A spectral sequence of the Floer cohomology of symplectomorphisms of trivial polarization class

TL;DR

This paper establishes a spectral sequence relating the Floer cohomology of a symplectomorphism and its square on certain exact symplectic manifolds, under specific fixed point conditions.

Contribution

It extends Seidel and Smith's localization theorem to new settings, providing a spectral sequence connecting Floer cohomologies of symplectomorphisms and their squares.

Findings

Spectral sequence from HF(φ^2) to HF(φ) established.

Conditions for the spectral sequence's existence are specified.

Application to symplectomorphisms with trivializable tangent bundle.

Abstract

Let $M$ be an exact symplectic manifold equal to a symplectization near infinity and having stably trivializable tangent bundle, and $\phi$ be an exact symplectomorphism of $M$ which, near infinity, is equal to either the identity or the symplectization of a contactomorphism $\hat{\phi}$ such that neither $\hat{\phi}$ nor $\hat{\phi}^2$ has fixed points. We give conditions under which Seidel and Smith's localization theorem for Lagrangian Floer cohomology implies the existence of a spectral sequence from $\mathit{HF}(\phi^2)\otimes \mathbb Z_2((\theta))$ to $\mathit{HF}(\phi)\otimes \mathbb Z_2((\theta))$.