Floer Homology for Symplectomorphism

TL;DR

This paper introduces a Floer-type homology for symplectomorphisms on compact symplectic manifolds, generalizing previous fixed point Floer homology to broader settings and establishing its algebraic properties.

Contribution

It defines a new Floer homology for symplectomorphisms that extends existing theories to non-monotone manifolds and analyzes its algebraic structure and invariance.

Findings

Homology groups depend only on the symplectomorphism up to Hamiltonian isotopy.

The homology is a module over a Novikov ring.

Generalizes Floer homology for fixed points to broader classes of symplectomorphisms.

Abstract

Let (M,\omega) be a compact symplectic manifold, and \phi be a symplectic diffeomorphism on M, we define a Floer-type homology FH_*(\phi) which is a gen- eralization of Floer homology for symplectic fixed points defined by Dostoglou and Salamon for monotone symplectic manifolds. These homology groups are modules over a suitable Novikov ring and depend only on \phi up to a Hamiltonian isotopy.