Communications in Information and Systems
Volume 13 (2013)
Special Issue in Honor of Marshall Slemrod: Part 4 of 4
Localization of Floer homology of engulfed topological Hamiltonian loop
Pages: 399 – 443
Localization of Floer homology is first introduced by Floer [F12] in the context of Hamiltonian Floer homology. The author employed the notion in the Lagrangian context for the pair $(\phi^1_H (L), L)$ of compact Lagrangian submanifolds in tame symplectic manifolds $(M, \omega)$ in [Oh1, Oh2] for a compact Lagrangian submanifold $L$ and $C^2$-small Hamiltonian $H$. In this article, motivated by the study of topological Hamiltonian dynamics, we extend the localization process for any engulfed Hamiltonian path $\phi_H$ whose time-one map $\phi^1_H$ is sufficiently $C^0$-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 $\phi^t_H$ in $(M \times M, \omega \oplus -\omega)$ 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.
local Floer homology, engulfed topological Hamiltonian loop, $J_0$-convex domain, maximum principle, thick-thin dichotomy, handle sliding lemma