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 $ψ^1$ of a symplectic manifold $(M, ω)$ 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, $ψ^1|_L$ 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 $ψ^1|_L$ 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 $K(π, 1)$, $ψ^1|_L$ 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 $\mathbb{Z}/2$.

Theorems & Definitions (164)

• Theorem 1.2: Mei-LinYau
• Theorem 1.4: Hu-Lalonde-Leclercq
• Remark 1.5
• Remark 1.6
• Theorem 1.8
• Remark 1.9
• Corollary 1.10
• Lemma 1.11
• Lemma 1.12
• Lemma 1.13