\end{aligned}$$\end{document}and conclude that λ(Ψ(ϕ)(x),x~)·τSn\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \big (\Psi (\phi )(x),\tilde{x}\big )\cdot \tau _{S^n}$$\end{document} vanishes, so λ(Ψ(ϕ)(x),x~)∈im(∂n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda \big (\Psi (\phi )(x),\tilde{x}\big )\in \mathrm {im}(\partial _{n+1})$$\end{document} holds as claimed. This proves that the image of Ψ\documentclass[12pt]{minimal}
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\begin{document}$$\Psi $$\end{document} is contained in Mg\documentclass[12pt]{minimal}
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\begin{document}$$M_g$$\end{document}, and to show that it agrees with it, we consider the commutative diagram
whose vertical arrow Ng→ker(Ggext→GL(HVg))\documentclass[12pt]{minimal}
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\begin{document}$$N_g\rightarrow \ker (G_g^{\mathrm {ext}}\rightarrow \mathrm {GL}(H_{V_g}))$$\end{document} is induced by the isomorphism (30). By [41, Lem. 24], the vertical composition in the diagram agrees with the projection Ng→Mg\documentclass[12pt]{minimal}
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\begin{document}$$N_g\rightarrow M_g$$\end{document} in (28), so it is surjective. Consequently, Ψ\documentclass[12pt]{minimal}
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\begin{document}$$\Psi $$\end{document} is surjective as well and hence an isomorphism, which proves the second claim of the statement. To show the third, we observe that, given that Ψ\documentclass[12pt]{minimal}
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\begin{document}$$\Psi $$\end{document} is an isomorphism and Ng→Mg\documentclass[12pt]{minimal}
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\begin{document}$$N_g\rightarrow M_g$$\end{document} is surjective, the diagram implies that π0Diff(Vg)→Ggext\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}(V_g)\rightarrow G_g^{\mathrm {ext}}$$\end{document} is surjective and moreover that its kernel agrees with the kernel of Ng→Mg\documentclass[12pt]{minimal}
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\begin{document}$$N_g\rightarrow M_g$$\end{document}, which is HVg∨⊗SπnSO(n)\documentclass[12pt]{minimal}
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\begin{document}$$H_{V_g}^\vee \otimes S\pi _n\mathrm {SO}(n)$$\end{document} as claimed.
We now identify the actions as asserted. For the second extension, one can argue as follows: an automorphism φ∈GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\varphi \in \mathrm {GL}(H_{V_g})$$\end{document} acts on ker(Ggext→GL(HVg))\documentclass[12pt]{minimal}
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\begin{document}$$\ker (G_g^{\mathrm {ext}}\rightarrow \mathrm {GL}(H_{V_g}))$$\end{document} by conjugating with a choice of lift φ~∈Ggext\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{\varphi }\in G_g^{\mathrm {ext}}$$\end{document}, so the isomorphism Ψ\documentclass[12pt]{minimal}
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\begin{document}$$\Psi $$\end{document} is equivariant simply because of the identity (φ~(ϕ-id)φ~-1)=(φ~ϕφ~-1-id)\documentclass[12pt]{minimal}
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\begin{document}$$(\widetilde{\varphi }(\phi -\mathrm {id})\widetilde{\varphi }^{-1})=(\widetilde{\varphi }\phi \widetilde{\varphi }^{-1}-\mathrm {id})$$\end{document} in Hom(HVg,Kg)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Hom}(H_{V_g},K_g)$$\end{document} for all ϕ∈Hom(HVg,Kg)\documentclass[12pt]{minimal}
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\begin{document}$$\phi \in \mathrm {Hom}(H_{V_g},K_g)$$\end{document}. For the first extension, we use that the left vertical composition in the diagram above agrees with the projection Ng→Mg\documentclass[12pt]{minimal}
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\begin{document}$$N_g\rightarrow M_g$$\end{document}, so it suffices to show that the compositionker(π0DiffD2n(Vg)→GL(HVg))⟶≅Ng⟶(πnSO(n+1))HVg\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \ker (\pi _0\mathrm {Diff}_{D^{2n}}(V_g)\rightarrow \mathrm {GL}(H_{V_g}))\overset{\cong }{\longrightarrow } N_g\longrightarrow (\pi _n\mathrm {SO}(n+1))^{H_{V_g}}\end{aligned}$$\end{document}of (30) with the projection is equivariant. From Wall’s definition [40, p. 267] of the invariant βφ\documentclass[12pt]{minimal}
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\begin{document}$$\beta _\varphi $$\end{document} of an element φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document} in the kernel, we see that this composition can be described as follows: representing a homology class [e]∈HVg\documentclass[12pt]{minimal}
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\begin{document}$$[e]\in H_{V_g}$$\end{document} by an embedded sphere e:Sn→Vg\documentclass[12pt]{minimal}
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\begin{document}$$e:S^n\rightarrow V_g$$\end{document}, we can alter φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document} by an isotopy such that it preserves e pointwise. In this case, the derivative of φ\documentclass[12pt]{minimal}
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\begin{document}$$\nu (e)\cong \varepsilon ^{n+1}$$\end{document} which induces an element βφ([e])∈πnSO(n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\beta _\varphi ([e])\in \pi _n\mathrm {SO}(n+1)$$\end{document}. This uses a trivialisation of ν(e)\documentclass[12pt]{minimal}
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\begin{document}$$\nu (e)$$\end{document}, but the resulting element βφ([e])\documentclass[12pt]{minimal}
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\begin{document}$$\beta _\varphi ([e])$$\end{document} is independent of this choice. From this description, the claimed equivariance is straight-forward to check, and the identification of the action of the last sequence follows from that of the first two by chasing through the diagram obtained by extending the diagram above by taking vertical kernels.
To see that the second extension splits, note that the second Sn\documentclass[12pt]{minimal}
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\begin{document}$$S^n$$\end{document}-factors in the connected sum decomposition of Wg,1=♯gSn×Sn\int(D2n)⊂♮gDn+1×Sn=Vg\documentclass[12pt]{minimal}
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\begin{document}$$W_{g,1}=\sharp ^gS^n\times S^n\backslash \mathrm {int}(D^{2n})\subset \natural ^g D^{n+1}\times S^n=V_g$$\end{document} induce a splitting of the canonical map HWg,1→HVg\documentclass[12pt]{minimal}
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\begin{document}$$H_{W_{g,1}}\rightarrow H_{V_g}$$\end{document} and thus an isomorphism of the form Kg⊕HVg≅HWg,1\documentclass[12pt]{minimal}
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\begin{document}$$K_g\oplus H_{V_g}\cong H_{W_{g,1}}$$\end{document}. Using this splitting of the homology, we can define a morphism GL(HVg)→Ggext\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {GL}(H_{V_g})\rightarrow G_{g}^{\mathrm {ext}}$$\end{document} by assigning ϕ∈GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\phi \in \mathrm {GL}(H_{V_g})$$\end{document} the automorphismHWg,1≅Kg⊕HVg→(ϕ-1)∨⊕ϕKg⊕HVg≅HWg,1,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}H_{W_{g,1}}\cong K_g\oplus H_{V_g}\xrightarrow {(\phi ^{-1})^{\vee }\oplus \phi }K_g\oplus H_{V_g}\cong H_{W_{g,1}},\end{aligned}$$\end{document}which clearly preserves Kg\documentclass[12pt]{minimal}
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\begin{document}$$K_g$$\end{document}. Moreover, its induced automorphism of HWg,1\documentclass[12pt]{minimal}
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\begin{document}$$H_{W_{g,1}}$$\end{document} agrees with ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document} by construction, so we obtain a splitting as desired. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Remark 3.5
With respect to the basis HWg,1≅Z2g\documentclass[12pt]{minimal}
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\begin{document}$$H_{W_{g,1}}\cong \mathbf {Z}^{2g}$$\end{document} suggested by the connected sum decomposition Wg,1≅♯g(Sn×Sn)\int(D2n)\documentclass[12pt]{minimal}
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\begin{document}$$W_{g,1}\cong \sharp ^g(S^n\times S^n)\backslash \mathrm {int}(D^{2n})$$\end{document}, the subgroup Ggext⊂GL(HWg,1)≅GL2g(Z)\documentclass[12pt]{minimal}
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\begin{document}$$G_g^{\mathrm {ext}}\subset \mathrm {GL}(H_{W_{g,1}})\cong \mathrm {GL}_{2g}(\mathbf {Z})$$\end{document} agrees with the group of block matrices of the formAM0(A-1)T\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\left( \begin{matrix} A&{}M\\ 0 &{}(A^{-1})^T\end{matrix}\right) \end{aligned}$$\end{document}with M∈Mg\documentclass[12pt]{minimal}
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\begin{document}$$M\in M_g$$\end{document}, using the matrix description of Mg\documentclass[12pt]{minimal}
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\begin{document}$$M_g$$\end{document} explained in Remark 3.2. From this point of view, the splitting GLg(Z)→Ggext\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {GL}_g(\mathbf {Z})\rightarrow G_g^{\mathrm {ext}}$$\end{document} described in the proof of Theorem 3.3 is the obvious one that sends a matrix A∈GLg(Z)\documentclass[12pt]{minimal}
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\begin{document}$$M=0$$\end{document}.
Stable framings
Choosing the canonical map sfr:EO→BO\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {sfr}:\mathrm {EO}\rightarrow \mathrm {BO}$$\end{document} as a stable tangential structure in the sense of Section 1.8.1, the space of sfr\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {sfr}$$\end{document}-structuresBunD2n(τVgs,sfr∗γ;ℓ0)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {Bun}_{D^{2n}}(\tau _{V_g}^s,\mathrm {sfr}^*\gamma ;\ell _0)\end{aligned}$$\end{document}as defined in that section is the space of stable framings of Vg\documentclass[12pt]{minimal}
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\begin{document}$$V_g$$\end{document} relative to a fixed stable framing ℓ0:τVgs|D2n→sfr∗γ\documentclass[12pt]{minimal}
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\begin{document}$$\ell _0:\tau _{V_g}^s|_{D^{2n}}\rightarrow \mathrm {sfr}^*\gamma $$\end{document}. We denote the stabiliser of the canonical action of π0DiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_g)$$\end{document} on the set of components π0BunD2n(τVgs,sfr∗γ;ℓ0)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Bun}_{D^{2n}}(\tau _{V_g}^s,\mathrm {sfr}^*\gamma ;\ell _0)$$\end{document} and its image in Ggext\documentclass[12pt]{minimal}
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\begin{document}$$G_g^{\mathrm {ext}}$$\end{document} byπ0DiffD2n(Vg)ℓ⊂π0DiffD2n(Vg)andGg,ℓext⊂Ggext.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\pi _0\mathrm {Diff}_{D^{2n}}(V_g)_\ell \subset \pi _0\mathrm {Diff}_{D^{2n}}(V_g)\quad \text {and}\quad G_{g,\ell }^{\mathrm {ext}}\subset G_g^{\mathrm {ext}}.\end{aligned}$$\end{document}Note that π0DiffD2n(Vg)ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_g)_\ell $$\end{document} agrees with the image of the canonical map BDiffD2nsfr(Vg)→BDiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BDiff}^{\mathrm {sfr}}_{D^{2n}}(V_g)\rightarrow \mathrm {BDiff}_{D^{2n}}(V_g)$$\end{document} on fundamental groups based at the point induced by the stable framing ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}, or equivalently, with the kernel of the crossed homomorphismπ0DiffD2n(Vg)⟶HVg∨⊗πnSO\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _0\mathrm {Diff}_{D^{2n}}(V_g)\longrightarrow H_{V_g}^\vee \otimes \pi _n\mathrm {SO}\end{aligned}$$\end{document}given by acting on ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}. This uses the identificationHVg∨⊗πnSO≅Hom(HVg,πnSO)≅π0MapsD2n(Vg,SO)≅π0BunD2n(τVgs,θ∗γ;ℓ0)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}H_{V_g}^\vee \otimes \pi _n\mathrm {SO}\cong & {} \mathrm {Hom}(H_{V_g},\pi _n\mathrm {SO})\\ {}\cong & {} \pi _0\mathrm {Maps}_{D^{2n}}(V_g,\mathrm {SO})\cong \pi _0\mathrm {Bun}_{D^{2n}}(\tau _{V_g}^s,\theta ^*\gamma ;\ell _0)\end{aligned}$$\end{document}whose first two isomorphisms are the evident ones and whose third is induced by the choice of stable framing ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}, using that π0BunD2n(τVgs,sfr∗γ;ℓ0)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Bun}_{D^{2n}}(\tau _{V_g}^s,\mathrm {sfr}^*\gamma ;\ell _0)$$\end{document} is a torsor over π0MapsD2n(Vg,SO)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Maps}_{D^{2n}}(V_g,\mathrm {SO})$$\end{document} with pointwise multiplication.
Remark 3.6
As the stabilisation map SO(2n+1)→SO\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {SO}(2n+1)\rightarrow \mathrm {SO}$$\end{document} is 2n-connected, the induced map MapsD2n(Vg,SO(2n+1))→MapsD2n(Vg,SO)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Maps}_{D^{2n}}(V_g,\mathrm {SO}(2n+1))\rightarrow \mathrm {Maps}_{D^{2n}}(V_g,\mathrm {SO})$$\end{document} is n-connected, which implies that there is no difference between equivalence classes of stable and unstable framings, so the discussion of this subsection applies equally well to unstable framings instead of stable ones.
To relate the subgroups π0DiffD2n(Vg)ℓ⊂π0DiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_g)_\ell \subset \pi _0\mathrm {Diff}_{D^{2n}}(V_g)$$\end{document} and Gg,ℓext⊂Ggext\documentclass[12pt]{minimal}
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\begin{document}$$G_{g,\ell }^{\mathrm {ext}}\subset G_{g}^{\mathrm {ext}}$$\end{document} to the sequences of Theorem 3.3, we define the GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {GL}(H_{V_g})$$\end{document}-submodule
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\begin{document}$$N_g^{\mathrm {sfr}}\subset N_g$$\end{document} as the intersection of Ng\documentclass[12pt]{minimal}
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\begin{document}$$N_g$$\end{document} with Mg⊕⟨τSn+1⟩HVg\documentclass[12pt]{minimal}
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\begin{document}$$M_g\oplus \langle \tau _{S^{n+1}} \rangle ^{H_{V_g}}$$\end{document}, where ⟨τSn+1⟩\documentclass[12pt]{minimal}
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\begin{document}$$\tau _{S^{n+1}}$$\end{document}
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\begin{document}$$M_g^{\mathrm {sfr}}\subset M_g$$\end{document} as the collection of (-1)n+1\documentclass[12pt]{minimal}
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\begin{document}$$(-1)^{n+1}$$\end{document}-symmetric bilinear forms μ∈(HVg⊗HVg)∨\documentclass[12pt]{minimal}
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\begin{document}$$\mu \in (H_{V_g}\otimes H_{V_g})^{\vee }$$\end{document} that are even, i.e. μ(x,x)∈2·Z\documentclass[12pt]{minimal}
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\begin{document}$$\mu (x,x)\in 2\cdot \mathbf {Z}$$\end{document} for all x∈HVg\documentclass[12pt]{minimal}
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\begin{document}$$x\in H_{V_g}$$\end{document}, which is automatic if n is even.
Standard arguments involving the long exact sequences in homotopy groups of the usual fibration SO(d)→SO(d+1)→Sd\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {SO}(d)\rightarrow \mathrm {SO}(d+1)\rightarrow S^d$$\end{document} (cf. [29, §1B)]) show that the sequence (28) restricts to an exact sequence of GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {GL}(H_{V_g})$$\end{document}-modules0⟶HVg∨⊗ker(SπnSO(n)→πnSO)⟶Ngsfr⟶Mgsfr⟶0.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}0\longrightarrow H_{V_g}^\vee \otimes \ker \big (S\pi _n\mathrm {SO}(n)\rightarrow \pi _n\mathrm {SO}\big )\longrightarrow N_g^{\mathrm {sfr}}\longrightarrow M_g^{\mathrm {sfr}}\longrightarrow 0.\end{aligned}$$\end{document}
The calculation of im(∂n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {im}(\partial _{n+1})$$\end{document} in Remark 3.2 shows that the inclusion Mgsfr⊂Mg\documentclass[12pt]{minimal}
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\begin{document}$$M_g^{\mathrm {sfr}}\subset M_g$$\end{document} is an equality for n≠1,3,7\documentclass[12pt]{minimal}
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\begin{document}$$n\ne 1,3,7$$\end{document}, so the same holds for Gg,ℓext⊂Ggext\documentclass[12pt]{minimal}
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\begin{document}$$G_{g,\ell }^{\mathrm {ext}}\subset G^{\mathrm {ext}}_g$$\end{document} as a result of Proposition 3.7.
Proof of Proposition 3.7
The first sequence of Theorem 3.3 fits into a diagram
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\begin{document}$$\tau $$\end{document} is the crossed homomorphism (32), which is up to isomorphism given by the action of π0DiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_g)$$\end{document} on ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}, so the first part of the statement is equivalent to the surjectivity of τ\documentclass[12pt]{minimal}
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\begin{document}$$\tau $$\end{document}. The diagonal arrow is the morphism which assigns an element (μ,β)∈Ng\documentclass[12pt]{minimal}
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\begin{document}$$(\mu ,\beta )\in N_g$$\end{document} the composition of β:HVg→πnSO(n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _n\mathrm {SO}(n+1)\rightarrow \pi _n\mathrm {SO}$$\end{document}. As τSn+1∈πnSO(n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\tau _{S^{n+1}}\in \pi _n\mathrm {SO}(n+1)$$\end{document} is stably trivial, it follows from the second defining property of Ng\documentclass[12pt]{minimal}
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\begin{document}$$N_g$$\end{document} that this composition is additive, so indeed defines an element of HVg∨⊗πnSO≅Hom(HVg,πnSO)\documentclass[12pt]{minimal}
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\begin{document}$${H_{V_g}}^\vee \otimes \pi _n\mathrm {SO}\cong \mathrm {Hom}(H_{V_g},\pi _n\mathrm {SO})$$\end{document}. As a next step, observe that the diagonal map is surjective, since the projectionNg⊂Mg⊕(πnSO(n+1))HVg→(πnSO(n+1))HVg\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}N_g\subset M_g\oplus (\pi _n\mathrm {SO}(n+1))^{H_{V_g}}\rightarrow (\pi _n\mathrm {SO}(n+1))^{H_{V_g}}\end{aligned}$$\end{document}has a section over the subspace of linear maps HVg∨⊗πnSO(n+1)⊂(πnSO(n+1))HVg\documentclass[12pt]{minimal}
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\begin{document}$${H_{V_g}}^\vee \otimes \pi _n\mathrm {SO}(n+1)\subset (\pi _n\mathrm {SO}(n+1))^{H_{V_g}}$$\end{document} by setting the Mg\documentclass[12pt]{minimal}
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\begin{document}$$M_g$$\end{document}-coordinate to zero and because the stabilisation map πnSO(n+1)→πnSO\documentclass[12pt]{minimal}
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\begin{document}$$\pi _n\mathrm {SO}(n+1)\rightarrow \pi _n\mathrm {SO}$$\end{document} is surjective. This reduces the first claim of the statement to the commutativity of the triangle, which follows from the geometric description of the composition (31) we gave in the proof of Theorem 3.3 in a straight-forward manner.
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_g)_\ell $$\end{document}, so the surjectivity of the diagonal arrow in the diagram also shows that this stabiliser surjects onto GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {GL}(H_{V_g})$$\end{document} and that the kernel of the restriction ker(τ)→GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\ker (\tau )\rightarrow \mathrm {GL}(H_{V_g})$$\end{document} agrees with the kernel of the diagonal arrow. But this kernel is exactly the submodule Ngsfr⊂Ng\documentclass[12pt]{minimal}
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\begin{document}$$N_g^{\mathrm {sfr}}\subset N_g$$\end{document}, because the kernel of the stabilisation map πnSO(n+1)→πnSO\documentclass[12pt]{minimal}
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\begin{document}$$\pi _n\mathrm {SO}(n+1)\rightarrow \pi _n\mathrm {SO}(n+2)$$\end{document}, i.e. the subgroup generated by τSn+1\documentclass[12pt]{minimal}
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\begin{document}$$\tau _{S^{n+1}}$$\end{document}. This establishes the first sequence of the second claim. For the second, note that the surjectivity of the morphismπ0DiffD2n(Vg)ℓ⟶GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\pi _0\mathrm {Diff}_{D^{2n}}(V_g)_\ell \longrightarrow \mathrm {GL}(H_{V_g})\end{aligned}$$\end{document}implies that also the morphismGg,ℓext⟶GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}G_{g,\ell }^{\mathrm {ext}}\longrightarrow \mathrm {GL}(H_{V_g})\end{aligned}$$\end{document}is surjective. Comparing the first two sequences of Theorem 3.3, we see that its kernel agrees with the image of Ngsfr\documentclass[12pt]{minimal}
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\begin{document}$$N_g^{\mathrm {sfr}}$$\end{document} in Mg\documentclass[12pt]{minimal}
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\begin{document}$$M_g$$\end{document} under the projection Ng→Mg\documentclass[12pt]{minimal}
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\begin{document}$$N_g\rightarrow M_g$$\end{document}, i.e. with those bilinear forms μ∈Mg\documentclass[12pt]{minimal}
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\begin{document}$$\mu \in M_g$$\end{document} that satisfy μ(x,x)∈im(∂n+1⟨τSn+1⟩)\documentclass[12pt]{minimal}
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\begin{document}$$S^{n+1}$$\end{document} (see [29, §1B)]), so the kernel in question agrees with Mgsfr\documentclass[12pt]{minimal}
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\begin{document}$$M_g^{\mathrm {sfr}}$$\end{document} as claimed. To establish the last sequence, one first observes thatπ0DiffD2n(Vg)ℓ⟶Gg,ℓext\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\pi _0\mathrm {Diff}_{D^{2n}}(V_g)_\ell \longrightarrow G_{g,\ell }^{\mathrm {ext}}\end{aligned}$$\end{document}is surjective by definition of the target, and then compares the first two sequences in the claim to see that its kernel agrees with the kernel of Ngsfr→Mgsfr\documentclass[12pt]{minimal}
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\begin{document}$$N_g^{\mathrm {sfr}}\rightarrow M_g^{\mathrm {sfr}}$$\end{document}, which agrees with HVg∨⊗ker(SπnSO(n)→πnSO)\documentclass[12pt]{minimal}
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\begin{document}$$H_{V_g}^\vee \otimes \ker (S\pi _n\mathrm {SO}(n)\rightarrow \pi _n\mathrm {SO})$$\end{document} as already noted in the discussion prior to this proof. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Lemma 3.9
The negative of the identity -id∈GL(HWg)\documentclass[12pt]{minimal}
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\begin{document}$$-\mathrm {id}\in \mathrm {GL}(H_{W_g})$$\end{document} is contained in the subgroup Gg,ℓext⊂GL(HWg)\documentclass[12pt]{minimal}
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\begin{document}$$G_{g,\ell }^{\mathrm {ext}}\subset \mathrm {GL}(H_{W_g})$$\end{document} for all stable framings ℓ:τVgs→sfr∗γ\documentclass[12pt]{minimal}
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\begin{document}$$\ell :\tau _{V_g}^s\rightarrow \mathrm {sfr}^*\gamma $$\end{document}.
