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

TL;DR

This paper proves that Hamiltonian isotopies fixing a relatively exact Lagrangian act trivially on certain generalized homology groups, with implications for the homotopy type of the isotopy on the Lagrangian and its free loop space.

Contribution

It extends previous results by showing trivial action of Hamiltonian isotopies on generalized homology for a broader class of Lagrangians, including homotopy spheres and spaces with specific fundamental groups.

Findings

Hamiltonian isotopies act trivially on generalized homology groups of Lagrangians.

If L is a homotopy sphere, the isotopy is homotopic to the identity.

For surfaces or K(π,1) spaces, the isotopy is homotopic to the identity.

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 $R_*(L)$, where $R_*$ 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 $R_*(\mathcal{L} L)$, 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 $\mathbb{Z}/2$.