Reidemeister torsion in Floer-Novikov theory and counting pseudo-holomorphic tori, I

Details Reidemeister torsion in Floer-Novikov theory and counting pseudo-holomorphic tori, I Reidemeister torsion in Floer-Novikov theory and counting pseudo-holomorphic tori, I

2001-11-30

arxiv.org/abs/math/0111313v3

Mathematics

Differential Geometry

88 pages. the previous version is splitted into two parts. This is the first part

Scientific paper

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.

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