Proof
We first restrict to the case g=1\documentclass[12pt]{minimal}
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\begin{document}$$g=1$$\end{document} and a particular choice of stable framing, namely that induced by the standard embedding Dn+1×Sn⊂Rn+1×Rn+1\documentclass[12pt]{minimal}
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\begin{document}$$D^{n+1}\times S^n\subset \mathbf {R}^{n+1}\times \mathbf {R}^{n+1}$$\end{document}. The diffeomorphismDn+1×Sn⟶Dn+1×Sn[0.3em]((x1,…,xn+1),(y1,…,yn+1))⟼((-x1,x2,…,xn+1),(-y1,y2,…,yn+1))\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{array}{rcl} D^{n+1}\times S^n &{} \longrightarrow &{} D^{n+1}\times S^n \\ [0.3em] ((x_1,\ldots ,x_{n+1}),(y_1,\ldots ,y_{n+1})) &{} \longmapsto &{} ((-x_1,x_2,\ldots ,x_{n+1}),(-y_1,y_2,\ldots ,y_{n+1})) \end{array}\end{aligned}$$\end{document}is orientation preserving, maps to -id∈GL(HW1)\documentclass[12pt]{minimal}
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\begin{document}$$-\mathrm {id}\in \mathrm {GL}(H_{W_1})$$\end{document}, and has constant derivative, so it preserves the stable framing as required. It does not preserve a disc in the boundary, but by Lemma 3.1, we can rectify this by an isotopy. To extend this argument to higher genera, we take the boundary connected sum of two copies of this diffeomorphism using the fixed discs in the boundary to obtain a diffeomorphism of V2\documentclass[12pt]{minimal}
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\begin{document}$$V_2$$\end{document} whose image in GL(HW2)\documentclass[12pt]{minimal}
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\begin{document}$$-\mathrm {id}$$\end{document}, which we can again isotope so it preserves a disc as required. Continuing like this yields a sequence of isotopy classes in π0DiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_g)$$\end{document} for all g≥0\documentclass[12pt]{minimal}
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\begin{document}$$g\ge 0$$\end{document} that satisfy the requirements of the claim. This establishes the statement for one specific stable framing for each g, but implies the general case, the reason being that -id∈GL(HWg)\documentclass[12pt]{minimal}
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\begin{document}$$-\mathrm {id}\in \mathrm {GL}(H_{W_g})$$\end{document} is central and all subgroups Gg,ℓext⊂Ggext\documentclass[12pt]{minimal}
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\begin{document}$$G_{g,\ell }^{\mathrm {ext}}\subset G_g^{\mathrm {ext}}$$\end{document} are conjugate, because the different stabilisers π0DiffD2n(Vg)ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_g)_\ell $$\end{document} are, as the action is transitive by Proposition 3.7. This concludes the proof. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
The homotopy mapping class group
Not only the smooth mapping class groups π0DiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_\partial (W_{g,1})$$\end{document} act on the homology HWg,1\documentclass[12pt]{minimal}
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\begin{document}$$H_{W_{g,1}}$$\end{document}, also their homotopical cousins π0hAutD2n(Vg,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {hAut}_{D^{2n}} (V_g,W_{g,1})$$\end{document} and π0hAut∂(Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {hAut}_\partial (W_{g,1})$$\end{document} do. Restricting these actions to the imagesπ0hAutD2n≅(Vg,Wg,1)⊂π0hAutD2n(Vg,Wg,1)andπ0hAut∂≅(Wg,1)⊂π0hAut∂(Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _0\mathrm {hAut}^{\cong }_{D^{2n}}(V_g,W_{g,1})\subset & {} \pi _0\mathrm {hAut}_{D^{2n}}(V_g,W_{g,1})\quad \text { and }\\ \pi _0\mathrm {hAut}^{\cong }_\partial (W_{g,1})\subset & {} \pi _0\mathrm {hAut}_\partial (W_{g,1})\end{aligned}$$\end{document}of the canonical mapsπ0DiffD2n(Vg,Wg,1)→π0hAutD2n(Vg,Wg,1)andπ0Diff∂(Wg,1)→π0hAut∂(Wg,1),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _0\mathrm {Diff}_{D^{2n}}(V_g,W_{g,1})\rightarrow & {} \pi _0\mathrm {hAut}_{D^{2n}}(V_g,W_{g,1})\quad \text {and}\\ \pi _0\mathrm {Diff}_\partial (W_{g,1})\rightarrow & {} \pi _0\mathrm {hAut}_\partial (W_{g,1}),\end{aligned}$$\end{document}they land in the subgroups Ggext\documentclass[12pt]{minimal}
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\begin{document}$$G_g^{\mathrm {ext}}$$\end{document} and Gg\documentclass[12pt]{minimal}
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\begin{document}$$G_g$$\end{document} of GL(HWg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {GL}(H_{W_{g,1}})$$\end{document}, respectively.
Lemma 3.10
Let n≥3\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 3$$\end{document}. The morphisms induced by the action on HWg,1\documentclass[12pt]{minimal}
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\begin{document}$$H_{W_{g,1}}$$\end{document}π0hAut∂≅(Wg,1)⟶Ggandπ0hAutD2n≅(Vg,Wg,1)⟶Ggext\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\pi _0\mathrm {hAut}_{\partial }^{\cong }(W_{g,1})\longrightarrow G_g\quad \text { and }\quad \pi _0\mathrm {hAut}_{D^{2n}}^{\cong }(V_g,W_{g,1})\longrightarrow G_g^{\mathrm {ext}}\end{aligned}$$\end{document}are surjective. Moreover, their kernels are finite and p-torsion free as long as n<2p-4\documentclass[12pt]{minimal}
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\begin{document}$$n<2p-4$$\end{document}.
Proof
In the course of this proof, we shall make frequent use of the fact that the homotopy groups πn+k(∨hSn)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{n+k}(\vee ^hS^n)$$\end{document} with n≥3\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 3$$\end{document} are p-torsion free for k<2p-3\documentclass[12pt]{minimal}
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\begin{document}$$k<2p-3$$\end{document} as a result of the Hilton–Milnor theorem (see (37) below) and the case h=1\documentclass[12pt]{minimal}
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\begin{document}$$h=1$$\end{document} due to Serre [38, p. 498, Prop. 5].
To begin with the actual proof, note that the two morphisms of the statement are certainly surjective, since this holds already for π0Diff∂(Wg,1)→Gg\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{\partial }(W_{g,1})\rightarrow G_g$$\end{document} and π0Diff(Vg)→Ggext\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}(V_g)\rightarrow G_g^{\mathrm {ext}}$$\end{document} by Theorem 3.3 and the discussion it precedes. Evidently, the kernel of the first morphism of the statement is contained in that of π0hAut∂(Wg,1)→GL(HWg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {hAut}_{\partial }(W_{g,1})\rightarrow \mathrm {GL}(H_{W_{g,1}})$$\end{document}, which enjoys the claimed finiteness and torsion property by an application of the fibre sequence induced by restriction along the boundary inclusionhAut∂(Wg,1)⟶hAut∗(Wg,1)⟶Maps∗(∂Wg,1,Wg,1),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {hAut}_{\partial }(W_{g,1})\longrightarrow \mathrm {hAut}_{*}(W_{g,1})\longrightarrow \mathrm {Maps}_{*}(\partial W_{g,1},W_{g,1}),\end{aligned}$$\end{document}using the isomorphism π0hAut∗(Wg,1)≅GL(HWg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {hAut}_{*}(W_{g,1})\cong \mathrm {GL}(H_{W_{g,1}})$$\end{document} induced by the homology action and the fact that the group π1Maps∗(∂Wg,1,Wg,1)≅π2n(∨2gSn)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _1\mathrm {Maps}_{*}(\partial W_{g,1},W_{g,1})\cong \pi _{2n}(\vee ^{2g}S^n)$$\end{document} is finite and without p-torsion for n<2p-3\documentclass[12pt]{minimal}
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\begin{document}$$n<2p-3$$\end{document}. To deduce the claim for the kernel of second morphism from this, we consult the fibre sequence hAut∂(Vg)→hAutD2n(Vg,Wg,1)→hAut∂(Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hAut}_{\partial }(V_{g})\rightarrow \mathrm {hAut}_{D^{2n}}(V_{g},W_{g,1})\rightarrow \mathrm {hAut}_{\partial }(W_{g,1})$$\end{document} to realise that it suffices to show that π0hAut∂(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$n<2p-4$$\end{document}. Using the fibre sequence hAut∂(Vg)→hAut∗(Vg)→Maps(Wg,Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hAut}_{\partial }(V_{g})\rightarrow \mathrm {hAut}_{*}(V_{g})\rightarrow \mathrm {Maps}(W_g,V_g)$$\end{document} and the observation that its fibre inclusion is trivial on path components since π0hAut∂(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {hAut}_{\partial }(V_{g})$$\end{document} acts trivially on homology and π0hAut∗(Vg)≅GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {hAut}_{*}(V_{g})\cong \mathrm {GL}(H_{V_g})$$\end{document}, we see that π0hAut∂(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {hAut}_{\partial }(V_{g})$$\end{document} is a quotient of the fundamental group π1Maps∗(Wg,Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _1\mathrm {Maps}_*(W_g,V_g)$$\end{document} based at the inclusion ι\documentclass[12pt]{minimal}
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\begin{document}$$\iota $$\end{document}. Finally, note that the fibre sequence Maps∗(Wg,Vg)→Maps∗(Wg,1,Vg)→Maps∗(∂Wg,1,Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\pi _{2n+1}(\vee ^gS^n)^{\oplus g}\longrightarrow \pi _1(\mathrm {Maps}_*(W_g,V_g);\iota )\longrightarrow \pi _{n+1}(\vee ^gS^n)^{\oplus 2g}\end{aligned}$$\end{document}whose outer groups are finite and p-torsion free for n<2p-4\documentclass[12pt]{minimal}
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\begin{document}$$n<2p-4$$\end{document}, so we can conclude the claim. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Stabilisation
To relate the automorphism spaces of the handlebody Vg=♮gDn+1×Sn\documentclass[12pt]{minimal}
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\begin{document}$$V_g=\natural ^gD^{n+1}\times S^n$$\end{document} relative to the disc D2n⊂∂Vg\documentclass[12pt]{minimal}
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\begin{document}$$V_{g+1}$$\end{document}, it is convenient to modify Vg\documentclass[12pt]{minimal}
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\begin{document}$$V_g$$\end{document} by introducing codimension 2 corners at the boundary of the disc D2n⊂Vg\documentclass[12pt]{minimal}
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\begin{document}$$D^{2n}\subset V_g$$\end{document} in the boundary so that there is smooth boundary preserving embedding c:(-1,0]×D2n→Vg\documentclass[12pt]{minimal}
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\begin{document}$$c:(-1,0]\times D^{2n}\rightarrow V_g$$\end{document} whose restriction to {0}×D2n\documentclass[12pt]{minimal}
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\begin{document}$${\{0\}\times D^{2n}}$$\end{document} agrees with the chosen disc. Abusing common terminology, we call such an embedding a collar. Fixing another disc D2n-1⊂∂D2n\documentclass[12pt]{minimal}
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\begin{document}$$D^{2n-1}\subset \partial D^{2n}$$\end{document}, we considerH:=([0,1]×D2n)♮(Dn+1×Sn),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}H:=([0,1]\times D^{2n})\natural (D^{n+1}\times S^n),\end{aligned}$$\end{document}where the boundary connected sum is performed away from the unionD:={0,1}×D2n∪[0,1]×D2n-1⊂[0,1]×D2n,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}D:=\{0,1\}\times D^{2n}\cup [0,1]\times D^{2n-1}\subset [0,1]\times D^{2n},\end{aligned}$$\end{document}and think of Vg+1\documentclass[12pt]{minimal}
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\begin{document}$$V_{g+1}$$\end{document} as being obtained by gluing Vg\documentclass[12pt]{minimal}
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\begin{document}$$V_g$$\end{document} to H along the two collared discs D2n⊂Vg\documentclass[12pt]{minimal}
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\begin{document}$$D^{2n}\subset V_g$$\end{document} and {0}×D2n⊂H\documentclass[12pt]{minimal}
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\begin{document}$$\{0\}\times D^{2n}\subset H$$\end{document}, where we declare {1}×D2n⊂H⊂Vg+1\documentclass[12pt]{minimal}
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\begin{document}$$\{1\}\times D^{2n}\subset H\subset V_{g+1}$$\end{document} to the new distinguished disc in the boundary, which comes with a preferred collar. Given a tangential structure θ:B→BO(d)\documentclass[12pt]{minimal}
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\begin{document}$$\theta :B\rightarrow \mathrm {BO}(d)$$\end{document}, a choice of bundle map ℓ0:ε⊕τD2n→θ∗γ2n+1\documentclass[12pt]{minimal}
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\begin{document}$$\ell _0:\varepsilon \oplus \tau _{D^{2n}}\rightarrow \theta ^*\gamma _{2n+1}$$\end{document} induces canonical θ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document}-structures on τVg|D2n\documentclass[12pt]{minimal}
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\begin{document}$$\tau _{V_g}|_{D^{2n}}$$\end{document} and τVg+1|D2n\documentclass[12pt]{minimal}
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\begin{document}$$\tau _{V_{g+1}}|_{D^{2n}}$$\end{document} by using the fixed collars, and also one on τH|D\documentclass[12pt]{minimal}
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\begin{document}$$\tau _{H}|_{D}$$\end{document} by making use of the canonical trivialisation of τ[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$\tau _{[0,1]}$$\end{document}. With respect to these θ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document}-structures, generically denoted by ℓ0\documentclass[12pt]{minimal}
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\begin{document}$$\ell _0$$\end{document}, there is an evident gluing map for tangential structuresBunD2n(τVg,θ∗γ2n+1;ℓ0)×BunD(τH,θ∗γ2n+1;ℓ0)⟶BunD2n(τVg+1,θ2n+1∗γ;ℓ0)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\mathrm {Bun}_{D^{2n}}(\tau _{V_g},\theta ^*\gamma _{2n+1};\ell _0)\times \mathrm {Bun}_{D}(\tau _{H},\theta ^*\gamma _{2n+1};\ell _0)\longrightarrow \mathrm {Bun}_{D^{2n}}(\tau _{V_{g+1}},\theta _{2n+1}^*\gamma ;\ell _0)\end{aligned}$$\end{document} that is equivariant with respect to the gluing morphismDiffD2n(Vg)×DiffD(H)⟶DiffD2n(Vg+1)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {Diff}_{D^{2n}}(V_g)\times \mathrm {Diff}_{D}(H)\longrightarrow \mathrm {Diff}_{D^{2n}}(V_{g+1})\end{aligned}$$\end{document}for diffeomorphisms. Taking homotopy orbits, this induces a map of the formBDiffD2nθ(Vg;ℓ0)×BDiffDθ(H;ℓ0)⟶BDiffD2nθ(Vg+1;ℓ0)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {BDiff}_{D^{2n}}^\theta (V_g;\ell _0)\times \mathrm {BDiff}_{D}^\theta (H;\ell _0)\longrightarrow \mathrm {BDiff}_{D^{2n}}^\theta (V_{g+1};\ell _0)\end{aligned}$$\end{document}and hence a homotopy class of stabilisation mapsBDiffD2nθ(Vg;ℓ0)→BDiffD2nθ(Vg+1;ℓ0)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {BDiff}_{D^{2n}}^\theta (V_g;\ell _0)\rightarrow \mathrm {BDiff}_{D^{2n}}^\theta (V_{g+1};\ell _0)\end{aligned}$$\end{document}that in general depends on the choice of a component of BDiffDθ(H;ℓ0)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BDiff}_{D}^\theta (H;\ell _0)$$\end{document}, but not in any of the cases we shall be interested in, because of the following.
Lemma 3.11
If B is n-connected, then the space BDiffDθ(H;ℓ0)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BDiff}_{D}^\theta (H;\ell _0)$$\end{document} is nonempty and connected.
Proof
Up to smoothing corners, the pairs (H, D) and (V1,D2n)\documentclass[12pt]{minimal}
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\begin{document}$$(V_1,D^{2n})$$\end{document} are diffeomorphic, so there we have an equivalence BDiffDθ(H;ℓ0)≃BDiffD2nθ(V1;ℓ0)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BDiff}_{D}^\theta (H;\ell _0)\simeq \mathrm {BDiff}_{D^{2n}}^\theta (V_1;\ell _0)$$\end{document}, which shows that the claim is equivalent to the transitivity of the action of π0DiffD2n(V1)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_1)$$\end{document} on π0BunD2n(τV1,θ∗γ2n+1;ℓ0)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Bun}_{D^{2n}}(\tau _{V_1},\theta ^*\gamma _{2n+1};\ell _0)$$\end{document}, using that the latter set is nonempty as V1\documentclass[12pt]{minimal}
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\begin{document}$$V_1$$\end{document} is parallelizable. As B is n-connected, it is a consequence of obstruction theory that every θ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document}-structure on Vg\documentclass[12pt]{minimal}
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\begin{document}$$V_g$$\end{document} is induced by a framing, so it suffices to consider the case θ:EO(2n+1)→BO(2n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\theta :\mathrm {EO}(2n+1)\rightarrow \mathrm {BO}(2n+1)$$\end{document}, which we have already settled as the first part of Proposition 3.7 (see also Remark 3.6). □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
A similar discussion results in analogous stabilisation maps of the formBDiff~D2nΞ(Vg;ℓ0)⟶BDiff~D2nΞ(Vg+1;ℓ0)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {B\widetilde{Diff}}^\Xi _{D^{2n}}(V_g;\ell _0)\longrightarrow \mathrm {B\widetilde{Diff}}^\Xi _{D^{2n}}(V_{g+1};\ell _0)\end{aligned}$$\end{document}for stable tangential structures Ξ:B→BO\documentclass[12pt]{minimal}
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\begin{document}$$\ell _0:\tau _{V_g}^s\rightarrow \Xi ^*\gamma $$\end{document}, which are compatible with the non-block variants defined above and are again unique up to homotopy if B is n-connected. Moreover, using the splittings of the inclusions HWg,1→HWg+1,1\documentclass[12pt]{minimal}
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\begin{document}$$V_{g+1}=V_g\cup H$$\end{document}, there are stabilisation maps for the respective linear groups on HWg,1\documentclass[12pt]{minimal}
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\begin{document}$$H_{V_g}$$\end{document} given by extending automorphisms by the identity and these are related to the stabilisation maps described above via the action on the homology of Vg\documentclass[12pt]{minimal}
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\begin{document}$$W_{g,1}$$\end{document} (see Section 3.2), so there is a commutative diagram of compatible stabilisation maps that has the form
Relative homotopy automorphisms of handlebodies
Theorem 2.2 illustrates that the space of block diffeomorphisms Diff~D2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{\mathrm {Diff}}_{D^{2n}}(V_g)$$\end{document} is closely related to the space hAut~D2n(Vg,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {\widetilde{hAut}}_{D^{2n}}(V_g,W_{g,1})$$\end{document} of relative block homotopy automorphisms or, equivalently, to its non-block variant hAutD2n(Vg,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hAut}_{D^{2n}}(V_g,W_{g,1})$$\end{document} (see Section 1.5). To access the homology of the classifying space of this space of homotopy automorphisms, one might try to study the Serre spectral sequence of the fibration sequence induced by taking componentsBhAutD2nid(Vg,Wg,1)⟶BhAutD2n(Vg,Wg,1)⟶Bπ0hAutD2n(Vg,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {BhAut}^{\mathrm {id}}_{D^{2n}}(V_g,W_{g,1})\longrightarrow \mathrm {BhAut}_{D^{2n}}(V_g,W_{g,1})\longrightarrow \mathrm {B}\pi _0\mathrm {hAut}_{D^{2n}}(V_g,W_{g,1})\end{aligned}$$\end{document}for which one ought to know at least the homology of the fibre as a module over the group π0hAutD2n(Vg,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {hAut}_{D^{2n}}(V_g,W_{g,1})$$\end{document}. This is what this section aims to compute—p-locally and in a range of degrees—by first calculating the p-local homotopy groups in a range using some tools from rational homotopy theory combined with an ad-hoc extension to the p-local setting tailored to our situation, and then pass from homotopy to homology groups.
Conventions on gradings
Essentially all objects in this section carry a Z\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {Z}$$\end{document}-grading, and we shall keep track of it throughout. For instance, we consider the (reduced) homology of a space X always with its natural grading, even if it is supported in a single degree.
