Reidemeister Torsion in Floer--Novikov Theory and Counting Pseudo-holomorphic Tori, I

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.].