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

We prove that (under appropriate orientation conditions, depending on R) a Hamiltonian isotopy ψ 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 et al. (Geom Topol 15:1617–1650, 2011), who proved an analogous result over Z/2 and over Z under stronger orientation assumptions. However the differences in our approaches let us deduce that if L is a homotopy sphere, ψ| 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 ψ| acts trivially on R(LL), where LL is the free loop space of L. From this we deduce that when L is a surface or a K(π,1), ψ| 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 Z/2.