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\begin{document}$$A\otimes B$$\end{document} is defined in the usual way. Note that (skA)⊗B=sk(A⊗B)=A⊗(skB)\documentclass[12pt]{minimal}
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\begin{document}$$A_{(p)}$$\end{document} respectively, and we view it as a graded Q\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {Q}$$\end{document}- respectively Z(p)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {Z}_{(p)}$$\end{document}-module.
Lie algebras and their derivations
We consider differential graded (short dg) Lie algebras over a commutative ring R. However, most of the dg Lie algebras which we shall encounter actually have trivial differential. Examples include the free graded Lie algebra L(V)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {L}(V)$$\end{document} on a graded R-module A or the onefold shift of the homotopy groups π∗+1X\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{*+1}X$$\end{document} of a based space X with its canonical Lie algebra structure over Z\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {Z}$$\end{document} given by the Whitehead bracket (except for a 2-torsion subtlety that will not play a role for us). Given a dg Lie algebra L, we write [L,L]⊂L\documentclass[12pt]{minimal}
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\begin{document}$$[L,L]\subset L$$\end{document} for the graded subspace generated by brackets. An important principle in this section is that the homotopy type of mapping spaces is closely related to certain chain complexes of f-derivations by which we mean the following: for a morphism f:(L,dL)→(L′,d′)\documentclass[12pt]{minimal}
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\begin{document}$$f:(L,d_L)\rightarrow (L',d')$$\end{document} of dg Lie algebras, an f-derivation of degreek is a linear map θ:L→L′\documentclass[12pt]{minimal}
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\begin{document}$$\theta :L\rightarrow L'$$\end{document} that raises the degree by k and satisfiesθ([x,y])=[θ(x),f(y)]+(-1)k|x|[f(x),θ(y].\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\theta ([x,y])=[\theta (x),f(y)]+(-1)^{k|x|}[f(x),\theta (y].\end{aligned}$$\end{document}These derivations form the degree k piece of the chain complex Derf(L,L′)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Der}^f(L,L')$$\end{document} of f-derivations over R whose differential is defined as d(θ)=dL′θ-(-1)|θ|θdL\documentclass[12pt]{minimal}
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\begin{document}$$d(\theta )=d_{L'}\theta -(-1)^{|\theta |}\theta d_L$$\end{document}, so it vanishes if both L and L′\documentclass[12pt]{minimal}
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\begin{document}$$L'$$\end{document} have trivial differential. Given a cycle ω∈L\documentclass[12pt]{minimal}
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\begin{document}$$\omega \in L$$\end{document}, we denote the subcomplex of f-derivations that vanish on ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} by Derωf(L,L′)⊂Derf(L,L′)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Der}^f_{\omega }(L,L')\subset \mathrm {Der}^f(L,L')$$\end{document}. In the case L=L′\documentclass[12pt]{minimal}
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\begin{document}$$L=L'$$\end{document} and f=id\documentclass[12pt]{minimal}
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\begin{document}$$f=\mathrm {id}$$\end{document}, we abbreviate the complex of id\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {id}$$\end{document}-derivations Derid(L,L)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Der}^{\mathrm {id}}(L,L)$$\end{document} by Der(L)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Der}(L)$$\end{document}.
Rational homotopy theory, Quillen style
Recall from [33] Quillen’s functor λ\documentclass[12pt]{minimal}
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\begin{document}$$\lambda $$\end{document}, which assigns a simply connected based space X a dg Lie algebra λ(X)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda (X)$$\end{document} over the rationals, one of whose many properties is that it captures the rationalised homotopy Lie algebra of X via a natural isomorphism H∗(λ(X))≅π∗+1(X)Q\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {H}_*(\lambda (X))\cong \pi _{*+1}(X)_\mathbf {Q}$$\end{document} of graded Lie algebras, where H∗(λ(X))=ker(dλ(X))/im(dλ(X))\documentclass[12pt]{minimal}
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\begin{document}$$\lambda (X)$$\end{document}. A Lie model of X is a rational dg Lie algebra LQX\documentclass[12pt]{minimal}
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\begin{document}$$L_\mathbf {Q}^X$$\end{document} quasi-isomorphic to λ(X)\documentclass[12pt]{minimal}
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\begin{document}$$\lambda (X)$$\end{document}. Such a model is called free if the underlying graded Lie algebra of LQX\documentclass[12pt]{minimal}
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\begin{document}$$L_\mathbf {Q}^X$$\end{document} is isomorphic to a free graded Lie algebra L(V)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {L}(V)$$\end{document} on a graded Q\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {Q}$$\end{document}-vector space V and minimal if it is free and has decomposable differential, i.e. d(LQX)⊂[LQX,LQX]\documentclass[12pt]{minimal}
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\begin{document}$$d(L_\mathbf {Q}^X)\subset [L_\mathbf {Q}^X,L_\mathbf {Q}^X]$$\end{document}. Any simply connected based space has a minimal Lie model LQX\documentclass[12pt]{minimal}
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\begin{document}$$L_\mathbf {Q}^X$$\end{document}, unique up to (non-canonical) isomorphism, and a based map between such spaces f:X→Y\documentclass[12pt]{minimal}
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\begin{document}$$f:X\rightarrow Y$$\end{document} gives rise to a map f:LQX→LQY\documentclass[12pt]{minimal}
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\begin{document}$$f:L_\mathbf {Q}^X \rightarrow L_\mathbf {Q}^Y$$\end{document} between their minimal models.
Derivations and mapping spaces
As mentioned earlier, the homotopy theory of mapping spaces is tightly connected to derivations of dg Lie algebras. In the rational setting, this is made precise for instance by a result of Lupton–Smith [30, Thm 3.1]. The version of their result we shall need is marginally stronger than stated in [30], but follows from the given proof in a straight-forward way (see also [8, Thm 3.6]).
Theorem 4.1
(Lupton–Smith) Let f:X→Y\documentclass[12pt]{minimal}
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\begin{document}$$f:X\rightarrow Y$$\end{document} be a map between simply connected finite based CW-complexes, with minimal Lie model f:LQX→LQY\documentclass[12pt]{minimal}
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\begin{document}$$f:L_\mathbf {Q}^X\rightarrow L_\mathbf {Q}^Y$$\end{document}. There is an isomorphismπ∗(Maps∗(X,Y);f)Q⟶≅H∗(Derf(LQX,LQY))\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\pi _*(\mathrm {Maps}_*(X,Y);f)_\mathbf {Q}\overset{\cong }{\longrightarrow }\mathrm {H}_*(\mathrm {Der}^{f}(L_\mathbf {Q}^X,L_\mathbf {Q}^Y))\end{aligned}$$\end{document}for ∗≥2\documentclass[12pt]{minimal}
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\begin{document}$$*\ge 2$$\end{document}, which is natural in both X and Y. For X a co-H-space, this also holds for ∗=1\documentclass[12pt]{minimal}
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\begin{document}$$*=1$$\end{document}.
A p-local generalisation
From the point of view of Quillen’s approach to rational homotopy theory, the spaces we shall be applying Theorem 4.1 to are of the simplest nature possible: they are homotopy equivalent to boquets of equidimensional spheres. The minimal model of such a space X≃∨gSn\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 2$$\end{document} agrees with the free graded Lie algebraLQX:=L(s-1HQX)≅π∗+1XQonHQX:=H~∗(X;Q),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} L_\mathbf {Q}^{X}:=\mathbf {L}(s^{-1}H_\mathbf {Q}^X)\cong \pi _{*+1}X_\mathbf {Q}\quad \text {on}\quad H_\mathbf {Q}^X:=\widetilde{\mathrm {H}}_*(X;\mathbf {Q}),\end{aligned}$$\end{document}equipped with the trivial differential. Given a map between spaces of this kind, the induced map on minimals models is simply given by the induced map on rational homotopy groups. It is a consequence of the Hilton–Milnor theorem that, p-locally in small degrees with respect to p, the homotopy Lie algebra π∗+1X\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{*+1}X$$\end{document} is free even before rationalisation. To make this precise, we abbreviate the integral and p-local analogue of (36) byLX:=L(s-1HX)andL(p)X:=L(s-1H(p)X)whereHX:=H~∗(X;Z)andH(p)X:=H~∗(X;Z(p)).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&L^X:=\mathbf {L}(s^{-1}H^{X})\text { and }L_{(p)}^X:=\mathbf {L}(s^{-1}H_{(p)}^{X})\text { where }\\&H^X:=\widetilde{\mathrm {H}}_*(X;\mathbf {Z})\text { and }H^X_{(p)}:=\widetilde{\mathrm {H}}_*(X;\mathbf {Z}_{(p)}).\end{aligned}$$\end{document}The inverse of the Hurewicz map Hn(X;Z(p))≅πnX(p)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {H}_n(X;\mathbf {Z}_{(p)})\cong \pi _nX_{(p)}$$\end{document} induces a map L(p)X→π∗+1X(p)\documentclass[12pt]{minimal}
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\begin{document}$$L_{(p)}^X\rightarrow \pi _{*+1}X_{(p)}$$\end{document} of graded Lie algebras over Z(p)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {Z}_{(p)}$$\end{document}, which turns out to be an isomorphism in a range of degrees.
Lemma 4.2
For an odd prime p and a based space X that is homotopy equivalent to ∨gSn\documentclass[12pt]{minimal}
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\begin{document}$$ \vee ^gS^n$$\end{document} with n≥2\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 2$$\end{document}, the morphismL(p)X⟶π∗+1X(p)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}L_{(p)}^X\longrightarrow \pi _{*+1}X_{(p)}\end{aligned}$$\end{document}is an isomorphism on torsion free quotients. Moreover, the right hand side is torsion free in degrees ∗<2p-4+n\documentclass[12pt]{minimal}
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\begin{document}$$*<2p-4+n$$\end{document}, so the map is an isomorphism in this range.
Proof
As a preparation to the proof, note that by specialising the Hilton–Milnor theorem to ∨gSn\documentclass[12pt]{minimal}
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\begin{document}$$\vee ^gS^n$$\end{document}, we have an isomorphismπi+1(∨gSn)≅⨁ω∈Lgπi+1(Sl(ω)(n-1)+1)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \textstyle {\pi _{i+1}(\vee ^gS^n)\cong \bigoplus _{\omega \in L_g}\pi _{i+1}(S^{l(\omega )(n-1)+1})}\end{aligned}$$\end{document}where Lg\documentclass[12pt]{minimal}
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\begin{document}$$l(\omega )$$\end{document} is the word-length of ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}. Here the map πi+1(Sl(ω)(n-1)+1)→πi+1(∨gSn)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{i+1}(S^{l(\omega )(n-1)+1})\rightarrow \pi _{i+1}(\vee ^gS^n)$$\end{document} corresponding to ω∈Lg\documentclass[12pt]{minimal}
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\begin{document}$$1\le i\le g$$\end{document} represented by the inclusions of the summands as guided by the Lie word ω∈Lg\documentclass[12pt]{minimal}
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\begin{document}$$\omega \in L_g$$\end{document}. A proof can be extracted from [47]: combine XI.6.6 and the subsequent discussion with VII.2.6 and X.7.10.
To prove the asserted claim, we use that source and domain of the morphism in the statement are both degreewise finitely generated and that the rationalisation of this morphism agrees with (36), so to prove the first part of the claim, it suffices to show that all classes in π∗+1X(p)\documentclass[12pt]{minimal}
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\begin{document}$$L_{(p)}^X\rightarrow \pi _{*+1}X_{(p)}$$\end{document} is closed under taking brackets, this implies the first part of the claim. The second part follows from Serre’s result [38, p. 498, Prop. 5] that πkSm\documentclass[12pt]{minimal}
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\begin{document}$$\pi _kS^m$$\end{document} is p-torsion free for k-m<2p-3\documentclass[12pt]{minimal}
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\begin{document}$$k-m<2p-3$$\end{document} together with another application of (37). □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
As a result of Lemma 4.2, every map f:X→Y\documentclass[12pt]{minimal}
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\begin{document}$$f:X\rightarrow Y$$\end{document} between bouquets of equidimensional spheres induces a morphism f∗:L(p)X→L(p)Y\documentclass[12pt]{minimal}
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\begin{document}$$f_*:L^X_{(p)}\rightarrow L^Y_{(p)}$$\end{document} by taking torsion free quotients of p-local homotopy groups, so the following extension of Theorem 4.1 might not come as a surprise.
Proposition 4.3
For an odd prime p and a map f:X→Y\documentclass[12pt]{minimal}
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\begin{document}$$f:X\rightarrow Y$$\end{document} between based spaces X≃∨gSn\documentclass[12pt]{minimal}
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\begin{document}$$n,m\ge 2$$\end{document} , the map of Theorem 4.1 fits into a commutative square
which is natural in X and Y and whose upper arrow is an isomorphism for ∗<2p-3-(n-m)\documentclass[12pt]{minimal}
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\begin{document}$$*<2p-3-(n-m)$$\end{document}.
Remark 4.4
Dwyer’s tame homotopy theory [13] provides a p-local generalisation of Quillen’s rational homotopy theory for primes p that are just large enough with respect to the degree to prevent stable k-invariants from appearing. It is not unlikely that Theorem 4.1 could be generalised to this setting, but our layman extension Proposition 4.3 for bouquets of spheres suffices for the applications we have in mind.
Proof of Proposition 4.3
We begin with a twofold simplification of the statement. Firstly, the claimed naturality is automatic, since the vertical maps are evidently natural, the bottom map is natural by Theorem 4.1, and the right vertical map is injective, so it suffices to construct a top arrow with the desired properties for X=∨gSn\documentclass[12pt]{minimal}
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\begin{document}$$X=\vee ^gS^n$$\end{document}. Secondly, there is a commutative diagram
induced by restricting derivations to generators, which shows that it is enough to produce a dashed arrow making the diagramcommute. To do so, we consider the compositionπ∗(Maps∗(X,Y);f)→≅(-f)∗π∗(Maps∗(X,Y);∗)≅Hom(s-1HX,π∗+1Y)+\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _*(\mathrm {Maps}_*(X,Y);f)\xrightarrow [\cong ]{(-f)_*}\pi _*(\mathrm {Maps}_*(X,Y);*)\cong \mathrm {Hom}(s^{-1}H^X,\pi _{*+1}Y)^+\end{aligned}$$\end{document} whose first isomorphism is given by acting with the inverse of f, using the loop space structure on Maps∗(X,Y)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Maps}_*(X,Y)$$\end{document}, and whose second isomorphism is induced by mapping a class in πk(Maps∗(X,Y);∗)\documentclass[12pt]{minimal}
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\begin{document}$$ \pi _k(\mathrm {Maps}_*(X,Y);*)$$\end{document} represented by a pointed map g:Sk→Maps∗(X,Y)\documentclass[12pt]{minimal}
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\begin{document}$$g:S^k\rightarrow \mathrm {Maps}_*(X,Y)$$\end{document} to the compositionHn(X)≅Hn+k(Sk∧X)≅πn+k(Sk∧X)⟶g∗πn+k(Y),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {H}_n(X)\cong \mathrm {H}_{n+k}(S^k\wedge X)\cong \pi _{n+k}(S^k\wedge X)\overset{g_*}{\longrightarrow }\pi _{n+k}(Y),\end{aligned}$$\end{document}involving the suspension isomorphism, the inverse of the Hurewicz map, and the adjoint of g. Postcomposing (39) with the map given by p-localising and taking torsion free quotients results by Lemma 4.2 in a dashed map with the claimed connectivity property, so we are left to show that this choice does make the diagram (38) commute, i.e. that the rationalisation of (39) agrees with the bottom map of (38). The adjoint of a map h:Sk→Maps∗(X,Y)\documentclass[12pt]{minimal}
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\begin{document}$$h:S^k\rightarrow \mathrm {Maps}_*(X,Y)$$\end{document} representing a class in π∗(Maps∗(X,Y);f)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _*(\mathrm {Maps}_*(X,Y);f)$$\end{document} forms the top arrow of the diagramwhose middle diagonal arrow is induced by h via the canonical homeomorphism (Sk×X)/(Sk∨∗)≅S+k∧X\documentclass[12pt]{minimal}
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\begin{document}$$(S^k\times X)/(S^k\vee *)\cong S^k_+\wedge X$$\end{document} and whose vertical equivalence is given as the compositionS+k∧X→idS+k∧∇S+k∧X∨S+k∧X→cX∨Sk∧X\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}S^k_+\wedge X\xrightarrow {\mathrm {id}_{S^k_+}\wedge \nabla } S^k_+\wedge X\vee S^k_+ \wedge X\xrightarrow {c}X\vee S^k \wedge X\end{aligned}$$\end{document}using the co-H-space structure ∇\documentclass[12pt]{minimal}
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\begin{document}$$\nabla $$\end{document} of X and the evident collapse map c. The map (-f)∗(h)\documentclass[12pt]{minimal}
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\begin{document}$$(-f)_*(h)$$\end{document} is the adjoint of a representative of the image of h under the first map in (39), so (40) commutes up to changing h within its class in πk(Maps∗(X,Y);f)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _k(\mathrm {Maps}_*(X,Y);f)$$\end{document}. We thus obtain a rational model for the top arrow in (40) as the compositionLs-1HQX⊕s-1+kHQX⊕s-1HQSk,d⟶LQSk∨Sk∧X→π∗+1(f∨(-f)∗(g))⊗QLQY,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned}&\left( \mathbf {L}\left( s^{-1}H^X_{\mathbf {Q}}\oplus s^{-1+k}H_\mathbf {Q}^X\oplus s^{-1}H_\mathbf {Q}^{S^k}\right) ,d \right) \\ {}&\quad \longrightarrow L_\mathbf {Q}^{S^k\vee S^k\wedge X}\xrightarrow {\pi _{*+1}(f\vee (-f)_*(g))\otimes \mathbf {Q}} L_{\mathbf {Q}}^Y,\end{aligned}\end{aligned}$$\end{document}where the source is the Lie model of Sk×X\documentclass[12pt]{minimal}
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\begin{document}$$S^k\times X$$\end{document} described in [30, Cor. 2.2], i.e. its differential d is trivial on s-1HQX⊕s-1HQSk\documentclass[12pt]{minimal}
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\begin{document}$$s^{-1}H^X_{\mathbf {Q}}\oplus s^{-1}H_\mathbf {Q}^{S^k}$$\end{document} and is on s-1+kHQX\documentclass[12pt]{minimal}
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\begin{document}$$s^{-1+k}H_\mathbf {Q}^X$$\end{document} given ass-1+kHQX≅s-1HQX→(-1)k-1[z,-]Ls-1HQX⊕s-1+kHQX⊕s-1HQSk\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} s^{-1+k}H_\mathbf {Q}^X\cong s^{-1}H^X_{\mathbf {Q}}\xrightarrow {(-1)^{k-1}[z,-]}\mathbf {L}\left( s^{-1}H^X_{\mathbf {Q}}\oplus s^{-1+k}H_\mathbf {Q}^X\oplus s^{-1}H_\mathbf {Q}^{S^k}\right) \end{aligned}$$\end{document}where the first isomorphism is the canonical identification as ungraded vector spaces induced by the identity and z∈s-1HQSk\documentclass[12pt]{minimal}
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\begin{document}$$z\in s^{-1}H_\mathbf {Q}^{S^k}$$\end{document} denotes the standard generator. The first map in the composition (41) takes the quotient by the dg Lie ideal generated by the subspace s-1HQSk\documentclass[12pt]{minimal}
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\begin{document}$$s^{-1}H_\mathbf {Q}^{S^k}$$\end{document} and the second map is defined as indicated. Using this particular choice of rational model in the definition of the isomorphism of Theorem 4.1 in [30, p. 176–177], the image of the class defined by h under the bottom horizontal composition in (38) is precisely its image under (39) after rationalisation, so the claim follows. □\documentclass[12pt]{minimal}
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The theory set up in the previous paragraphs will allow us to compute the p-local homotopy groups of the classifying space BhAutD2nid(Vg,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BhAut}^{\mathrm {id}}_{D^{2n}}(V_g,W_{g,1})$$\end{document} of the identity component of the topological monoid of relative homotopy automorphisms as defined in Sect. 3.1 as a module over the groupπ1BhAutD2n(Vg,Wg,1)≅π0hAutD2n(Vg,Wg,1).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\pi _1\mathrm {BhAut}_{D^{2n}}(V_g,W_{g,1})\cong \pi _0\mathrm {hAut}_{D^{2n}}(V_g,W_{g,1}).\end{aligned}$$\end{document}More generally, we will compute the p-local homotopy groups of the spaces participating in the fibration sequence (see Section 3.1 for the notation)BhAut∂(Vg)⟶BhAutD2n(Vg,Wg,1)⟶BhAut∂ext(Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {BhAut}_{\partial }(V_g)\longrightarrow \mathrm {BhAut}_{D^{2n}}(V_g,W_{g,1})\longrightarrow \mathrm {BhAut}^{\mathrm {ext}}_{\partial }(W_{g,1}) \end{aligned}$$\end{document}induced by restriction, together with the induced action of π0hAutD2n(Vg,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\partial W_{g,1}$$\end{document} involved, which are homotopy equivalent to bouquets of spheres. We do however omit the g-superscripts to increase readability, so writeH(p)W=H~∗(Wg,1;Z(p)),L(p)W=L(s-1H(p)W),H(p)V=H~∗(Vg;Z(p)),andL(p)V=L(s-1H(p)V)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} H_{(p)}^W= & {} \widetilde{\mathrm {H}}_*(W_{g,1};\mathbf {Z}_{(p)}), L_{(p)}^W=\mathbf {L}(s^{-1}H_{(p)}^W),\\ H_{(p)}^V= & {} \widetilde{\mathrm {H}}_*(V_{g};\mathbf {Z}_{(p)}),\quad \text {and} \quad L_{(p)}^V=\mathbf {L}(s^{-1}H_{(p)}^V) \end{aligned}$$\end{document}and omit the (p)-subscripts to denote the integral analogues. Moreover, we generically write ι\documentclass[12pt]{minimal}
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\begin{document}$$\partial W_{g,1}\subset V_g$$\end{document} is trivial, since it factors over Wg=Wg,1∪∂Wg,1D2n\documentclass[12pt]{minimal}
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\begin{document}$$W_g=W_{g,1}\cup _{\partial W_{g,1}}D^{2n}$$\end{document}.
