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 .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 τ F of the Floer-Novikov complex of (possibly non-Hamiltonian) symplectomorphisms.τ 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 ζ 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 := τ F ζ 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.

Locations

• Journal of Symplectic Geometry - View - PDF
• Project Euclid (Cornell University) - View - PDF