Mathematics – Symplectic Geometry
Scientific paper
2011-11-25
Mathematics
Symplectic Geometry
31 pages, incorrect usage of area in the localization process is replaced by the usage of maximum principle, a coincidence the
Scientific paper
Localization of Floer homology is first introduced by Floer \cite{floer:fixed} in the context of Hamiltonian Floer homology. The author employed the notion in the Lagrangian context for the pair $(\phi_H^1(L),L)$ of compact Lagrangian submanifolds in tame symplectic manifolds $(M,\omega)$ in \cite{oh:newton,oh:imrn} for a compact Lagrangian submanifold $L$ and $C^2$-small Hamiltonian $H$. In this article, we extend the localization process for any engulfable Hamiltonian path $\phi_H$ whose time-one map $\phi_H^1$ is sufficiently $C^0$-close to the identity (and also to the case of triangle product), and prove that the value of local Lagrangian spectral invariant is the same as that of global one. Such a Hamiltonian path naturally occurs as an approximating sequence \cite{oh:homotopy} of engulfable topological Hamiltonian loop. We apply this localization to the graphs $\Graph \phi_H^t$ in $(M\times M, \omega\oplus -\omega)$ and localize the Hamiltonian Floer complex of such a Hamiltonian $H$. This study plays an important role in the proof of homotopy invariance of the spectral invariants of topological Hamiltonian flows proved in \cite{oh:homotopy}.
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