Remark 4.5
Note that HW\documentclass[12pt]{minimal}
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\begin{document}$$V_g$$\end{document}, which shall not be confused with the ungraded middle dimensional integral homology groups of these spaces that featured in Section 3.2.1 as HWg,1\documentclass[12pt]{minimal}
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\begin{document}$$H_{V_g}$$\end{document}.
Theorem 4.6
Let n≥2\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 2$$\end{document} and p an odd prime.
The inclusion π0Maps∂(Vg,Vg)⊂π0hAut∂(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{0}\mathrm {Maps}_{\partial }(V_{g},V_g)\subset \pi _{0}\mathrm {hAut}_{\partial }(V_{g})$$\end{document} is an equality. This group is abelian.
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\begin{document}$$0<*<2p-3-n$$\end{document}, the boundary map of the fibration (42) fits into a commutative diagram of graded Z(p)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {Z}_{(p)}$$\end{document}-modules with exact rows
which is π0hAutD2n(Vg,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {hAut}_{D^{2n}}(V_g,W_{g,1})$$\end{document}-equivariant with respect to the action on the leftmost column induced by (42) and by the action through HW\documentclass[12pt]{minimal}
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\begin{document}$$H^{V}$$\end{document} on the other columns.
Rationally, the conclusion of (ii) holds in all positive degrees.
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\begin{document}$$\partial $$\end{document} of the fibration (42) is p-locally split surjective in a range and we conclude the following.
Corollary 4.7
Let n≥2\documentclass[12pt]{minimal}
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\begin{document}$$0<*<2p-4-n$$\end{document}, the graded Z(p)[π0hAutD2n(Vg,Wg,1)]\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {Z}_{(p)}[\pi _0\mathrm {hAut}_{D^{2n}}(V_g,W_{g,1})]$$\end{document}-module π∗+1BhAutD2n(Vg,Wg,1)(p)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _{*+1}\mathrm {BhAut}_{D^{2n}}(V_g,W_{g,1})_{(p)}$$\end{document} is isomorphic to the common kernel of the mapss-(2n-1)H(p)W⊗L(p)W⟶-,-s-(2n-2)[L(p)W,L(p)W]ands-(2n-1)H(p)W⊗L(p)W⟶ι∗⊗ι∗s-(2n-1)H(p)V⊗L(p)V.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&s^{-(2n-1)}H^{W}_{(p)}\otimes L_{(p)}^{W}\overset{\left[ -,-\right] }{\longrightarrow } s^{-(2n-2)}[L_{(p)}^{W}, L_{(p)}^{W}] \quad \text {and}\\&s^{-(2n-1)}H^{W}_{(p)}\otimes L_{(p)}^{W}\overset{\iota _*\otimes \iota _*}{\longrightarrow }s^{-(2n-1)}H^{V}_{(p)}\otimes L_{(p)}^{V}.\end{aligned}$$\end{document}Rationally, this holds in all positive degrees.
In particular, Theorem 4.6 and Corollary 4.7 imply that the π0hAutD2n(Vg,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {hAut}_{D^{2n}}(V_g,W_{g,1})$$\end{document}-action on the p-local higher homotopy groups of the spaces participating in (42) factors in a range of degrees through the morphism (recall Kg=ker(Hn(Wg,1)→Hn(Vg))\documentclass[12pt]{minimal}
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\begin{document}$$K_g=\ker (\mathrm {H}_n(W_{g,1})\rightarrow \mathrm {H}_n(V_g))$$\end{document} from Section 3.2.1)π0hAutD2n(Vg,Wg,1)⟶{ϕ∈GL(HWg,1)∣ϕ(Kg)⊂Kg}\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\pi _0\mathrm {hAut}_{D^{2n}}(V_g,W_{g,1})\longrightarrow \{\phi \in \mathrm {GL}(H_{W_{g,1}})\mid \phi (K_g)\subset K_g\}\end{aligned}$$\end{document}induced by the action on the homology of Wg,1\documentclass[12pt]{minimal}
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\begin{document}$$W_{g,1}$$\end{document}. During the proof of Theorem 4.6 and the preceding Lemma 4.8, we frequently use Proposition 4.3 to implicitly identify p-local homotopy groups of path components of pointed mapping spaces between bouquets of equidimensional spheres with derivations of free graded Lie algebras in a range of degrees. We denote by Maps∗f(X,Y)\documentclass[12pt]{minimal}
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\begin{document}$$f:X\rightarrow Y$$\end{document} the corresponding component of the mapping space, pointed by f. Reminding the reader of our notation for spaces of derivations in Section 4.2, we begin with the following lemma, whose first part is rationally due to Berglund and Madsen [8, Prop. 5.6].
Lemma 4.8
Let n≥2\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 2$$\end{document} and p an odd prime.
In degrees ∗<2p-3-n\documentclass[12pt]{minimal}
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\begin{document}$$*<2p-3-n$$\end{document}, the morphism induced by relaxing the boundary condition π∗Maps∂id(Wg,1,Wg,1)(p)⟶π∗Maps∗id(Wg,1,Wg,1)(p)≅Der(L(p)W)+\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\pi _*\mathrm {Maps}_\partial ^{\mathrm {id}}(W_{g,1},W_{g,1})_{(p)}\longrightarrow \pi _*\mathrm {Maps}_*^{\mathrm {id}}(W_{g,1},W_{g,1})_{(p)}\cong \mathrm {Der}(L_{(p)}^{W})^+\end{aligned}$$\end{document} is injective and has image Derω(L(p)W)+⊂Der(L(p)W)+\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Der}_{\omega }(L_{(p)}^{W})^+\subset \mathrm {Der}(L_{(p)}^{W})^+$$\end{document}.
In the range ∗<2p-3-n\documentclass[12pt]{minimal}
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\begin{document}$$W_{g,1}\subset W_g$$\end{document}π∗Maps∗ι(Wg,Vg)(p)⟶π∗Maps∗ι(Wg,1,Vg)(p)≅Derι(L(p)W,L(p)V)+\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\pi _*\mathrm {Maps}_*^{\iota }(W_{g},V_g)_{(p)}\longrightarrow \pi _*\mathrm {Maps}_*^{\iota }(W_{g,1},V_g)_{(p)}\cong \mathrm {Der}^{\iota }(L_{(p)}^{W},L_{(p)}^{V})^+\end{aligned}$$\end{document} is injective and has image Derωι(L(p)W,L(p)V)+⊂Derι(L(p)W,L(p)V)+\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Der}^{\iota }_{\omega }(L_{(p)}^{W},L_{(p)}^{V})^+\subset \mathrm {Der}^{\iota }(L_{(p)}^{W},L_{(p)}^{V})^+$$\end{document}.
Proof
Restriction along the inclusion ∂Wg,1⊂Wg,1\documentclass[12pt]{minimal}
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\begin{document}$$\partial W_{g,1}\subset W_{g,1}$$\end{document} yields a fibrationMaps∗(Wg,1,Wg,1)⟶Maps∗(∂Wg,1,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {Maps}_*(W_{g,1},W_{g,1})\longrightarrow \mathrm {Maps}_*(\partial W_{g,1},W_{g,1})\end{aligned}$$\end{document}whose fibre at ι\documentclass[12pt]{minimal}
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\begin{document}$$\iota $$\end{document} is Maps∂(Wg,1,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Maps}_\partial (W_{g,1},W_{g,1})$$\end{document}. The induced maps on homotopy groups fits in the range 0<∗<2p-2-n\documentclass[12pt]{minimal}
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\begin{document}$$0<*<2p-2-n$$\end{document} into a diagram of the formwhose top square is provided by Proposition 4.3, so commutes. The bottom square is given as follows: the bottom right vertical map is the evaluation at the fundamental class∈s-1H(p)∂Wg,1≅Q[2n-2],\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\in s^{-1}H_{(p)}^{\partial W_{g,1}}\cong \mathbf {Q}[2n-2],\end{aligned}$$\end{document}which factors as a composition of isomorphismsDerι(L(p)∂Wg,1,L(p)W)+⟶≅Hom(s-1H(p)∂Wg,1,L(p)W)+⟶≅(s-(2n-2)L(p)W)+=(s-(2n-2)[L(p)W,L(p)W])+\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {Der}^{\iota }(L_{(p)}^{\partial W_{g,1}},L_{(p)}^{W})^+&\overset{\cong }{\longrightarrow }&\mathrm {Hom}(s^{-1}H_{(p)}^{\partial W_{g,1}}, L_{(p)}^{W})^+\\&\overset{\cong }{\longrightarrow }&\big (s^{-(2n-2)}L_{(p)}^{W}\big )^+=\big (s^{-(2n-2)}{[L_{(p)}^{W},L_{(p)}^{W}]}\big )^+\end{aligned}$$\end{document}where the first isomorphism restricts to generators, the second isomorphism evaluates at the fundamental class, and the final equality holds for degree reasons. The latter is because elements of degree >0\documentclass[12pt]{minimal}
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\begin{document}$$n-1$$\end{document} as HW\documentclass[12pt]{minimal}
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\begin{document}$$H_W$$\end{document} is supported in degree n. The bottom left vertical map is the restriction to positive degrees of the mapDer(L(p)W)⟶≅Hom(s-1H(p)W,L(p)W)≅L(p)W⊗(s-1H(p)W)∨≅s-(2n-1)L(p)W⊗H(p)W,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned}\mathrm {Der}(L_{(p)}^{W})\overset{\cong }{\longrightarrow }\mathrm {Hom}(s^{-1}H_{(p)}^{W},L_{(p)}^{W})\cong&L_{(p)}^{W}\otimes (s^{-1}H_{(p)}^{W})^{\vee }\\\cong&s^{-(2n-1)}L_{(p)}^{W}\otimes H_{(p)}^{W}, \end{aligned} \end{aligned}$$\end{document}where the first isomorphism is given by restricting to generators, the second is the canonical one, and the third is induced by the intersection form on s-1HWg,1\documentclass[12pt]{minimal}
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\begin{document}$$s^{-1}H_{W_{g,1}}$$\end{document} (see Example B.1). By construction, the compositionDer(L(p)W)+⟶(s-(2n-2)[L(p)W,L(p)W])+\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {Der}(L_{(p)}^{W})^+\longrightarrow \big (s^{-(2n-2)}[L_{(p)}^{W},L_{(p)}^{W}]\big )^+\end{aligned}$$\end{document}coincides with the evaluation at the class ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} that represents the inclusion ∂Wg,1⊂Wg,1\documentclass[12pt]{minimal}
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\begin{document}$$\partial W_{g,1}\subset W_{g,1}$$\end{document}, so it follows from Lemma 1 that this square commutes up to a sign, since ω=∑i=1g[ei,fi]∈LWg,1\documentclass[12pt]{minimal}
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\begin{document}$$\omega =\sum _{i=1}^{g}[e_i,f_i]\in L_{W_{g,1}}$$\end{document} agrees up to a sign with the element (76) from the appendix as ei#=fi\documentclass[12pt]{minimal}
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\begin{document}$$e_i^{\#}= f_i$$\end{document} and fi#=ei\documentclass[12pt]{minimal}
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\begin{document}$$f_i^{\#}= e_i$$\end{document} holds in the notation of the appendix up to a fixed sign depending on n (which does not play a role in the argument). As the bottom horizontal map is surjective as a consequence of the graded Jacobi identity, the middle horizontal arrow is surjective as well, and hence so is the top one. A consultation of the long exact sequence in homotopy groups induced by the fibration (44) thus proves (i) since the kernel of the middle horizontal arrow of the diagram is Derω(L(p)W)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Der}_{\omega }(L_{(p)}^{W})$$\end{document} as (47) is given by the evaluation at ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document}. This finishes the proof of (i). The proof of (ii) is completely analogous, based on the fibration sequence obtained by applying Maps∗(-,Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Maps}_*(-,V_g)$$\end{document} to the cofibration sequence ∂Wg,1→Wg,1→Wg\documentclass[12pt]{minimal}
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\begin{document}$$\partial W_{g,1}\rightarrow W_{g,1}\rightarrow W_g$$\end{document} instead of (44). The bottom square of the diagram corresponding to (45) is now given bywhose vertical arrows are given by the compositionDerι(L(p)∂Wg,1,L(p)V)+⟶≅Hom(s-1H(p)∂Wg,1,L(p)V)+⟶≅(s-(2n-2)L(p)V)+=(s-(2n-2)[L(p)V,L(p)V])+\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {Der}^{\iota }(L_{(p)}^{\partial W_{g,1}},L_{(p)}^{V})^+&\overset{\cong }{\longrightarrow }&\mathrm {Hom}(s^{-1}H_{(p)}^{\partial W_{g,1}},L_{(p)}^{V})^+\\&\overset{\cong }{\longrightarrow }&\big (s^{-(2n-2)}L_{(p)}^{V}\big )^+=\big (s^{-(2n-2)}{[L_{(p)}^{V},L_{(p)}^{V}]}\big )^+\end{aligned}$$\end{document}and the restriction to elements of positive degrees of the compositionDerι(L(p)W,L(p)V)→≅Hom(s-1H(p)W,L(p)V)≅L(p)V⊗(s-1H(p)W)∨≅s-(2n-1)L(p)V⊗H(p)W,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \mathrm {Der}^\iota (L_{(p)}^{W},L_{(p)}^{V})\xrightarrow {\cong }\mathrm {Hom}(s^{-1}H_{(p)}^{W},L_{(p)}^{V})\cong&L_{(p)}^{V}\otimes (s^{-1}H_{(p)}^{W})^{\vee }\\\cong&s^{-(2n-1)}L_{(p)}^{V}\otimes H_{(p)}^{W},\end{aligned}\end{aligned}$$\end{document}both completely analogous to the two compositions explained below (45). □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Proof of Theorem 4.6
We consider the map of horizontal fibration sequenceswhere the top right horizontal arrow is induced by restriction and the rightmost vertical arrow is given by extending a selfmap of Wg,1\documentclass[12pt]{minimal}
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\begin{document}$$W_{g,1}$$\end{document} relative to the boundary over the complement of Wg,1⊂Wg\documentclass[12pt]{minimal}
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\begin{document}$$W_{g,1}\subset W_{g}$$\end{document} by the identity, followed by postcomposition with the inclusion Wg⊂Vg\documentclass[12pt]{minimal}
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\begin{document}$$W_{g}\subset V_g$$\end{document}. The induced morphism π∗Maps∗id(Vg,Vg)→π∗Maps∗ι(Wg,Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _*\mathrm {Maps}^{\mathrm {id}}_*(V_g,V_g)\rightarrow \pi _*\mathrm {Maps}^{\iota }_*(W_g,V_g)$$\end{document} is injective because its composition with the morphism π∗Maps∗ι(Wg,Vg)→π∗Maps∗ι(Wg,1,Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _*\mathrm {Maps}^{\iota }_*(W_g,V_g)\rightarrow \pi _*\mathrm {Maps}^{\iota }_*(W_{g,1},V_g)$$\end{document} induced by restriction along the inclusion Wg,1⊂Wg\documentclass[12pt]{minimal}
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\begin{document}$$W_{g,1}\subset W_g$$\end{document} is a retract since the composition∨g(Sn∨Sn)≃Wg,1⊂Wg⊂Vg≃∨gSn\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\vee ^g(S^n\vee S^n)\simeq W_{g,1}\subset W_g\subset V_g\simeq \vee ^gS^n\end{aligned}$$\end{document}is a homotopy retraction. From the long exact sequence in homotopy groups of the bottom fibration, we see that the monoid π0Maps∂(Vg,Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Maps}_{\partial }(V_g,V_g)$$\end{document} receives a surjection from π1Maps∗ι(Wg,Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _1\mathrm {Maps}_*^{\iota }(W_g,V_g)$$\end{document}, and this is a monoid homomorphism as it agrees with the map induced on π0(-)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0(-)$$\end{document} by the homotopy fibre inclusion of the fibre sequence of A∞\documentclass[12pt]{minimal}
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\begin{document}$$A_\infty $$\end{document}-spaces(ΩMaps∗ι(Wg,Vg)≃hofibid(inc))⟶Maps∂(Vg,Vg)⟶incMaps∗(Vg,Vg).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\big (\Omega \mathrm {Maps}_*^{\iota }(W_g,V_g) \simeq \mathrm {hofib}_{\mathrm {id}}(\mathrm {inc})\big )\longrightarrow \mathrm {Maps}_{\partial }(V_g,V_g)\overset{\mathrm {inc}}{\longrightarrow }\mathrm {Maps}_{*}(V_g,V_g).\end{aligned}$$\end{document}The group π1Maps∗ι(Wg,Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _1\mathrm {Maps}_*^{\iota }(W_g,V_g)$$\end{document} is abelian since we haveπ1Maps∗ι(Wg,Vg)≅[S1∧Wg,Vg]∗≅[S1∧(∨2gSn∨S2n),Vg]∗≅πn+1(Vg)⊕2g⊕π2n+1(Vg),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _1\mathrm {Maps}_*^{\iota }(W_g,V_g)\cong [S^1\wedge W_g,V_g]_*\cong & {} [S^1\wedge (\vee ^{2g} S^n\vee S^{2n}),V_g]_*\\\cong & {} \pi _{n+1}(V_g)^{\oplus 2g}\oplus \pi _{2n+1}(V_g),\end{aligned}$$\end{document}using S1∧Wg≃S1∧(∨2gSn∨S2n)\documentclass[12pt]{minimal}
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\begin{document}$$S^1\wedge W_g\simeq S^1\wedge (\vee ^{2g} S^n\vee S^{2n})$$\end{document} due to the fact that the attaching map (43) of the top-dimensional cell in the usual CW-decomposition of Wg\documentclass[12pt]{minimal}
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\begin{document}$$W_g$$\end{document} is a sum of Whitehead-brackets and thus nullhomotopic after suspension. Being surjected upon by an abelian group, the monoid π0Maps∂(Vg,Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Maps}_{\partial }(V_g,V_g)$$\end{document} is itself an abelian group and hence agrees with π0hAut∂(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {hAut}_{\partial }(V_g)$$\end{document}, as claimed in (i). To prove (ii), we combine the injectivity we just observed with Lemma 4.8 (ii) and the long exact sequence of the bottom row in (50) to obtain a short exact sequence0⟶Der(L(p)V)+⟶(-)∘ι∗Derωι(L(p)W,L(p)V)+⟶π∗-1Maps∂id(Vg,Vg)(p)⟶0\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} 0\longrightarrow \mathrm {Der}(L_{(p)}^{V})^+&\overset{(-)\circ \iota _*}{\longrightarrow }&\mathrm {Der}^{\iota }_{\omega }(L_{(p)}^{W},L_{(p)}^{V})^+\nonumber \\\longrightarrow & {} \pi _{*-1}\mathrm {Maps}_\partial ^{\mathrm {id}}(V_g,V_g)_{(p)}\longrightarrow 0\end{aligned}$$\end{document}in the range ∗<2p-3-n\documentclass[12pt]{minimal}
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\begin{document}$$*<2p-3-n$$\end{document}. Combining this with Lemma 4.8 (i), we see that the boundary map in the long exact sequence in homotopy groups of the upper fibration of (50) fits in degrees ∗<2p-3-n\documentclass[12pt]{minimal}
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\begin{document}$$*<2p-3-n$$\end{document} into a commutative diagramwhere π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document} is the quotient map and the right vertical map is induced by (51). To finish the proof of (ii), we thus need to show that the bottom composition of (52) fits as the left vertical arrow in a diagram as in (ii). To see this, we first combine the bottom square of (45) with the compatible square (48) to obtain a commutative diagram with exact rowsNext, writingK:=ker(ι∗:HW→HV),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}K:=\ker (\iota _*:H^W\rightarrow H^V),\end{aligned}$$\end{document}we note that there is a chain of natural isomorphismsDer(L(p)V)≅Hom(H(p)V,L(p)V)≅L(p)V⊗(s-1H(p)V)∨≅s-(2n-1)L(p)V⊗K(p)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {Der}(L_{(p)}^{V}){\cong }\mathrm {Hom}(H_{(p)}^{V},L_{(p)}^{V}){\cong } L_{(p)}^{V}{\otimes } (s^{-1}H_{(p)}^{V})^\vee {\cong } s^{-(2n-1)}L_{(p)}^{V}{\otimes } K_{(p)}\qquad \end{aligned}$$\end{document}defined analogously to (and compatible with) (49), using that the isomorphism (HW)∨≅HW\documentclass[12pt]{minimal}
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\begin{document}$$(H^{W})^\vee \cong H^{W}$$\end{document} induced by the intersection form (neglecting grading shifts) sends (HV)∨⊂(HW)∨\documentclass[12pt]{minimal}
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\begin{document}$$(H^{V})^\vee \subset (H^{W})^\vee $$\end{document} to K⊂HW\documentclass[12pt]{minimal}
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\begin{document}$$K\subset H^{W}$$\end{document}. Except for the equivariance claim, (ii) now follows by combining the chain of isomorphisms (54) with the diagrams (52)–(53) and the chain of isomorphisms(s-(2n-1)L(p)V⊗H(p)W)/(s-(2n-1)L(p)V⊗K(p))≅s-(2n-1)L(p)V⊗(H(p)W/K(p))≅s-(2n-1)L(p)V⊗H(p)V.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\big (s^{-(2n-1)}L_{(p)}^{V}\otimes H_{(p)}^{W}\big )/\big ( s^{-(2n-1)}L_{(p)}^{V}\otimes K_{(p)} \big )\\&\quad \cong s^{-(2n-1)}L_{(p)}^{V}\otimes \big (H_{(p)}^{W}/ K_{(p)} \big )\cong s^{-(2n-1)}L_{(p)}^{V}\otimes H_{(p)}^{V}. \end{aligned}$$\end{document}This uses that the inclusion hAut∂ext(Wg,1)⊂Maps∂(Wg,1,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hAut}^{\mathrm {ext}}_\partial (W_{g,1})\subset \mathrm {Maps}_\partial (W_{g,1},W_{g,1})$$\end{document} is 0-coconnected and that we have hAut∂(Vg)=Maps∂(Vg,Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hAut}_\partial (V_g)=\mathrm {Maps}_\partial (V_g,V_g)$$\end{document} by (i). To see the equivariance, note that all vertical maps in the diagram of (ii) are equivariant by construction. Since they are also surjective (see the discussion after the statement), it suffices to show that the top row is equivariant. This is clear for the second map in the top row, so we are left with showing equivariance of the first mapπ∗+1BhAut∂(Wg,1)(p)⟶s2n-1H(p)W⊗L(p)W.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _{*+1}\mathrm {BhAut}_\partial (W_{g,1})_{(p)}\longrightarrow s^{2n-1}H_{(p)}^{W}\otimes L_{(p)}^{W}.\end{aligned}$$\end{document}With respect to the canonical isomorphisms in positive degreesπ∗+1BhAut∂(Wg,1)(p)≅π∗hAut∂(Wg,1)(p)≅π∗Maps∂(Wg,1,Wg,1)(p),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\pi _{*+1}\mathrm {BhAut}_\partial (W_{g,1})_{(p)}\cong \pi _{*}\mathrm {hAut}_\partial (W_{g,1})_{(p)}\cong \pi _*\mathrm {Maps}_\partial (W_{g,1},W_{g,1})_{(p)},\end{aligned}$$\end{document}the action on the domain of (55) is induced by conjugation. Going through the proof, we see that (55) arises as a composition of the formπ∗Maps∂id(Wg,1,Wg,1)(p)⟶π∗Maps∗id(Wg,1,Wg,1)(p)⟶Der(L(p)W)≅s2n-1H(p)W⊗L(p)W.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _{*}\mathrm {Maps}^{\mathrm {id}}_\partial (W_{g,1},W_{g,1})_{(p)}\longrightarrow & {} \pi _{*}\mathrm {Maps}^{\mathrm {id}}_*(W_{g,1},W_{g,1})_{(p)}\\\longrightarrow & {} \mathrm {Der}(L_{(p)}^{W})\cong s^{2n-1}H_{(p)}^{W}\otimes L_{(p)}^{W}.\end{aligned}$$\end{document}The first map relaxes the boundary condition, which is equivariant. The second map is given by the isomorphism in Proposition 4.3, and its equivariance follows from the naturality part of that proposition. The third map is provided by the chain of isomorphisms (46), which is equivariant by the naturality of the intersection form. This finishes the proof of (ii). To see (iii), note that all restrictions on the degree in the proof of (ii) originated from the assumption on the degree in Proposition 4.3. This proposition holds rationally without that assumption, so the proof of 4.6 also applies to 4.6. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
In a range of degrees, the particular shape of the p-local homotopy groups of the space BhAutD2nid(Vg,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BhAut}^{\mathrm {id}}_{D^{2n}}(V_g,W_{g,1})$$\end{document} ensured by Corollary 4.7 allows us to pass from homotopy to homology groups, which is what we are actually interested in. In the following statement, we consider the module Hn(Wg,1;Z(p))\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {H}_n(W_{g,1};\mathbf {Z}_{(p)})$$\end{document} as ungraded.
