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

Abstract

We prove that (under appropriate orientation conditions, depending on $R$) a Hamiltonian isotopy ${\psi }^1$ of a symplectic manifold $(M,\omega )$ fixing a relatively exact Lagrangian $L$ setwise must act trivially on $R_{\ast }(L)$, where $R_{\ast }$ is some generalised homology theory. We use a strategy inspired by that of Hu, Lalonde and Leclercq (\cite{Hu-Lalonde-Leclercq}), who proved an analogous result over $\mathbb{Z}/2$ and over $\mathbb{Z}$ under stronger orientation assumptions. However the differences in our approaches let us deduce that if $L$ is a homotopy sphere, ${\psi }^1{|}_L$ is homotopic to the identity. Our technical set-up differs from both theirs and that of Cohen, Jones and Segal (\cite{Cohen-Jones-Segal, Cohen}). We also prove (under similar conditions) that ${\psi }^1{|}_L$ acts trivially on $R_{\ast }(\mathcal{L}L)$, where $\mathcal{L}L$ is the free loop space of $L$. From this we deduce that when $L$ is a surface or a $K(\pi ,1)$, ${\psi }^1{|}_L$ is homotopic to the identity. Using methods of \cite{Lalonde-McDuff}, 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 $\mathbb{Z}/2$.

arXiv commentAccepted version. Appendix B added containing a new example; another appendix removed; an error in Section 3 is fixed. More small changes made suggestions of referees and thesis examiners