2022/02/19 by Noah Porcelli, Porcelli, Noah · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)
paper · pdf · doi:10.48550/arxiv.2202.09677
openalex publication_date 2022/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that (under appropriate orientation conditions, depending on R) a Hamiltonian isotopy ψ1 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 (\citeHu-Lalonde-Leclercq), 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, ψ1|L is homotopic to the identity. Our technical set-up differs from both theirs and that of Cohen, Jones and Segal (\citeCohen-Jones-Segal, Cohen). We also prove (under similar conditions) that ψ1|L acts trivially on R_*(L L), where L L is the free loop space of L. From this we deduce that when L is a surface or a K(π, 1), ψ1|L is homotopic to the identity. Using methods of \citeLalonde-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 ℤ/2.