Communications in Information and Systems, v.13, no.4, pp.399 - 443
International Press of Boston, Inc.
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
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