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 $$\psi ^1$$ 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 .