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_F). (See math.DG/0505013 for part II). 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_F) of the Floer-Novikov complex of (possibly non-hamiltonian) symplectomorphisms. (\tau_F) turns out NOT to be invariant under hamiltonian isotopies, but this failure may be fixed by introducing certain ``correction term'': We define a Floer-theoretic zeta function (\zeta_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_F:=\tau_F\zeta_F) is invariant under hamiltonian isotopies. In fact, (I_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_F), and the construction of (I_F) in different versions of Floer theories are discussed in sequels to this article.