Corollary 4.9
For n≥2\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 2$$\end{document}, there is an injection of graded π0hAutD2n(Vg,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {hAut}_{D^{2n}} (V_g,W_{g,1})$$\end{document}-modulesin degrees ∗<min(2n-1,2p-3-n)\documentclass[12pt]{minimal}
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\begin{document}$$*< \min (2n-1,2p-3-n)$$\end{document} for primes p.
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We may assume p>3\documentclass[12pt]{minimal}
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\begin{document}$$p> 3$$\end{document}, since otherwise the claim has no content. As a result of Corollary 4.7, there is an injective map of graded π0hAutD2n(Vg,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {hAut}_{D^{2n}}(V_g,W_{g,1})$$\end{document}-modulesin degrees 0<∗<2p-4-n\documentclass[12pt]{minimal}
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\begin{document}$$0<*< 2p-4-n$$\end{document}. Using that we have[s-1H(p)W,s-1H(p)W]⊂(s-1H(p)W)⊗2\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&[s^{-1}H^{W}_{(p)},s^{-1}H^{W}_{(p)}]\subset (s^{-1}H^{W}_{(p)})^{\otimes 2}\end{aligned}$$\end{document}by antisymmetrisation and that H(p)W=H~∗(Wg,1;Z(p))\documentclass[12pt]{minimal}
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\begin{document}$$H^{W}_{(p)}=\widetilde{\mathrm {H}}_*(W_{g,1};\mathbf {Z}_{(p)})$$\end{document} is concentrated in degree n, we obtains-(2n-1)H(p)W⊗L(s-1H(p)W)⊂s-(2n-1)H(p)W⊗(s-1H(p)W)⊗2=(Hn(Wg,1;Z(p))⊗3)[n-1],\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} s^{-(2n-1)}H^{W}_{(p)}\otimes \mathbf {L}(s^{-1}H^{W}_{(p)})\subset & {} s^{-(2n-1)}H^{W}_{(p)}\otimes (s^{-1}H^{W}_{(p)})^{\otimes 2}\\= & {} \big (\mathrm {H}_n(W_{g,1};\mathbf {Z}_{(p)})^{\otimes 3}\big )[n-1],\end{aligned}$$\end{document}in degrees ∗<2n-2\documentclass[12pt]{minimal}
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\begin{document}$$*<2n-2$$\end{document}, so the claim holds for homotopy instead of homology groups. This leaves us with showing that the p-local Hurewicz homomorphismπ∗BhAutD2nid(Vg,Wg,1)(p)⟶H~∗(BhAutD2nid(Vg,Wg,1)(p);Z(p))\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \pi _{*}\mathrm {BhAut}^{\mathrm {id}}_{D^{2n}}(V_g,W_{g,1})_{(p)}\longrightarrow \widetilde{\mathrm {H}}_*(\mathrm {BhAut}^{\mathrm {id}}_{D^{2n}}(V_g,W_{g,1})_{(p)};\mathbf {Z}_{(p)}) \end{aligned}$$\end{document}is an isomorphism in degree ∗<m:=min(2n-1,2p-3-n)\documentclass[12pt]{minimal}
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\begin{document}$$*< m:=\min (2n-1,2p-3-n)$$\end{document}. Since submodules of free Z(p)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {Z}_{(p)}$$\end{document}-modules are free, it follows from the first part of the proof that n-truncation induces a p-locally m-connected map of the formBhAutD2nid(Vg,Wg,1)⟶K(A,n)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {BhAut}^{\mathrm {id}}_{D^{2n}}(V_g,W_{g,1})\longrightarrow K(A,n)\end{aligned}$$\end{document}where A is a free Z(p)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {Z}_{(p)}$$\end{document}-module, so it suffices to show that K(A, n) has trivial Z(p)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {Z}_{(p)}$$\end{document}-homology in the range n<∗<m\documentclass[12pt]{minimal}
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\begin{document}$$n<*<m$$\end{document}. Since A is free, it is enough to show that H∗(K(Z(p),n);Z(p))≅H∗(K(Z,n);Z(p))\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {H}_*(K(\mathbf {Z}_{(p)},n);\mathbf {Z}_{(p)})\cong \mathrm {H}_*(K(\mathbf {Z},n);\mathbf {Z}_{(p)})$$\end{document} vanishes in this range, which is certainly true rationally, so we may instead prove that H∗(K(Z,n);Fp)\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
The proof of Theorem A
This section is devoted to the proof of the following refinement of Theorem A.
Theorem 5.1
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\begin{document}$$n>3$$\end{document}, there is a nilpotent space X and a zig–zagBC(D2n)⟶ϕX⟵ψΩ0∞+1K(Z)\documentclass[12pt]{minimal}
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\begin{document}$$\min (2n-4,2p-4-n)$$\end{document}-connected and ψ\documentclass[12pt]{minimal}
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\begin{document}$$\min (2n,2p-4)$$\end{document}-connected.
Remark 5.2
As remarked in the introduction, our proof is independent of Waldhausen’s work on pseudoisotopy theory. If one is willing to invest Dwyer–Weiss–Williams’ index theorem [12] (which relies in parts of Waldhausen’s work), then it takes little effort to also explicitly identify X and ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document} in terms of well-known infinite loop spaces, see Section 5.4.
As a first step towards proving Theorem 5.1, we replace BC(D2n)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BC}(D^{2n})$$\end{document} by an equivalent space that is more convenient to compare to the various automorphism spaces of high-dimensional handlebodies Vg=♮gDn+1×Sn\documentclass[12pt]{minimal}
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\begin{document}$$V_g=\natural ^gD^{n+1}\times S^n$$\end{document} we studied in the previous sections.
A choice of identification Dd×[0,1]≅Dd+1\documentclass[12pt]{minimal}
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\begin{document}$$D^{d}\times [0,1]\cong D^{d+1}$$\end{document} by smoothing corners induces an equivalence BC(Dd)≃BDiffDd(Dd+1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BC}(D^{d})\simeq \mathrm {BDiff}_{D^{d}}(D^{d+1})$$\end{document}, so the claim is equivalent to showing that the space of block diffeomorphisms Diff~Dd(Dd+1)\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{\mathrm {Diff}}_{D^{d}}(D^{d+1})$$\end{document} is contractible. This follows from Cerf’s result that every concordance of a manifold of dimension at least 5 is isotopic to the identity [11], together with the isomorphisms πk(Diff~Dd(Dd+1);id)≅π0DiffDd+k(Dd+k+1)≅π0C(Dd+k)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _k(\widetilde{\mathrm {Diff}}_{D^{d}}(D^{d+1});\mathrm {id})\cong \pi _0\mathrm {Diff}_{D^{d+k}}(D^{d+k+1})\cong \pi _0\mathrm {C}(D^{d+k})$$\end{document} which is most easily seen by using the combinatorial description of the homotopy groups of the Kan complex Diff~Dd(Dd+1)∙\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{\mathrm {Diff}}_{D^{d}}(D^{d+1})_\bullet $$\end{document} (see Section 1.4). □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
The alternative point of view on BC(D2n)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BC}(D^{2n})$$\end{document} as the homotopy fibreDiff~D2n(D2n+1)/DiffD2n(D2n+1)=Diff~D2n(V0)/DiffD2n(V0)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\widetilde{\mathrm {Diff}}_{D^{2n}}(D^{2n+1})/\mathrm {Diff}_{D^{2n}}(D^{2n+1})=\widetilde{\mathrm {Diff}}_{D^{2n}}(V_0)/\mathrm {Diff}_{D^{2n}}(V_0)\end{aligned}$$\end{document}is advantageous as it makes a stabilisation map of the formBC(D2n)≃Diff~D2n(V0)/DiffD2n(V0)⟶Diff~D2n(Vg)/DiffD2n(Vg),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {BC}(D^{2n})\simeq \widetilde{\mathrm {Diff}}_{D^{2n}}(V_0)/\mathrm {Diff}_{D^{2n}}(V_0)\longrightarrow \widetilde{\mathrm {Diff}}_{D^{2n}}(V_g)/\mathrm {Diff}_{D^{2n}}(V_g),\end{aligned}$$\end{document}apparent, which is induced by iterating the stabilisation maps for BDiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BDiff}_{D^{2n}}(V_g)$$\end{document} and its block analogue explained in Section 3.5. It is a consequence of Morlet’s lemma of disjunction that this map is highly connected:
Lemma 5.4
The stabilisation map (56) is (2n-4)\documentclass[12pt]{minimal}
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\begin{document}$$(2n-4)$$\end{document}-connected.
Proof
Taking vertical homotopy fibres in the diagram (see Section 3.1 for the notation)
of fibre sequences whose diagonal arrows are given by the iterated stabilisation maps results in a map of fibre sequences of the form
whose inner diagonal map is the map in question and whose rightmost equivalences follow from another application of Cerf’s result mentioned in the previous proof. As the manifolds Vg\documentclass[12pt]{minimal}
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\begin{document}$$(2n-4)$$\end{document}-connected by a form of Morlet’s lemma of disjunction [6, p. 29, Cor. 3.2], so the claim follows from the induced ladder of long exact sequences in homotopy groups. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Combining the previous two lemmas results in a (2n-4)\documentclass[12pt]{minimal}
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\begin{document}$$(2n-4)$$\end{document}-connected mapBC(D2n)→Diff~D2n(V∞)/DiffD2n(V∞):=hocolimgDiff~D2n(Vg)/DiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\mathrm {BC}(D^{2n})\rightarrow \widetilde{\mathrm {Diff}}_{D^{2n}}(V_\infty )/\mathrm {Diff}_{D^{2n}}(V_\infty ):=\mathrm {hocolim}_g\widetilde{\mathrm {Diff}}_{D^{2n}}(V_g)/\mathrm {Diff}_{D^{2n}}(V_g)\end{aligned}$$\end{document}to the homotopy colimit over the stabilisation maps, so in order to prove Theorem 5.1, it remains to establish a zig–zag with the claimed connectivity properties between this homotopy colimit and the zero component of the once looped algebraic K-theory space of the integers Ω0∞+1K(Z)\documentclass[12pt]{minimal}
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\begin{document}$$\Omega _0^{\infty +1}\mathrm {K}(\mathbf {Z})$$\end{document}, which we model as the plus-construction3Ω0∞+1K(Z)≃BGL∞(Z)+\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\Omega _0^{\infty +1}\mathrm {K}(\mathbf {Z})\simeq \mathrm {BGL}_\infty (\mathbf {Z})^+\end{aligned}$$\end{document}of the homotopy colimit BGL∞(Z):=hocolimgBGLg(Z)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BGL}_\infty (\mathbf {Z}):=\mathrm {hocolim}_g\mathrm {BGL}_g(\mathbf {Z})$$\end{document} over the stabilisation maps induced by the usual block inclusions GLg(Z)⊂GLg+1(Z)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {GL}_g(\mathbf {Z})\subset \mathrm {GL}_{g+1}(\mathbf {Z})$$\end{document}. The zig–zag we construct arises as part of a commutative zig–zag of horizontal homotopy fibre sequences of the formwhich we explain now. Denoting the tangential structure encoding stable framings by sfr:EO→BO\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {sfr}:\mathrm {EO}\rightarrow \mathrm {BO}$$\end{document}, the upper right corner is defined asBDiff~D2nsfr(V∞;ℓ0)ℓ:=hocolimgBDiff~D2nsfr(Vg;ℓ0)ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {B\widetilde{Diff}}^{\mathrm {sfr}}_{D^{2n}}(V_\infty ;\ell _0)_\ell :=\mathrm {hocolim}_g\mathrm {B\widetilde{Diff}}^{\mathrm {sfr}}_{D^{2n}}(V_g;\ell _0)_\ell \end{aligned}$$\end{document}along the stabilisation maps explained in Section 3.5, and the space BDiffD2nsfr(V∞;ℓ0)ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BDiff}^{\mathrm {sfr}}_{D^{2n}}(V_\infty ;\ell _0)_\ell $$\end{document} is defined analogously, using the unstable (2n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {sfr}:\mathrm {EO}\rightarrow \mathrm {BO}$$\end{document}, which we denote by the same symbol (see Section 1.8.1). The upper right horizontal map is the homotopy colimit of the comparison mapBDiffD2nsfr(Vg;ℓ0)ℓ⟶BDiff~D2nsfr(Vg;ℓ0)ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {BDiff}^{\mathrm {sfr}}_{D^{2n}}(V_g;\ell _0)_\ell \longrightarrow \mathrm {B\widetilde{Diff}}^{\mathrm {sfr}}_{D^{2n}}(V_g;\ell _0)_\ell \end{aligned}$$\end{document}whose homotopy fibre at the base point is canonically equivalent to Diff~D2n(V∞)/DiffD2n(V∞)\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{\mathrm {Diff}}_{D^{2n}}(V_\infty )/\mathrm {Diff}_{D^{2n}}(V_\infty )$$\end{document} in view of Lemma 1.8 and the fact that homotopy fibres commute with sequential homotopy colimits. Using a functorial model of the plus-construction (see e.g. [4, VII.6.2]), the upper right square of (58) is induced by the homotopy colimit of the compositionBDiffsfr(Vg;ℓ0)ℓ→BDiff~sfr(Vg;ℓ0)ℓ→BGL(HVg)=BGL(Hn(Vg;Z))≅BGLg(Z)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\mathrm {BDiff}^{\mathrm {sfr}}(V_g;\ell _0)_\ell \rightarrow \mathrm {B\widetilde{Diff}}^{\mathrm {sfr}}(V_g;\ell _0)_\ell \rightarrow \mathrm {BGL}(H_{V_g}) =\mathrm {BGL}(\mathrm {H}_n(V_g;\mathbf {Z}))\cong \mathrm {BGL}_g(\mathbf {Z}) \end{aligned}$$\end{document}along the stabilisation maps (see Section 3.5), where the second map is induced by the action on the middle homology of Vg\documentclass[12pt]{minimal}
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\begin{document}$$V_g$$\end{document}. The space X is defined as the homotopy fibre of the map BDiffsfr(V∞;ℓ0)ℓ+→BGL∞(Z)+\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BDiff}^{\mathrm {sfr}}(V_\infty ;\ell _0)_\ell ^+\rightarrow \mathrm {BGL}_\infty (\mathbf {Z})^+$$\end{document}, and it receives a map from the top left corner induced by the commutativity of the upper right square. The bottom row is induced by the inclusion of the basepoint in the universal cover BGL∞(Z)+⟨1⟩\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BGL}_\infty (\mathbf {Z})^+\langle 1\rangle $$\end{document} whose homotopy fibre agrees with the base point component of ΩBGL∞(Z)+\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \mathrm {BGL}_\infty (\mathbf {Z})^+$$\end{document}. This explains the diagram (58), aside from the map of fibre sequences from the bottom to the middle row, which is induced by the universal cover BGL∞(Z)+⟨1⟩→BGL∞(Z)+\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BGL}_\infty (\mathbf {Z})^+\langle 1\rangle \rightarrow \mathrm {BGL}_\infty (\mathbf {Z})^+$$\end{document} and the basepoint inclusion of BDiffsfr(V∞;ℓ0)ℓ+\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BDiff}^{\mathrm {sfr}}(V_\infty ;\ell _0)_\ell ^+$$\end{document}.
In the two following subsections, we continue the preparations of the proof of Theorem 5.1 by analysing the vertical maps and .
The stable homology of BDiffD2n(Vg)\documentclass[12pt]{minimal}
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Botvinnik and Perlmutter [10] have computed the stable homology of BDiffD2nθ(Vg;ℓ0)ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\theta :B\rightarrow \mathrm {BSO}(2n+1)$$\end{document} whose space B is n-connected. For us, their main result [10, Cor. 6.8.1, Prop. 6.14] is most conveniently expressed as an identification of the group completion of the disjoint unionMθ:=∐g≥0BDiffD2nθ(Vg;ℓ0)ℓ,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\textstyle {\mathcal {M}_{\theta }:=\coprod _{g\ge 0}\mathrm {BDiff}_{D^{2n}}^{\theta }(V_g;\ell _0)_\ell },\end{aligned}$$\end{document}which becomes a homotopy commutative topological monoid under boundary connected sum when choosing an appropriate point-set model (see [10, Prop. 6.11, Prop. 6.14]).
Theorem 5.5
(Botvinnik–Perlmutter)For n>3\documentclass[12pt]{minimal}
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\begin{document}$$\theta :B\rightarrow \mathrm {BSO}(2n+1)$$\end{document} for which B is n-connected, there is a homotopy equivalence of the formΩBMθ≃Ω∞Σ+∞B.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\Omega \mathrm {B}\mathcal {M}_{\theta }\simeq \Omega ^\infty \Sigma ^\infty _+B.\end{aligned}$$\end{document}
The equivalence they produce factors as a compositionΩBMθ⟶≃ΩBCθ∂⟶≃Ω∞Σ+∞B\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Omega \mathrm {B} \mathcal {M}_{\theta }\overset{\simeq }{\longrightarrow }\Omega \mathrm {B}\mathcal {C}^\partial _\theta \overset{\simeq }{\longrightarrow }\Omega ^\infty \Sigma ^\infty _+B\end{aligned}$$\end{document}whose intermediate terms is the group-completion of a topological category Cθ∂\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {C}_\theta ^\partial $$\end{document} of bordisms between 2n-manifolds with θ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document}-structures, possibly with boundary. This category was studied by Genauer [17], who established the final equivalence in (60) as a parametrised form of the Pontryagin–Thom construction. Botvinnik and Perlmutter showed that Mθ\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}_{\theta }$$\end{document} can be seen as a submonoid of the endomorphism monoid Cθ∂(D2n,D2n)\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document}-structure and that the chain of inclusions Mθ⊂Cθ∂(D2n,D2n)⊂Cθ∂\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}_{\theta }\subset \mathcal {C}_\theta ^\partial (D^{2n},D^{2n})\subset \mathcal {C}_\theta ^\partial $$\end{document} is an equivalence upon taking classifying spaces (see [10, Thm 6.3, Prop. 6.14]).
As Mθ\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}_\theta $$\end{document} is homotopy commutative, Randal-Williams’ elucidation of the group completion theorem [36, Cor. 1.2] moreover provides an equivalenceBDiffD2nθ(V∞;ℓ0)ℓ+≃Ω0BMθ,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {BDiff}_{D^{2n}}^{\theta }(V_\infty ;\ell _0)_\ell ^+\simeq \Omega _0\mathrm {B}\mathcal {M}_\theta ,\end{aligned}$$\end{document}which leads to the following consequence of Theorem 5.5 when specialised to the tangential tangential structure encoding stable framings.
