Localization of Floer homology of engulfed topological Hamiltonian loop

Abstract

Localization of Floer homology is first introduced by Floer [Fl2] in the context of Hamiltonian Floer homology. The author employed the notion in the Lagrangian context for the pair (φ1 H(L), L) of compact Lagrangian submanifolds in tame symplectic manifolds (M,ω) in [Oh1, Oh2] for a compact Lagrangian submanifold L and C2-small Hamiltonian H. In this article, motivated by the study of topological Hamiltonian dynamics, we extend the localization process for any engulfed Hamiltonian path φH whose time-one map φ1 H is sufficiently C0-close to the identity (and also to the case of triangle product), and prove that the local Lagrangian spectral invariant defined on a Darboux-Weinstein neighborhood of L is the same as the global one defined on the full cotangent bundle T.L. Such a Hamiltonian path naturally occurs as an approximating sequence of engulfed topological Hamiltonian loop. We also apply this localization to the graph of φt H in (M × M,ω ..ω) and localize the Hamiltonian Floer complex of such a Hamiltonian H. We expect that this study will play an important role in the study of homotopy invariance of the spectral invariants of topological Hamiltonian.