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June 2005 ReidemeisterTorsion in Floer--Novikov Theory and Counting Pseudo-holomorphicTori, I
Yi-Jen Lee
J. Symplectic Geom. 3(2): 221-311 (June 2005).

Abstract

This is the first part of an article in two parts, which builds the foundation of a Floer-theoretic invariant, $I_{\textrm{F}}$. The Floer homology can be trivial in many variants of the Floer theory; it is therefore interesting to consider more refined invariants of the Floer complex. We consider one such instance --- the Reidemeister torsion $\tau_{\textrm{F}}$ of the Floer--Novikov complex of (possibly non-Hamiltonian) symplectomorphisms. $\tau_{\textrm{F}}$ turns out not to be invariant under Hamiltonian \hbox{isotopies}, but this failure may be fixed by introducing certain "correction term'': We define a Floer-theoretic zeta function $\zeta_{\textrm{F}}$, by counting perturbed pseudo-holomorphic tori in a way very similar to the genus 1 Gromov invariant. The main result of this article states that under suitable monotonicity conditions, the product $I_{\textrm{F}}:=\tau_{\textrm{F}}\zeta_{\textrm{F}}$ is invariant under Hamiltonian isotopies. In fact, $I_{\textrm{F}}$ is invariant under general symplectic isotopies when the underlying symplectic manifold $M$ is monotone. Because the torsion invariant we consider is not a homotopy invariant, the continuation method used in typical invariance proofs of Floer theory does not apply; instead, the detailed bifurcation analysis is worked out. This is the first time such analysis appears in the Floer theory literature in its entirety. Applications of $I_{\textrm{F}}$, and the construction of $I_{\textrm{F}}$ in different versions of Floer theories are discussed in sequels to this article [Y.-J.L.].

Citation

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Yi-Jen Lee. "ReidemeisterTorsion in Floer--Novikov Theory and Counting Pseudo-holomorphicTori, I." J. Symplectic Geom. 3 (2) 221 - 311, June 2005.

Information

Published: June 2005
First available in Project Euclid: 13 April 2006

zbMATH: 1093.53091
MathSciNet: MR2199540

Rights: Copyright © 2005 International Press of Boston

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Vol.3 • No. 2 • June 2005
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