Corollary 5.6
The space BDiffD2nsfr(V∞;ℓ0)ℓ+\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BDiff}_{D^{2n}}^{\mathrm {sfr}}(V_\infty ;\ell _0)_\ell ^+$$\end{document} is nilpotent and p-locally min(2n,2p-4)\documentclass[12pt]{minimal}
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\begin{document}$$\min (2n,2p-4)$$\end{document}-connected as long as n>3\documentclass[12pt]{minimal}
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\begin{document}$$n>3$$\end{document}.
Proof
Connected H-spaces are nilpotent, so (61) settles the nilpotency claim. Regarding the connectivity part of the statement, note that the unstable (2n+1)\documentclass[12pt]{minimal}
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\begin{document}$$(2n+1)$$\end{document}-dimensional tangential structure (sfr)2n+1\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {sfr}:\mathrm {EO}\rightarrow \mathrm {BO}$$\end{document} (see Section 1.8.1) is equivalent to the inclusion O/O(2n+1)→BO(2n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {O}/\mathrm {O}(2n+1)\rightarrow \mathrm {BO}(2n+1)$$\end{document} of the homotopy fibre of the stabilisation BO(2n+1)→BO\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BO}(2n+1)\rightarrow \mathrm {BO}$$\end{document}, so Theorem 5.5 and the discussion preceding this corollary show that the space in question is equivalent to Ω0∞Σ+∞O/O(2n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {O}/\mathrm {O}(2n+1)$$\end{document} is 2n-connected, the homotopy groups of Ω0∞Σ+∞O/O(2n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^\infty _0\Sigma ^\infty _+\mathrm {O}/\mathrm {O}(2n+1)$$\end{document} agree in positive degrees less than 2n+1\documentclass[12pt]{minimal}
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\begin{document}$$2p-3$$\end{document} by a result of Serre [38, p. 498, Prop. 5], so the claim follows. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Said differently Corollary 5.6 shows that the base point inclusion in (58) is p-locally min(2n,2p-4)\documentclass[12pt]{minimal}
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\begin{document}$$n>3$$\end{document}.
The homology action and the map
As a consequence of Proposition 3.7 (ii), the map on fundamental group induced by is infinite abelian, so the map is as far from being highly connected as possible, even p-locally for any p. Nevertheless, it turns out that it does induce an isomorphism on p-local homology groups in a range, which we shall prove by separately studying the effect on homology of the two maps in the factorisationBDiff~D2nsfr(Vg;ℓ0)ℓ⟶BGg,ℓext⟶BGL(HVg),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {B\widetilde{Diff}}^{\mathrm {sfr}}_{D^{2n}}(V_g;\ell _0)_{\ell }\longrightarrow \mathrm {BG}^{\mathrm {ext}}_{g,\ell }\longrightarrow \mathrm {BGL}(H_{V_g}),\end{aligned}$$\end{document}of the second map in (59). Here Gg,ℓext⊂GL(HWg,1)\documentclass[12pt]{minimal}
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\begin{document}$$G_{g,\ell }^{\mathrm {ext}}\subset \mathrm {GL}(H_{W_{g,1}})$$\end{document} is the subgroup considered in Sect. 3.3.
Lemma 5.7
For n≥3\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 3$$\end{document} and any prime p, the induced mapH∗(BDiff~D2nsfr(Vg;ℓ0)ℓ;Z(p))⟶H∗(BGg,ℓext;Z(p))\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {H}_*\big (\mathrm {B\widetilde{Diff}}^{\mathrm {sfr}}_{D^{2n}}(V_g;\ell _0)_\ell ;\mathbf {Z}_{(p)}\big )\longrightarrow \mathrm {H}_*\big (\mathrm {BG}^{\mathrm {ext}}_{g,\ell };\mathbf {Z}_{(p)}\big )\end{aligned}$$\end{document}is an isomorphism for ∗<min(2n-1,2p-3-n)\documentclass[12pt]{minimal}
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\begin{document}$$*<\min (2n-1,2p-3-n)$$\end{document} and a surjection in that degree.
Proof
The claim is vacuous for p=2\documentclass[12pt]{minimal}
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\begin{document}$$p=2$$\end{document}, so we assume otherwise. Consider the factorisation of the map in questionBDiff~D2nsfr(Vg;ℓ0)ℓ→BhAut~D2n≅(Vg,Wg,1)ℓ→Bπ0hAut~D2n≅(Vg,Wg,1)ℓ→BGg,ℓext,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \mathrm {B\widetilde{Diff}}^{\mathrm {sfr}}_{D^{2n}}(V_g;\ell _0)_\ell&\rightarrow \mathrm {B\widetilde{hAut}}^{\cong }_{D^{2n}}(V_g,W_{g,1})_\ell \\ {}&\rightarrow \mathrm {B}\pi _0\mathrm {\widetilde{hAut}}^{\cong }_{D^{2n}}(V_g,W_{g,1})_\ell \rightarrow \mathrm {BG}^{\mathrm {ext}}_{g,\ell },\end{aligned} \end{aligned}$$\end{document}where the first map is that of Corollary 2.4 applied to the triad (Vg;D2n,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$(V_g;D^{2n},W_{g,1})$$\end{document}, the second map is induced by taking path components, and the third is given by acting on the homology of Wg,1⊂∂Vg\documentclass[12pt]{minimal}
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\begin{document}$$W_{g,1}\subset \partial V_g$$\end{document} (see Section 3.3). It follows from the corollary just mentioned that the first map induces an isomorphism in homology with Z(p)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {Z}_{(p)}$$\end{document}-coefficients in degrees ∗<2p-3-n\documentclass[12pt]{minimal}
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\begin{document}$$*<2p-3-n$$\end{document} and a surjection in that degree since Vg\documentclass[12pt]{minimal}
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\begin{document}$$V_g$$\end{document} is obtained from D2n\documentclass[12pt]{minimal}
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\begin{document}$$D^{2n}$$\end{document} by attaching n-handles and the triad (Vg;D2n,Wg,1)\documentclass[12pt]{minimal}
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\begin{document}$$W_{g,1}$$\end{document} and Vg\documentclass[12pt]{minimal}
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\begin{document}$$V_g$$\end{document} are simply connected for n≥2\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 2$$\end{document}. To study the remaining maps, we may replace the space of block homotopy automorphisms by its equivalent non-block analogue BhAutD2n≅(Vg,Wg,1)ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BhAut}^{\cong }_{D^{2n}}(V_g,W_{g,1})_\ell $$\end{document} (see Section 1.5), so the E2\documentclass[12pt]{minimal}
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\begin{document}$$E_2$$\end{document}-page of the p-local Serre spectral sequence of the second map has the formEk,l2≅Hk(π0hAutD2n≅(Vg,Wg,1)ℓ;Hl(BhAutD2nid(Vg,Wg,1);Z(p))).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} E_{k,l}^2\cong \mathrm {H}_k\Big (\pi _0\mathrm {hAut}^{\cong }_{D^{2n}}(V_g,W_{g,1})_\ell ;\mathrm {H}_l \big (\mathrm {BhAut}^{\mathrm {id}}_{D^{2n}}(V_g,W_{g,1});\mathbf {Z}_{(p)} \big ) \Big ).\end{aligned}$$\end{document}To compute Ek,l2\documentclass[12pt]{minimal}
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\begin{document}$$E_{k,l}^2$$\end{document}, we employ the Serre spectral sequence of the extension0⟶Lg⟶π0hAutD2n≅(Vg,Wg,1)ℓ⟶Gg,ℓext⟶0,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} 0\longrightarrow L_g\longrightarrow \pi _0\mathrm {hAut}^{\cong }_{D^{2n}}(V_g,W_{g,1})_\ell \longrightarrow G^{\mathrm {ext}}_{g,\ell }\longrightarrow 0,\end{aligned}$$\end{document}with coefficients in Hl(BhAutD2nid(Vg,Wg,1);Z(p))\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {H}_l(\mathrm {BhAut}^{\mathrm {id}}_{D^{2n}}(V_g,W_{g,1});\mathbf {Z}_{(p)})$$\end{document}, where Lg\documentclass[12pt]{minimal}
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\begin{document}$$L_g$$\end{document} is the kernel as indicated. The E2\documentclass[12pt]{minimal}
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\begin{document}$$E_2$$\end{document}-page of this spectral sequence has the formEs,t2≅Hs(Gg,ℓext;Ht(Lg;Hl(BhAutD2nid(Vg,Wg,1);Z(p)))).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}E_{s,t}^2\cong \mathrm {H}_s\Big (G^{\mathrm {ext}}_{g,\ell };\mathrm {H}_t\big (L_g;\mathrm {H}_l(\mathrm {BhAut}^{\mathrm {id}}_{D^{2n}}(V_g,W_{g,1});\mathbf {Z}_{(p)})\big )\Big ).\end{aligned}$$\end{document}The kernel Lg\documentclass[12pt]{minimal}
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\begin{document}$$L_g$$\end{document} of the extension (65) is a subgroup of the kernel of the analogous extension without the ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}-subscripts, and as the latter is finite and p-torsion free for n<2p-4\documentclass[12pt]{minimal}
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\begin{document}$$n<2p-4$$\end{document} by Lemma 3.10, so is the former. The group Lg\documentclass[12pt]{minimal}
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\begin{document}$$L_g$$\end{document} thus has no nontrivial homology in positive degrees with coefficients in a Z(p)[Lg]\documentclass[12pt]{minimal}
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\begin{document}$$n<2p-4$$\end{document}, so in this case the spectral sequence Es,t2\documentclass[12pt]{minimal}
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\begin{document}$$E_{s,t}^2$$\end{document} is concentrated in the bottom row and we conclude thatEk,l2≅Hk(Gg,ℓext;Hl(BhAutD2nid(Vg,Wg,1);Z(p))Lg)forn<2p-4,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} E_{k,l}^2\cong \mathrm {H}_k\Big (G^{\mathrm {ext}}_{g,\ell };\mathrm {H}_l\big (\mathrm {BhAut}^{\mathrm {id}}_{D^{2n}}(V_g,W_{g,1});\mathbf {Z}_{(p)}\big )_{L_g}\Big )\quad \text {for }n<2p-4,\end{aligned}$$\end{document}where (-)Lg\documentclass[12pt]{minimal}
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\begin{document}$$(-)_{L_g}$$\end{document} stands for taking coinvariants. By Lemma 3.9, the group Gg,ℓext\documentclass[12pt]{minimal}
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\begin{document}$$-1$$\end{document} as long as 0<l<m:=min(2n-1,2p-3-n)\documentclass[12pt]{minimal}
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\begin{document}$$-1$$\end{document} on Hn(Wg,1;Z)⊗3\documentclass[12pt]{minimal}
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\begin{document}$$(-1)^3=(-1)$$\end{document}. This allows us to apply the “centre kills”-trick4 to conclude that the E2\documentclass[12pt]{minimal}
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\begin{document}$$0<l<m$$\end{document} (which implies n<2p-4\documentclass[12pt]{minimal}
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\begin{document}$$n<2p-4$$\end{document}) and is therefore trivial as p was assumed to be odd. In this range, the spectral sequence (64) is therefore concentrated at the bottom row, so the second map in (63) induces an isomorphism on Z(p)\documentclass[12pt]{minimal}
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\begin{document}$$*<m$$\end{document} and a surjection in that degree. This leaves us with arguing that the final map in (63) also has this property. But we already observed that the kernel in (65) is p-locally acyclic for n<2p-4\documentclass[12pt]{minimal}
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\begin{document}$$n<2p-4$$\end{document}, so in this case the third map is in fact a p-local homology isomorphism. As the statement is vacuous unless 1<m\documentclass[12pt]{minimal}
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\begin{document}$$1<m$$\end{document} and thus n<2p-4\documentclass[12pt]{minimal}
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\begin{document}$$n<2p-4$$\end{document}, this finishes the proof. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
In contrast to the first map in (62), the second map might not induce a p-local homology isomorphism in a range of degrees, but we will see below that its homotopy colimit BG∞,ℓext→BGL(HV∞)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BG}^{\mathrm {ext}}_{\infty ,\ell }\rightarrow \mathrm {BGL}(H_{V_\infty })$$\end{document} with respect to the stabilisation maps explained in Section 3.5 does.
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\begin{document}$$\begin{aligned}0\longrightarrow M_g^{\mathrm {sfr}}\longrightarrow G_{g,\ell }^{\mathrm {ext}}\longrightarrow \mathrm {GL}(H_{V_g})\longrightarrow 0\end{aligned}$$\end{document}established in Proposition 3.7 (ii), it suffices to show that the colimit induced by stabilisationcolimgH∗(GL(HVg);H~∗(Mgsfr;Z(p)))≅colimgH∗(GL(HVg);H~∗(Mgsfr⊗Z(p);Z(p)))\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&\mathrm {colim}_g\mathrm {H}_*\big (\mathrm {GL}(H_{V_g});\widetilde{\mathrm {H}}_*(M_g^{\mathrm {sfr}};\mathbf {Z}_{(p)})\big )\\&\quad \cong \mathrm {colim}_g\mathrm {H}_*\big (\mathrm {GL}(H_{V_g});\widetilde{\mathrm {H}}_*(M_g^{\mathrm {sfr}}\otimes \mathbf {Z}_{(p)};\mathbf {Z}_{(p)})\big )\end{aligned}$$\end{document}vanishes; the isomorphism can be viewed as being induced by the p-localisation of K(Mgsfr,1)\documentclass[12pt]{minimal}
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\begin{document}$$K(M_g^{\mathrm {sfr}},1)$$\end{document}, see Section 1.2. Recall from Section 3.3 that the GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\mu \in (H_{V_g}\otimes H_{V_g})^\vee $$\end{document}, so as p is assumed to be odd, its p-localisation Mgsfr⊗Z(p)\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda ^2(H_{V_g})^\vee \otimes \mathbf {Z}_{(p)}$$\end{document} if n is even. In particular, by antisymmetrisation, the GL(HVg)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {GL}(H_{V_g})$$\end{document}-module Mgsfr⊗Z(p)\documentclass[12pt]{minimal}
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\begin{document}$$M_g^{\mathrm {sfr}}\otimes \mathbf {Z}_{(p)}$$\end{document} is a direct summand of (HVg⊗HVg)∨⊗Z(p)\documentclass[12pt]{minimal}
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\begin{document}$$(H_{V_g}\otimes H_{V_g})^{\vee }\otimes \mathbf {Z}_{(p)}$$\end{document}. Choosing a basis HVg≅Zg\documentclass[12pt]{minimal}
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\begin{document}$$H_{V_g}\cong \mathbf {Z}^g$$\end{document} compatible with the stabilisation maps, this shows that it suffices to show that the stable GLg(Z)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {H}_k((\mathbf {Z}^{g}\otimes \mathbf {Z}^{g})^{\vee };\mathbf {Z})\cong \Lambda ^{k}(\mathbf {Z}^{g}\otimes \mathbf {Z}^{g})^{\vee }$$\end{document} vanishes for k>0\documentclass[12pt]{minimal}
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\begin{document}$$k>0$$\end{document}. Pulling back this module along the automorphism of GLg(Z)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {GL}_g(\mathbf {Z})$$\end{document} given by taking transpose inverse, we see that we may replace this module by its dual Λk(Zg⊗Zg)\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda ^{k}(\mathbf {Z}^{g}\otimes \mathbf {Z}^{g})$$\end{document} whose GLg(Z)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {GL}_g(\mathbf {Z})$$\end{document}-homology for large g with respect to k does indeed vanish by an application of a result due to Betley [1, Thm 3.1]. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Corollary 5.9
Let n≥3\documentclass[12pt]{minimal}
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\begin{document}$$n\ge 3$$\end{document} and p a prime. The map on homology induced by the map ,H∗(BDiff~D2nsfr(V∞;ℓ0)ℓ;Z(p))⟶H∗(BGL∞(Z)+;Z(p)),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {H}_*(\mathrm {B\widetilde{Diff}}^{\mathrm {sfr}}_{D^{2n}}(V_\infty ;\ell _0)_\ell ;\mathbf {Z}_{(p)})\longrightarrow \mathrm {H}_*(\mathrm {BGL}_\infty (\mathbf {Z})^+;\mathbf {Z}_{(p)}),\end{aligned}$$\end{document}is an isomorphism for ∗<min(2n-1,2p-3-n)\documentclass[12pt]{minimal}
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\begin{document}$$*<\min (2n-1,2p-3-n)$$\end{document} and a surjection in that degree.
Proof
This is free of content if p is even and follows for p odd from a combination of Lemmas 5.7 and 5.8, using that the canonical map BGL∞(Z)→BGL∞(Z)+\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BGL}_\infty (\mathbf {Z})\rightarrow \mathrm {BGL}_\infty (\mathbf {Z})^+$$\end{document} is acyclic. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Proof of Theorem 5.1
As in the previous proofs, we assume p>2\documentclass[12pt]{minimal}
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\begin{document}$$p>2$$\end{document}; the claim is vacuous otherwise. Our goal is to demonstrate that the precomposition of the left column in (58) with the (2n-4)\documentclass[12pt]{minimal}
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\begin{document}$$(2n-4)$$\end{document}-connected map (57) provides a zig–zag as promised by Theorem 5.1. As a result of the discussion in Section 5.1, the space X is the homotopy fibre of a map out of an infinite loop space which is moreover surjective on fundamental groups as a result of Proposition 3.7, so it follows from Lemma 1.4 that X is nilpotent. That the map is p-locally min(2n,2p-4)\documentclass[12pt]{minimal}
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\begin{document}$$\min (2n,2p-4)$$\end{document}-connected is a consequence of having this property by Corollary 5.6 and the 1-connected cover BGL∞(Z)+⟨1⟩→BGL∞(Z)+\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BGL}_\infty (\mathbf {Z})^+\langle 1 \rangle \rightarrow \mathrm {BGL}_\infty (\mathbf {Z})^+$$\end{document} being a p-local equivalence as π1BGL∞(Z)+≅Z/2\documentclass[12pt]{minimal}
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\begin{document}$$\pi _1\mathrm {BGL}_\infty (\mathbf {Z})^+\cong \mathbf {Z}/2$$\end{document} and p is odd. This leaves us with showing that the compositionBC(D2n)⟶Diff~D2n(V∞)/Diff~D2n(V∞)⟶X\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {BC}(D^{2n})\longrightarrow \widetilde{\mathrm {Diff}}_{D^{2n}}(V_\infty )/\widetilde{\mathrm {Diff}}_{D^{2n}}(V_\infty )\longrightarrow X\end{aligned}$$\end{document}is p-locally min(2n-4,2p-4-n)\documentclass[12pt]{minimal}
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\begin{document}$$\min (2n-4,2p-4-n)$$\end{document}-connected, which we can test on p-local homology groups by Lemma 1.1 as source and target are nilpotent. The first map in this composition is (2n-4)\documentclass[12pt]{minimal}
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\begin{document}$$(2n-4)$$\end{document}-connected by Lemma 5.4, so we may focus on the second and show that it induces an isomorphism in p-local homology in the required range. Since plus-constructions do not affect homology groups and is an isomorphism in homology with Z(p)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {Z}_{(p)}$$\end{document}-coefficients in degrees less than min(2n-1,2p-3-n)\documentclass[12pt]{minimal}
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\begin{document}$$\min (2n-1,2p-3-n)$$\end{document} and an epimorphism in that degree by Corollary 5.9, the claim follows from an application of Zeeman’s comparison theorem (see e.g. [19, Thm. 3.2]) to the map of Serre spectral sequences induced by the first two rows of (58), provided we ensure that the actions of the fundamental groups of the bases of both fibre sequences on the Z(p)\documentclass[12pt]{minimal}
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\begin{document}$$\mathbf {Z}_{(p)}$$\end{document}-homology of the respective fibres are trivial in this range. For the second row, this follows from the p-local high connectivity of the map established in the first part of the proof, using that the canonical action of π1BGL∞(Z)+\documentclass[12pt]{minimal}
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\begin{document}$$\pi _1\mathrm {BGL}_\infty (\mathbf {Z})^+$$\end{document} on the homology of ΩBGL∞(Z)+\documentclass[12pt]{minimal}
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\begin{document}$$\Omega \mathrm {BGL}_\infty (\mathbf {Z})^+$$\end{document} is trivial as BGL∞(Z)+≃Ω0B(∐gBGLg(Z))\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BGL}_\infty (\mathbf {Z})^+\simeq \Omega _0 \mathrm {B}(\coprod _g\mathrm {BGL}_g(\mathbf {Z}))$$\end{document} is a loop space (in fact, an infinite one). For the first row, the triviality of the action is a consequence of Lemma 5.4 together with the observation that any element in π0DiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _0\mathrm {Diff}_{D^{2n}}(V_g)$$\end{document} can be represented by a diffeomorphism that fixes D2n+1=V0⊂Vg\documentclass[12pt]{minimal}
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\begin{document}$$D^{2n+1}=V_0\subset V_g$$\end{document} pointwise, so commutes with any diffeomorphism in the image of the iterated stabilisation map C(D2n)≃DiffD2n(D2n+1)→DiffD2n(Vg)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {C}(D^{2n})\simeq \mathrm {Diff}_{D^{2n}}(D^{2n+1})\rightarrow \mathrm {Diff}_{D^{2n}}(V_g)$$\end{document}. This finishes the proof of Theorem 5.1, which in particular implies Theorem A.
