Families of relatively exact Lagrangians, free loop spaces and generalised homology

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2024年2月3日

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Noah W. Porcelli

摘要

We prove that (under appropriate orientation conditions, depending on R) a Hamiltonian isotopy $\psi ^1$ $ψ^{1}$ of a symplectic manifold $(M, \omega )$ $(M, ω)$ fixing a relatively exact Lagrangian L setwise must act trivially on $R_*(L)$ $R_{*} (L)$ , where $R_*$ $R_{*}$ is some generalised homology theory. We use a strategy inspired by that of Hu et al. (Geom Topol 15:1617–1650, 2011), who proved an analogous result over ${\mathbb {Z}}/2$ $Z / 2$ and over ${\mathbb {Z}}$ $Z$ under stronger orientation assumptions. However the differences in our approaches let us deduce that if L is a homotopy sphere, $\psi ^1|_L$ $ψ^{1} |_{L}$ is homotopic to the identity. Our technical set-up differs from both theirs and that of Cohen et al. (in: Algebraic topology, Springer, Berlin, 2019) and Cohen (in: The Floer memorial volume, Birkhäuser, Basel). We also prove (under similar conditions) that $\psi ^1|_L$ $ψ^{1} |_{L}$ acts trivially on $R_*({\mathcal {L}}L)$ $R_{*} (L L)$ , where ${\mathcal {L}}L$ $L L$ is the free loop space of L. From this we deduce that when L is a surface or a $K(\pi , 1)$ $K (π, 1)$ , $\psi ^1|_L$ $ψ^{1} |_{L}$ is homotopic to the identity. Using methods of Lalonde and McDuff (Topology 42:309–347, 2003), we also show that given a family of Lagrangians all of which are Hamiltonian isotopic to L over a sphere or a torus, the associated fibre bundle cohomologically splits over ${\mathbb {Z}}/2$ $Z / 2$ .