Projective twists and the Hopf correspondence

Abstract

This dissertation is the fruit of a research project on a class of symplectic automorphisms called $projective$ $twists$. In the first part of the thesis (Chapters 3,4) we use Picard$-$Lefschetz theory to introduce a new local model for the planar projective twists $\tau_{\mathbb{A}\mathbb{P}^2} \in \mathrm{Symp}{ct}(T^\mathbb{A}\mathbb{P}^2), \ \mathbb{A} \in { \mathbb{R}, \mathbb{C} }$. In each case, we construct an exact Lefschetz fibration $\pi\colon T^\mathbb{A}\mathbb{P}^2\to \mathbb{C}$ with three singular fibres, and define a compactly supported symplectomorphism $\varphi \in \mathrm{Symp}{ct}(T^*\mathbb{A}\mathbb{P}^2)$ on the total space. Given two disjoint Lefschetz thimbles $\Delta_{\alpha},\Delta_{\beta} \subset T^*\mathbb{A}\mathbb{P}^2$, we compute the Floer cohomology groups $\operatorname{HF}(\varphi^k(\Delta_{\alpha}), \Delta_{\beta};\mathbb{Z}/2\mathbb{Z})$ and verify (partially for $\mathbb{C}\mathbb{P}^2$) that $\varphi$ is indeed isotopic to (a power of) the standard local projective twist. The constructions we present are governed by $generalised$ $lantern$ $relations$, which provide an isotopy between the global monodromy of a Lefschetz fibration and a fibred twist along an $S^1$-fibred coisotropic submanifold of the smooth fibre. We also use these relations to study two classes of monotone Lagrangian submanifolds of $(T^\mathbb{C}\mathbb{P}^2, d\lambda_{T^\mathbb{C}\mathbb{P}^2})$. In the second part of the thesis, starting from Chapter 5, we investigate the properties of projective twists within the symplectic mapping class group of Liouville/Stein manifolds. We define the $Hopf$ $correspondence$, a Lagrangian correspondence (in the sense of Wehrheim$-$Woodward) aimed at assigning Lagrangian spheres $L_1, \dots , L_m$ of a Liouville manifold $(Y, \Omega)$ to given Lagrangian (real, complex) projective spaces $K_1, \dots , K_m$ of a Liouville manifold $(W, \omega)$. When this correspondence can be established, it intertwines the (real, complex) projective twists $\tau_{K_i} \in \pi_0(\mathrm{Symp}{ct}(W))$ (and the induced autoequivalences of the compact Fukaya category $\operatorname{\mathcal{F}uk}(W)$) with the Dehn twists $\tau{L_i} \in \pi_{0}(\mathrm{Symp}_{ct}(Y))$ (and the corresponding autoequivalences of $\operatorname{\mathcal{F}uk}(Y)$), for $i=1, \dots m$. Using the Hopf correspondence, we obtain a free generation result for projective twists in a clean plumbing of projective spaces and a result about products of positive powers of real projective twists in Liouville manifolds. The same techniques are also used to show that in infinitely many dimensions $n$, the Hamiltonian class of the local projective twist in $\mathrm{Symp}{ct}(T^\mathbb{C}\mathbb{P}^n)$ does depend on a choice of framing, i.e a choice of smooth parametrisation of the Lagrangian projective space used to define the twist. Another application of the Hopf correspondence delivers smooth homotopy complex projective spaces $K\simeq \mathbb{C}\mathbb{P}^n$, that do not admit Lagrangian embeddings into $(T^\mathbb{C}\mathbb{P}^n, d\lambda{T^*\mathbb{C}\mathbb{P}^n})$, for $n=4,7$.