A reformulation
Although not necessary for the proof of Theorem 5.1 or Theorem A, we shall explain in Section 5.4.2 below how an instance of the Dwyer–Weiss–Williams index theorem [12] shows that the plus constructed stable homology actionBDiffD2nsfr(V∞)ℓ+⟶BGL∞(Z)+\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {BDiff}^{\mathrm {sfr}}_{D^{2n}}(V_\infty )_\ell ^+\longrightarrow \mathrm {BGL}_{\infty }(\mathbf {Z})^+\end{aligned}$$\end{document}featuring in (58) agrees with respect to the equivalence BDiffD2nsfr(V∞)ℓ+≃Ω0∞Σ+∞O/O(2n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BDiff}^{\mathrm {sfr}}_{D^{2n}}(V_\infty )_\ell ^+\simeq \Omega ^\infty _0\Sigma _+^\infty \mathrm {O}/\mathrm {O}(2n+1)$$\end{document} explained in Section 5.1 with the map obtained by applying Ω0∞(-)\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^\infty _0(-)$$\end{document} to the compositionΣ+∞O/O(2n+1)⟶prS⟶ιK(Z)⟶(-1)nK(Z),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \Sigma _+^\infty \mathrm {O}/\mathrm {O}(2n+1)\overset{\mathrm {pr}}{\longrightarrow }\mathbf {S}{\overset{\iota }{\longrightarrow }}K(\mathbf {Z})\overset{(-1)^n}{\longrightarrow }K(\mathbf {Z}), \end{aligned}$$\end{document}where pr\documentclass[12pt]{minimal}
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\begin{document}$$\iota $$\end{document} the unit. This identifies the homotopy fibre X in (58) as the spaceX≃Ω+∞hofib(Σ+∞O/O(2n+1)→prS→ιK(Z)),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}X\simeq \Omega ^\infty _+\mathrm {hofib}\big (\Sigma _+^\infty \mathrm {O}/\mathrm {O}(2n+1)\xrightarrow {\mathrm {pr}}\mathbf {S}\xrightarrow {\iota } K(\mathbf {Z})\big ),\end{aligned}$$\end{document}so we may postcompose the map ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document} from Theorem 5.1 with the (2n+1)\documentclass[12pt]{minimal}
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\begin{document}$$(2n+1)$$\end{document}-connected map hofib(ι∘pr)→hofib(ι)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hofib}(\iota \circ \mathrm {pr})\rightarrow \mathrm {hofib}(\iota )$$\end{document} induced by the projection to arrive at a cleaner formulation of Theorem A: there is a p-locally min(2n-4,2p-4-n)\documentclass[12pt]{minimal}
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\begin{document}$$\min (2n-4,2p-4-n)$$\end{document}-connected mapBC(D2n)⟶Ω0∞hofib(S→ιK(Z)).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {BC}(D^{2n})\longrightarrow \Omega _0^\infty \mathrm {hofib}(\mathbf {S}\xrightarrow {\iota } K(\mathbf {Z})).\end{aligned}$$\end{document}
Relation to Waldhausen’s work
It is reasonable to expect that the map (69) agrees up to equivalences with the compositionBC(D2n)⟶Ω0∞WhDiff(∗)=Ω0∞hofib(S→K(S))⟶Ω0∞hofib(S→K(Z))\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned}\mathrm {BC}(D^{2n})&\longrightarrow \Omega _0^\infty \mathrm {Wh}^{\mathrm {Diff}}(*)=\Omega _0^\infty \mathrm {hofib}\big (\mathbf {S}\rightarrow K(\mathbf {S})\big )\\ {}&\longrightarrow \Omega _0^\infty \mathrm {hofib}\big (\mathbf {S}\rightarrow K(\mathbf {Z})\big )\end{aligned} \end{aligned}$$\end{document}of the map known from Waldhausen’s work [42] with the map induced by linearisation. While it is not hard to show that the map (69) does indeed factor over the second map in (70), a convincing comparison between (69) and (70) appears to be more laborious and we will not go into this matter at this point (however, see the final remark of the introduction).
An application of the index theorem
To justify the claimed identification of (67) with the map resulting from applying Ω0∞(-)\documentclass[12pt]{minimal}
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\begin{document}$$\Omega ^\infty _0(-)$$\end{document} of (68), one can argue as follows: firstly, it suffices to show that these maps agree when precomposed with an arbitrary map B→BDiffD2nsfr(V∞)ℓ+\documentclass[12pt]{minimal}
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\begin{document}$$B\rightarrow \mathrm {BDiff}^{\mathrm {sfr}}_{D^{2n}}(V_\infty )_\ell ^+$$\end{document} that classifies a smooth Vg\documentclass[12pt]{minimal}
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\begin{document}$$V_g$$\end{document}-bundle π:E→B\documentclass[12pt]{minimal}
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\begin{document}$$\pi :E\rightarrow B$$\end{document} for some g≥0\documentclass[12pt]{minimal}
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\begin{document}$$g\ge 0$$\end{document} together with a trivial D2n\documentclass[12pt]{minimal}
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\begin{document}$$D^{2n}$$\end{document}-subbundle and stably framed vertical tangent bundle TπE\documentclass[12pt]{minimal}
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\begin{document}$$T_\pi E$$\end{document}. Tracing through the equivalences featuring in Section 5.1, one finds that the compositionB⟶BDiffD2nsfr(V∞)ℓ+≃Ω0∞Σ+∞(O/O(2n+1))\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} B\longrightarrow \mathrm {BDiff}^{\mathrm {sfr}}_{D^{2n}}(V_\infty )_\ell ^+\simeq \Omega ^\infty _0\Sigma ^\infty _+(\mathrm {O}/\mathrm {O}(2n+1)) \end{aligned}$$\end{document}represents the class∘BG(π)-χ(Vg)∈[Σ+∞B,Σ+∞(O/O(2n+1)]\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\circ \mathrm {BG}(\pi )-\chi (V_g)\in [\Sigma _+^\infty B, \Sigma ^\infty _+(\mathrm {O}/\mathrm {O}(2n+1)]\end{aligned}$$\end{document}where χ(-)\documentclass[12pt]{minimal}
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\begin{document}$$\chi (-)$$\end{document} is the Euler characteristic, BG(π)∈[Σ+∞B,Σ+∞E]\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {BG}(\pi )\in [\Sigma _+^\infty B, \Sigma ^\infty _+E]$$\end{document} is the Becker–Gottlieb transfer of the bundle π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document} described in terms of the Pontryagin–Thom construction [2, 3], and the class [TπE]∈[Σ+∞E,Σ+∞O/O(2n+1)]\documentclass[12pt]{minimal}
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\begin{document}$$[T_\pi E]\in [\Sigma _+^\infty E, \Sigma ^\infty _+\mathrm {O}/\mathrm {O}(2n+1)]$$\end{document} is induced by the vertical tangent bundle of π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document} together with its stable framing. It thus suffices to show that the class(-1)n(ι∘pr)∗(BG(π)-χ(Vg))∈[Σ+∞B,K(Z)]\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} (-1)^n(\iota \circ \mathrm {pr})^* (\mathrm {BG}(\pi )-\chi (V_g))\in [\Sigma _+^\infty B, K(\mathbf {Z})]\end{aligned}$$\end{document}agrees with the class represented by the compositionB⟶BDiffD2nsfr(V∞)ℓ+⟶BGL∞(Z)+≃Ω0∞K(Z).\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}B\longrightarrow \mathrm {BDiff}^{\mathrm {sfr}}_{D^{2n}}(V_\infty )_\ell ^+\longrightarrow \mathrm {BGL}_{\infty }(\mathbf {Z})^+\simeq \Omega ^{\infty }_0K(\mathbf {Z}).\end{aligned}$$\end{document}The latter can be identified with the class[Hn(π)]-g∈[Σ+∞B,K(Z)]\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}{}[H_n(\pi )]-g\in [\Sigma ^\infty _+B,K(\mathbf {Z})]\end{aligned}$$\end{document}where Hn(π)\documentclass[12pt]{minimal}
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\begin{document}$$H_n(\pi )$$\end{document} is the local system over B of fibrewise middle-dimensional homology groups of π\documentclass[12pt]{minimal}
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\begin{document}$$\pi $$\end{document}. Dwyer–Weiss–Williams’ improved Riemann–Roch theorem [12, p. 2] gives(ι∘pr)∗BG(π)=1+(-1)n·[Hn(π)]∈[Σ+∞B,K(Z)],\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}(\iota \circ \mathrm {pr})^* \mathrm {BG}(\pi )=1+(-1)^n\cdot [H_n(\pi )]\in [\Sigma ^\infty _+B,K(\mathbf {Z})],\end{aligned}$$\end{document}so using χ(Vg)=1+(-1)ng\documentclass[12pt]{minimal}
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\begin{document}$$\chi (V_g)=1+(-1)^ng$$\end{document} it follows that (72) and (73) indeed agree.
Appendix A. Stable tangential bundle maps are stable bundle maps
Fix a d-dimensional vector bundle ξ\documentclass[12pt]{minimal}
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\begin{document}$$\xi $$\end{document} over a space X, a stable vector bundle {ψk→Bk}k≥0\documentclass[12pt]{minimal}
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\begin{document}$$\{\psi _k\rightarrow B_k\}_{k\ge 0}$$\end{document}, subcomplexes A,C⊂X\documentclass[12pt]{minimal}
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\begin{document}$$A,C\subset X$$\end{document}, and a bundle map ℓ0:ξ|A⊕εk→ψd+k\documentclass[12pt]{minimal}
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\begin{document}$$\ell _0:\xi |_A\oplus \varepsilon ^k\rightarrow \psi _{d+k}$$\end{document} covering a map ℓ¯0:X→Bd+k\documentclass[12pt]{minimal}
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\begin{document}$$\bar{\ell }_0:X\rightarrow B_{d+k}$$\end{document} for some k≥0\documentclass[12pt]{minimal}
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\begin{document}$$k\ge 0$$\end{document}. In addition to the notation for various types of bundle maps in Section 1, we abbreviateMap~A(X,B;ℓ¯0)∙:=colimm≥kMap~A(X,Bm;ℓ¯0)∙,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\widetilde{\mathrm {Map}}_A(X,B;\bar{\ell }_0)_\bullet :=\mathrm {colim}_{m\ge k}\widetilde{\mathrm {Map}}_A(X,B_m;\bar{\ell }_0)_\bullet ,\end{aligned}$$\end{document}where Map~A(X,Bm;ℓ¯0)∙\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{\mathrm {Map}}_A(X,B_m;\bar{\ell }_0)_\bullet $$\end{document} is the semi-simplicial set whose p-simplices are block maps Δp×X→Δp×Bm\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p\times X\rightarrow \Delta ^p\times B_{m}$$\end{document} which agree with idΔp×ℓ¯0\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {id}_{\Delta ^p}\times \bar{\ell }_0$$\end{document} on Δp×A\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p\times A$$\end{document}. The colimit is taken over the maps induced by post-composition with the maps Bm→Bm+1\documentclass[12pt]{minimal}
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\begin{document}$$B_m\rightarrow B_{m+1}$$\end{document} underlying the structure maps of ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document}. The sub semi-simplicial set of maps Δp×X→Δp×Bm\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p\times X\rightarrow \Delta ^p\times B_m$$\end{document} over Δp\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p$$\end{document} is denoted by MapsA(X,B;ℓ¯0)∙⊂Map~A(X,B;ℓ¯0)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Maps}_A(X,B;\bar{\ell }_0)_\bullet \subset \widetilde{\mathrm {Map}}_A(X,B;\bar{\ell }_0)_\bullet $$\end{document}.
Lemma A.1
If the base X of ξ\documentclass[12pt]{minimal}
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\begin{document}$$\xi $$\end{document} is a finite CW-complex, then the mapsBun~A(ξs,ψ;ℓ0)∙τ⟶Map~A(X,B;ℓ¯0)∙andhAut~A(ξs;C)∙τ⟶hAut~A(X;C)∙\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {\widetilde{Bun}}_A(\xi ^s,\psi ;\ell _0)^\tau _\bullet\longrightarrow & {} \widetilde{\mathrm {Map}}_A(X,B;\bar{\ell }_0)_\bullet \quad \text {and}\quad \\\mathrm {\widetilde{hAut}}_A(\xi ^s;C)^\tau _\bullet\longrightarrow & {} \mathrm {\widetilde{hAut}}_A(X;C)_\bullet \end{aligned}$$\end{document}induced by forgetting bundle maps are Kan fibrations.
Proof
For ψ=ξs\documentclass[12pt]{minimal}
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\begin{document}$$\psi =\xi ^s$$\end{document} and ℓ0=inc\documentclass[12pt]{minimal}
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\begin{document}$$\ell _0=\mathrm {inc}$$\end{document} the inclusion, the second map agrees with the pullback of the first map along the inclusion hAut~A(X;C)∙⊂Map~A(X,X;inc)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {\widetilde{hAut}}_A(X;C)_\bullet \subset \widetilde{\mathrm {Map}}_A(X,X;\mathrm {inc})_\bullet $$\end{document}, so it suffices to show that the first map is a Kan fibration. We shall do so in the case where A is empty; the argument for general A is similar. The upper horizontal map in a lifting problem
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\begin{document}$$\begin{aligned}\phi _i:\tau _{\Delta ^p_i}\times \xi \oplus \varepsilon ^k\longrightarrow \tau _{\Delta ^p_i}\times \psi _{d+k}\end{aligned}$$\end{document}for i≠j\documentclass[12pt]{minimal}
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\begin{document}$$i\ne j$$\end{document} that agree on their faces and cover the restriction of the map ϕ¯:Δp×X→Δp×Bd+k\documentclass[12pt]{minimal}
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\begin{document}$$\bar{\phi }:\Delta ^p\times X\rightarrow \Delta ^p\times B_{d+k}$$\end{document} induced by the bottom horizontal arrow to Δip×X\documentclass[12pt]{minimal}
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\begin{document}$$\Delta ^p_i\times X$$\end{document}. By replacing ξ\documentclass[12pt]{minimal}
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\begin{document}$$\xi $$\end{document} with its stabilisation ξ⊕εk\documentclass[12pt]{minimal}
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\begin{document}$$\xi \oplus \varepsilon ^k$$\end{document}, we may assume k=0\documentclass[12pt]{minimal}
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\begin{document}$$k=0$$\end{document}. There is an extension of ϕi\documentclass[12pt]{minimal}
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\begin{document}$$\phi _i$$\end{document} to τΔp|Δip\documentclass[12pt]{minimal}
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\begin{document}$$\tau _{\Delta ^p}|_{\Delta ^p_i}$$\end{document} given by the compositionτΔp|Δip×ξ→≅ε⊕τΔip×ξ→idε⊕ϕiε⊕τΔip×ψd→≅τΔp|Δip×ψd\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\tau _{\Delta ^p}|_{\Delta ^p_i} \times \xi \xrightarrow {\cong }\varepsilon \oplus \tau _{\Delta ^p_i} \times \xi \xrightarrow {\mathrm {id}_\varepsilon \oplus \phi _i } \varepsilon \oplus \tau _{\Delta ^p_i}\times \psi _{d}\xrightarrow {\cong }\tau _{\Delta ^p}|_{\Delta ^p_i} \times \psi _{d}\end{aligned}$$\end{document}whose outer isomorphisms are induced by the differential of the diffeomorphism ci,ϵ\documentclass[12pt]{minimal}
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\begin{document}$$c_{i,\epsilon }$$\end{document} of Section 1.3. The condition (4) in the definition of tangential bundle maps is made precisely such that these extensions agree on their intersections, so they assemble to a bundle mapϕΛjp:τΔp|Λjp×ξ⟶τΔp|Λjp×ψd\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\phi _{\Lambda ^p_j}:\tau _{\Delta ^p}|_{\Lambda ^p_j}\times \xi \longrightarrow \tau _{\Delta ^p}|_{\Lambda ^p_j}\times \psi _{d} \end{aligned}$$\end{document}that covers the restriction of ϕ¯\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda ^p_j\times X$$\end{document} and we are left to argue that this bundle map is the restriction of a p-simplex in Bun(ξs,ψ)∙τ\documentclass[12pt]{minimal}
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\begin{document}$$\bar{\phi }$$\end{document}. As the inclusion Λjp×X⊂Δp×X\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda ^p_j\times X\subset \Delta ^p\times X$$\end{document} is a trivial cofibration, obstruction theory provides an extension of ϕΛjp\documentclass[12pt]{minimal}
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\begin{document}$$\phi _{\Lambda ^p_j}$$\end{document} to a bundle mapϕΔp:τΔp×ξ⟶τΔp×ψd\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\phi _{\Delta ^p}:\tau _{\Delta ^p}\times \xi \longrightarrow \tau _{\Delta ^p}\times \psi _{d}\end{aligned}$$\end{document}that covers ϕ¯\documentclass[12pt]{minimal}
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\begin{document}$$\bar{\phi }$$\end{document}, but this extension might violate condition (4) on the jth face, i.e. the mapϕj:ε⊕τΔjp×ξ⟶ε⊕τΔjp×ψd\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\phi _j:\varepsilon \oplus \tau _{\Delta ^p_j}\times \xi \longrightarrow \varepsilon \oplus \tau _{\Delta ^p_j}\times \psi _{d}\end{aligned}$$\end{document}obtained by restricting ϕΔp\documentclass[12pt]{minimal}
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\begin{document}$$\phi _{\Delta ^p}$$\end{document} to Δjp×X\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\phi _j:\tau _{\Delta ^p_j}\times \xi \longrightarrow \tau _{\Delta ^p_j}\times \psi _{d}.\end{aligned}$$\end{document}Nevertheless, its restriction to ∂Δjp×X\documentclass[12pt]{minimal}
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\begin{document}$$\partial \Delta ^p_j\times X$$\end{document} does have this form and we will argue that after adding a trivial bundle of sufficiently large dimension, say n, we can alter ϕj\documentclass[12pt]{minimal}
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\begin{document}$$\phi _j$$\end{document} by a homotopy of bundle maps relative to ∂Δip×X\documentclass[12pt]{minimal}
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\begin{document}$$\bar{\phi }|_{\Delta ^p_i}$$\end{document} so that it does have the required form. This would show the claim because we can use this homotopy to change ϕΔp⊕idεn\documentclass[12pt]{minimal}
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\begin{document}$$\phi _{\Delta ^p}\oplus \mathrm {id}_{\varepsilon ^n}$$\end{document} to a p-simplex in Bun(ξ⊕εn,ψd⊕εn)∙τ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Bun}(\xi \oplus \varepsilon ^n,\psi _{d}\oplus \varepsilon ^n)^\tau _\bullet $$\end{document} that covers ϕ¯\documentclass[12pt]{minimal}
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\begin{document}$$\bar{\phi }$$\end{document} and extends ϕΛjp⊕idεn\documentclass[12pt]{minimal}
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\begin{document}$$\phi _{\Lambda ^p_j}\oplus \mathrm {id}_{\varepsilon ^n}$$\end{document} and thus provides a lift as required. To this end, we denote by Iso(ν,η)→Y\documentclass[12pt]{minimal}
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\begin{document}$$\nu $$\end{document} and η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document} over a space Y the fibre bundle whose sections correspond to bundle morphisms ν→η\documentclass[12pt]{minimal}
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\begin{document}$$\nu \rightarrow \eta $$\end{document} over the identity, so the fibre over y∈Y\documentclass[12pt]{minimal}
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\begin{document}$$\nu _y\rightarrow \eta _y$$\end{document} between the fibres of ν\documentclass[12pt]{minimal}
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\begin{document}$$\nu $$\end{document} and η\documentclass[12pt]{minimal}
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\begin{document}$$\eta $$\end{document}. Abbreviating ν:=τΔjp×ξ\documentclass[12pt]{minimal}
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\begin{document}$$\nu :=\tau _{\Delta _j^p}\times \xi $$\end{document} and η:=ϕ¯∗(τΔjp×ψd)\documentclass[12pt]{minimal}
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\begin{document}$$\eta :=\bar{\phi }^*(\tau _{\Delta _j^p}\times \psi _d)$$\end{document}, we consider the diagram
in which the solid diagonal arrow is induced by ϕj⊕idεn\documentclass[12pt]{minimal}
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\begin{document}$$\phi _j\oplus \mathrm {id}_{\varepsilon ^n}$$\end{document} and the upper left horizontal one by its restriction to ∂Δjp×X\documentclass[12pt]{minimal}
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\begin{document}$$\partial \Delta ^p_j\times X$$\end{document}, which makes the subdiagram of solid arrows commute strictly. From this point of view, the task we set us is equivalent to constructing a dashed arrow for some n that makes the upper leftmost triangle commutes strictly and the one formed by the two diagonal arrows up to homotopy relative ∂Δjp×X\documentclass[12pt]{minimal}
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\begin{document}$$\partial \Delta ^p_j\times X$$\end{document}. Using that X is a finite CW complex and that the connectivity of the map on vertical homotopy fibres of the right square increases in n as it agrees with the inclusion O(p+d+n)⊂O(p+d+n+1)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {O}(p+d+n)\subset \mathrm {O}(p+d+n+1)$$\end{document} up to equivalence, this follows from obstruction theory. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Remark
We learnt the “stabilisation trick” of the previous proof from Appendix D of [8], which contains results similar to those of this appendix.
Corollary A.2
If the base X of ξ\documentclass[12pt]{minimal}
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\begin{document}$$\xi $$\end{document} is a finite CW-complex, then the following semi-simplicial sets satisfy the Kan property.
It is straight-forward to see that Map~A(X,B;ℓ¯0)∙\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{\mathrm {Map}}_A(X,B;\bar{\ell }_0)_\bullet $$\end{document} is Kan, so the first part follows from Lemma A.1 and the fact that the domain of a Kan fibration over a Kan complex is Kan. The same reasoning applies to the second semi-simplicial set, using that BunA(ξs,ψ;ℓ0)∙τ→MapsA(X,B;ℓ¯0)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Bun}_A(\xi ^s,\psi ;\ell _0)^\tau _\bullet \rightarrow \mathrm {Maps}_A(X,B;\bar{\ell }_0)_\bullet $$\end{document} is a Kan fibration because it is the pullback of the first map of Lemma A.1 along the inclusion MapsA(X,B;ℓ¯0)∙⊂Map~A(X,B;ℓ¯0)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Maps}_A(X,B;\bar{\ell }_0)_\bullet \subset \widetilde{\mathrm {Map}}_A(X,B;\bar{\ell }_0)_\bullet $$\end{document}. Also the semi-simplicial sets hAutA(X;C)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {hAut}_A(X;C)_\bullet $$\end{document} and hAut~A(X;C)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {\widetilde{hAut}}_A(X;C)_\bullet $$\end{document} are easily seen to be Kan (see Section 1.5), so the remaining claims can be proved in the same way. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Lemma A.3
If the base X of ξ\documentclass[12pt]{minimal}
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\begin{document}$$\xi $$\end{document} is a finite CW-complex, then the inclusionsBunA(ξs,ψ;ℓ0)∙τ⊂Bun~A(ξs,ψ;ℓ0)∙τandhAutA(ξs;C)∙τ⊂hAut~A(ξs;C)∙τ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {Bun}_A(\xi ^s,\psi ;\ell _0)_\bullet ^\tau\subset & {} \mathrm {\widetilde{Bun}}_A(\xi ^s,\psi ;\ell _0)_\bullet ^\tau \quad \text {and}\\ \mathrm {hAut}_A(\xi ^s;C)_\bullet ^\tau\subset & {} \mathrm {\widetilde{hAut}}_A(\xi ^s;C)_\bullet ^\tau \end{aligned}$$\end{document}are equivalences.
Proof
These inclusions are pullbacks of the inclusionsMapsA(X,B;ℓ¯0)∙⊂Map~A(X,B;ℓ¯0)∙andhAutA(X;C)∙⊂hAut~A(X;C)∙\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned}&\mathrm {Maps}_A(X,B;\bar{\ell }_0)_\bullet \subset \widetilde{\mathrm {Map}}_A(X,B;\bar{\ell }_0)_\bullet \quad \text { and }\\&\quad \mathrm {hAut}_A(X;C)_\bullet \subset \mathrm {\widetilde{hAut}}_A(X;C)_\bullet \end{aligned}\end{aligned}$$\end{document}along the two Kan fibrations discussed in Lemma A.1, so the claim follows from showing that the inclusions (74) are equivalences. As already mentioned in the previous proof, it is straight-forward to show that these semi-simplicial sets are Kan. Using the combinatorial description of their homotopy groups, the claim follows from the contractibility of the mapping space Maps∂Δp(Δp,Δp)\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Maps}_{\partial \Delta ^p}(\Delta ^p,\Delta ^p)$$\end{document} (cf. Section 1.5). □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Lemma A.4
If the base X of ξ\documentclass[12pt]{minimal}
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\begin{document}$$\xi $$\end{document} is a finite CW-complex, then the extension maps of (7)BunA(ξs,ψ;ℓ0)∙→BunA(ξs,ψ;ℓ0)∙τandhAutA(ξs;C)∙→hAutA(ξs;C)∙τ\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\mathrm {Bun}_A(\xi ^s,\psi ;\ell _0)_\bullet\rightarrow & {} \mathrm {Bun}_A(\xi ^s,\psi ;\ell _0)_\bullet ^\tau \quad \text {and}\\ \mathrm {hAut}_A(\xi ^s;C)_\bullet\rightarrow & {} \mathrm {hAut}_A(\xi ^s;C)_\bullet ^\tau \end{aligned}$$\end{document}are equivalences.
Proof
The proof for the two maps are essentially identical; we restrict our attention to the first map. Its source and target have the Kan property (the source because it is the singular complex of a space, the target by Corollary A.2), so we may test the claim on semi-simplicial homotopy groups. The two semi-simplicial sets involved have the same 0-simplices, so the map is clearly surjective on path components. To show that it is surjective on homotopy groups in positive degrees, we fix a semi-simplicial base point in BunA(ξs,ψ;ℓ0)∙\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Bun}_A(\xi ^s,\psi ;\ell _0)_\bullet $$\end{document} by choosing a 0-simplex ℓ:ξ⊕εl→ψd+l\documentclass[12pt]{minimal}
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\begin{document}$$\ell :\xi \oplus \varepsilon ^l\rightarrow \psi _{d+l}$$\end{document} and taking products with the tangent bundles τΔp\documentclass[12pt]{minimal}
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\begin{document}$$\tau _{\Delta ^p}$$\end{document}. Letϕ:ξ⊕εk×τΔp⟶ψd+k×τΔp\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\phi :\xi \oplus \varepsilon ^k\times \tau _{\Delta ^p} \longrightarrow \psi _{d+k}\times \tau _{\Delta ^p}\end{aligned}$$\end{document}be a p-simplex that represents a class in πp(BunA(ξs,ψ;ℓ0)∙τ;ℓ)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _p(\mathrm {Bun}_A(\xi ^s,\psi ;\ell _0)^\tau _\bullet ;\ell )$$\end{document}. Up to changing ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document} within its homotopy class, we may assume that the underlying map ϕ¯:X×Δp→Bd+k×Δp\documentclass[12pt]{minimal}
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\begin{document}$$\bar{\phi }:X\times \Delta ^p\rightarrow B_{d+k}\times \Delta ^p$$\end{document} satisfies the collaring condition from Section 1.3 for some 0<ϵ<1/2\documentclass[12pt]{minimal}
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\begin{document}$$0<\epsilon <1/2$$\end{document}. By replacing ξ\documentclass[12pt]{minimal}
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\begin{document}$$\xi $$\end{document} with ξ⊕ϵk\documentclass[12pt]{minimal}
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\begin{document}$$k=0$$\end{document}. Fixing a trivialisation F:τΔp≅Rp×Δp\documentclass[12pt]{minimal}
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\begin{document}$$F:\tau _{\Delta ^p}\cong \mathbf {R}^p\times \Delta ^p$$\end{document}, our candidate for a preimage in πp(BunA(ξs,ψ;ℓ0)∙;ℓ)\documentclass[12pt]{minimal}
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\begin{document}$$\pi _p(\mathrm {Bun}_A(\xi ^s,\psi ;\ell _0)_\bullet ;\ell )$$\end{document} is the class defined by the compositionξ⊕εp×Δp⟶idξ×F-1ξ×τΔp⟶ϕψd×τΔp⟶idψd×Fψd⊕εp×Δp⟶ψd+p×Δp,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \begin{aligned} \xi \oplus \varepsilon ^p\times \Delta ^p&\overset{\mathrm {id}_{\xi }\times F^{-1}}{\longrightarrow }\xi \times \tau _{\Delta ^p}\overset{\phi }{\longrightarrow }\psi _{d}\times \tau _{\Delta ^p}\\ {}&\overset{\mathrm {id}_{\psi _{d}}\times F}{\longrightarrow } \psi _d\oplus \varepsilon ^p\times \Delta ^p\longrightarrow \psi _{d+p}\times \Delta ^p,\end{aligned}\end{aligned}$$\end{document}where the last arrow is induced by the structure map of ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document}. To justify this, we will show the existences of a (p+1)\documentclass[12pt]{minimal}
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\begin{document}$$(p+1)$$\end{document}-simplex in BunA(ξ⊕εp⊕ε2,ψd+p+2;ℓ0)∙τ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Bun}_A(\xi \oplus \varepsilon ^p\oplus \varepsilon ^2,\psi _{d+p+2};\ell _0)^\tau _\bullet $$\end{document} which on the pth face agrees with the (p+2)\documentclass[12pt]{minimal}
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\begin{document}$$(p+2)$$\end{document}-fold stabilisation of ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document}, on the (p+1)\documentclass[12pt]{minimal}
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\begin{document}$$(p+1)$$\end{document}st face with the image ϕF∈BunA(ξ⊕ϵp⊕ϵ2,ψd+p+2;ℓ0)pτ\documentclass[12pt]{minimal}
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\begin{document}$$\phi ^F\in \mathrm {Bun}_A(\xi \oplus \epsilon ^p\oplus \epsilon ^2,\psi _{d+p+2};\ell _0)_p^\tau $$\end{document} of the 2-fold stabilisation of the composition (75) and on the remaining faces with the basepoint. To this end, we consider the linear mapΛ:Rp×R×R×Rp⟶Rp×R×R×Rp[0.3em](x,u,v,y)⟼(y,u,v,x)ifpis even(y,v,u,x)ifpis odd,,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}\Lambda :\begin{array}{rcl} \mathbf {R}^p\times \mathbf {R}\times \mathbf {R}\times \mathbf {R}^p &{} \longrightarrow &{} \mathbf {R}^p\times \mathbf {R}\times \mathbf {R}\times \mathbf {R}^p \\ [0.3em] (x,u,v,y) &{} \longmapsto &{} {\left\{ \begin{array}{ll}(y,u,v,x)&{}\text {if }p\text { is even}\\ (y,v,u,x)&{}\text {if }p\text { is odd}, \end{array}\right. } \end{array},\end{aligned}$$\end{document}which has determinant 1 (this is the reason we introduced the additional (R×R)\documentclass[12pt]{minimal}
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\begin{document}$$(\mathbf {R}\times \mathbf {R})$$\end{document}-coordinate), so there exists a path γ:[0,1]→GL(Rp×R×R×Rp)\documentclass[12pt]{minimal}
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\begin{document}$$\gamma :[0,1]\rightarrow \mathrm {GL}(\mathbf {R}^p\times \mathbf {R}\times \mathbf {R}\times \mathbf {R}^p)$$\end{document} from the identity to Λ\documentclass[12pt]{minimal}
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\begin{document}$$\Lambda $$\end{document}, which we may choose to be constant in a neighborhood of [0,ϵ]∪[1-ϵ,1]\documentclass[12pt]{minimal}
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\begin{document}$$[0,\epsilon ]\cup [1-\epsilon ,1]$$\end{document}. In terms of this path, we define a homotopy Ht\documentclass[12pt]{minimal}
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\begin{document}$$H_t$$\end{document} of bundle automorphisms of εp⊕τΔp\documentclass[12pt]{minimal}
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\begin{document}$$\varepsilon ^p\oplus \tau _{\Delta ^p}$$\end{document} covering the identity byεp⊕ε2⊕τΔp→idRp×R2×FRp×R2×Rp×Δp→γt×idΔpRp×R2×Rp×Δp→idRp×R2×F-1εp⊕τΔp.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varepsilon ^p\oplus \varepsilon ^2\oplus \tau _{\Delta ^p}&\xrightarrow {\mathrm {id}_{\mathbf {R}^p\times \mathbf {R}^2}\times F}&\mathbf {R}^p\times \mathbf {R}^2\times \mathbf {R}^p\times \Delta ^p\\&\xrightarrow {\gamma _t\times \mathrm {id}_{\Delta ^p}}&\mathbf {R}^p\times \mathbf {R}^2\times \mathbf {R}^p\times \Delta ^p\xrightarrow {\mathrm {id}_{\mathbf {R}^p\times \mathbf {R}^2}\times F^{-1}}\varepsilon ^p\oplus \tau _{\Delta ^p}.\end{aligned}$$\end{document}Using H, we define a homotopy H~\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{H}$$\end{document} of bundle maps as the compositionξ⊕εp⊕ε2×τΔp⟶idξ×Htξ⊕εp⊕ε2×τΔp→ϕψd⊕εp⊕ε2×τΔp⟶idψd×Ht-1ψd⊕εp⊕ε2×τΔp→ψd+p+2×τΔp,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \xi \oplus \varepsilon ^p\oplus \varepsilon ^2\times \tau _{\Delta ^p}&\overset{\mathrm {id}_{\xi }\times H_t}{\longrightarrow }&\xi \oplus \varepsilon ^p\oplus \varepsilon ^2\times \tau _{\Delta ^p}\xrightarrow {\phi }\psi _{d}\oplus \varepsilon _p\oplus \varepsilon ^2 \times \tau _{\Delta ^p}\\&\overset{\mathrm {id}_{\psi _d}\times H_t^{-1}}{\longrightarrow }&\psi _{d}\oplus \varepsilon _p\oplus \varepsilon ^2\times \tau _{\Delta ^p}\rightarrow \psi _{d+p+2}\times \tau _{\Delta ^p},\end{aligned}$$\end{document}where the last map is induced by the structure map of ψ\documentclass[12pt]{minimal}
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\begin{document}$$\psi $$\end{document}. Going through the definitions, one checks that this is a homotopy from the stabilisation of ϕ\documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document} to ϕF\documentclass[12pt]{minimal}
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\begin{document}$$\phi ^F$$\end{document} and that its underlying homotopy of maps of spaces is constantly ϕ¯\documentclass[12pt]{minimal}
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\begin{document}$$\bar{\phi }$$\end{document}. Using the canonical trivialisation of τ[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$\tau _{[0,1]}$$\end{document}, this homotopy gives rise to a bundle map H~:ξ⊕εp⊕ε2×τΔp×[0,1]→ψd+p+2×τΔp×[0,1]\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{H}:\xi \oplus \varepsilon ^p\oplus \varepsilon ^2\times \tau _{\Delta ^p\times [0,1]}\rightarrow \psi _{d+p+2}\times \tau _{\Delta ^p\times [0,1]}$$\end{document} which one checks to descend uniquely to a dashed arrow making the diagram
commute, where c is the mapc:Δp×[0,1]⟶Δp+1[0.3em]((x0,…,xp),s)⟼(x0,…,xp-1,s·xp,(1-s)·xp)\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}c:\begin{array}{rcl} \Delta ^p\times [0,1] &{} \longrightarrow &{} \Delta ^{p+1} \\ [0.3em] ((x_0,\ldots ,x_p),s) &{} \longmapsto &{} (x_0,\ldots ,x_{p-1},s\cdot x_p,(1-s)\cdot x_{p}) \end{array}\end{aligned}$$\end{document}whose derivative is surjective (though not fibrewise). The resulting bundle map H¯\documentclass[12pt]{minimal}
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\begin{document}$$\bar{H}$$\end{document} has the correct behaviour on all faces, so almost provides a (p+1)\documentclass[12pt]{minimal}
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\begin{document}$$(p+1)$$\end{document}-simplex in BunA(ξ⊕εp⊕ε2,ψd+p+2;ℓ0)∙τ\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {Bun}_A(\xi \oplus \varepsilon ^p\oplus \varepsilon ^2,\psi _{d+p+2};\ell _0)^\tau _\bullet $$\end{document} as wished. The only problem is that it does not satisfy the collaring condition (4) for i=p\documentclass[12pt]{minimal}
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\begin{document}$$i=p$$\end{document}, but this can be rectified as follows: choosing a block diffeomorphism α:Δp+1→Δp+1\documentclass[12pt]{minimal}
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\begin{document}$$\alpha :\Delta ^{p+1}\rightarrow \Delta ^{p+1}$$\end{document} which agrees with the identity on Δp+1,δp+1\documentclass[12pt]{minimal}
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\begin{document}$$\delta >0$$\end{document} (see Section 1.3 for the notation) and makes the diagram
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\begin{document}$$\delta '>\epsilon $$\end{document}, the compositionξ⊕εp⊕ε2×τΔp+1⟶idξ⊕εp⊕ε2×dαξ⊕εp⊕ε2×τΔp+1⟶H¯ψd+p+2×τΔp+1⟶idξ⊕εp⊕ε2×dα-1ψd+p+2×τΔp+1\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \xi \oplus \varepsilon ^p\oplus \varepsilon ^2\times \tau _{\Delta ^{p+1}}&\overset{\mathrm {id}_{\xi \oplus \varepsilon ^p\oplus \varepsilon ^2}\times \mathrm {d}\alpha }{\longrightarrow }\xi \oplus \varepsilon ^p\oplus \varepsilon ^2\times \tau _{\Delta ^{p+1}}\overset{\bar{H}}{\longrightarrow }\psi _{d+p+2}\times \tau _{\Delta ^{p+1}}\\&\overset{\mathrm {id}_{\xi \oplus \varepsilon ^p\oplus \varepsilon ^2}\times \mathrm {d}\alpha ^{-1}}{\longrightarrow }\psi _{d+p+2}\times \tau _{\Delta ^{p+1}}\end{aligned}$$\end{document}defines a (p+1)\documentclass[12pt]{minimal}
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\begin{document}$$(p+1)$$\end{document}-simplex as required. This finishes the proof of surjectivity of the map on homotopy groups and injectivity follows from a relative version of the argument. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Appendix B. A lemma on symplectic derivations
This appendix serves to record a minor generalisation of Proposition 3.9 in [8]. We adopt the notation and conventions introduced in Sections 4.1 and 4.2 and fix a principal ideal domain R with 2∈R×\documentclass[12pt]{minimal}
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\begin{document}$$\lambda (v,-)$$\end{document} is an isomorphism. We moreover assume that λ\documentclass[12pt]{minimal}
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\begin{document}$$x,y\in V$$\end{document}.
Example B.1
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\begin{document}$$D(-)$$\end{document} denotes the Poincaré duality isomorphism.
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\begin{document}$$2\in R^\times $$\end{document}. Under this identification, one verifiesω=12∑i[ai#,ai]∈[V,V].\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \textstyle {\omega =\tfrac{1}{2}\sum _i[a_i^{\#},a_i]\in [V,V]}.\end{aligned}$$\end{document}as in p. 91 loc.cit. Given a dg Lie algebra K over R and a map f:V→K\documentclass[12pt]{minimal}
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\begin{document}$$f:V\rightarrow K$$\end{document} of graded R-modules, we consider the diagram of graded R-moduleswhose vertical isomorphism is given by the compositionDerf(L(V),K)⟶≅Hom(V,K)⟶≅K⊗V∨⟶≅K⊗s-kV=s-kK⊗V,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathrm {Der}^f(\mathbf {L}(V),K)\overset{\cong }{\longrightarrow }\mathrm {Hom}(V,K)&\overset{\cong }{\longrightarrow }&K\otimes V^\vee \\ {}&\overset{\cong }{\longrightarrow }&K\otimes s^{-k}V=s^{-k}K\otimes V, \end{aligned}$$\end{document}where the first map is given by restriction to generators, the second map is the canonical isomorphism, and the third map is induced by the inverse of the adjoint map of λ\documentclass[12pt]{minimal}
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\begin{document}$$R=\mathbf {Q}$$\end{document}, K=L(V)\documentclass[12pt]{minimal}
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\begin{document}$$K=\mathbf {L}(V)$$\end{document}, and f=incV⊂L(V)\documentclass[12pt]{minimal}
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\begin{document}$$f=\mathrm {inc}_{V\subset \mathbf {L}(V)}$$\end{document}.
Lemma 1
The diagram (77) is commutative.
Proof
Following the beginning of the proof of [8, Prop. 3.9], one sees that a derivation θ∈Derf(L(V),K)\documentclass[12pt]{minimal}
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\begin{document}$$\theta \in \mathrm {Der}^f(\mathbf {L}(V),K)$$\end{document} evaluates on the special element ω\documentclass[12pt]{minimal}
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\begin{document}$$\omega $$\end{document} as θ(ω)=∑i[θ(ai#),f(ai)]\documentclass[12pt]{minimal}
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\begin{document}$$\theta (\omega )=\sum _i[\theta (a_i^{\#}),f(a_i)]$$\end{document}. We will use this to show the equivalent claim that the inverse of the vertical map in (77) followed by evω\documentclass[12pt]{minimal}
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\begin{document}$$\mathrm {ev}_\omega $$\end{document} agrees with [-,f(-)]\documentclass[12pt]{minimal}
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\begin{document}$$[-,f(-)]$$\end{document}. This inverse assigns an element k⊗v∈s-kK⊗V\documentclass[12pt]{minimal}
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\begin{document}$$k\otimes v\in s^{-k}K\otimes V$$\end{document} the unique f-derivation θk,v\documentclass[12pt]{minimal}
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\begin{document}$$\theta _{k,v}$$\end{document} that extends the linear map λ(v,-)·k:V→K\documentclass[12pt]{minimal}
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\begin{document}$$\lambda (v,-)\cdot k:V\rightarrow K$$\end{document}. Using that for v∈V\documentclass[12pt]{minimal}
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\begin{document}$$v\in V$$\end{document}, we have v=∑iλ(v,ai#)·ai\documentclass[12pt]{minimal}
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\begin{document}$$v=\sum _i\lambda (v,a_i^{\#})\cdot a_i$$\end{document} by the definition of the dual basis, we computeθk,v(ω)=∑i[θk,v(ai#),f(ai)]=∑i[λ(v,ai#)·k,f(ai)]=[k,f(∑iλ(v,ai#)·ai)]=[k,v],\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \theta _{k,v}(\omega )= & {} \sum _i[\theta _{k,v}(a_i^{\#}),f(a_i)]=\sum _i[\lambda (v,a_i^{\#})\cdot k,f(a_i)]\\= & {} [k,f(\sum _i\lambda (v,a_i^{\#})\cdot a_i)]=[k,v],\end{aligned}$$\end{document}which implies the claim. □\documentclass[12pt]{minimal}
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\begin{document}$$\square $$\end{document}
Acknowledgements
My thanks go to Oscar Randal-Williams for several valuable discussions and to a referee for their constructive feedback. I was partially supported by O. Randal-Williams’ Philip Leverhulme Prize from the Leverhulme Trust and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 756444).
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See also Igusa’s corrections in [21].
In [6, App. A §3 a], the authors attempt to construct degeneracy maps for certain semi-simplicial sets of block embeddings, which would in particular enhance Diff~∂(W)∙\documentclass[12pt]{minimal}
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\begin{document}$$\widetilde{\mathrm {Diff}}_\partial (W)_\bullet $$\end{document} to a simplicial group and thus imply that it satisfies the Kan property as every simplicial group does. However, the argument in [6] is flawed: on two of the faces, the proposed degeneracy maps violate the collaring condition described on p. 116 loc. cit. The more laborious argument in [18] appears to be correct.
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