Symplectic Floer homology of area-preserving surface diffeomorphisms

TL;DR

This paper presents an algorithm to compute symplectic Floer homology for surface symplectomorphisms, enhancing understanding of fixed points through holomorphic cylinder counts, especially for pseudo-Anosov and reducible classes.

Contribution

It provides a complete algorithm for computing HF_*(f) for surface symplectomorphisms in pseudo-Anosov or reducible classes, extending Seidel's work to all orientation-preserving mapping classes.

Findings

Algorithm successfully computes HF_*(f) for specified classes

Completes the computation of Seidel's HF_*(h) for all orientation-preserving classes

Advances understanding of fixed points via holomorphic cylinders in symplectic topology

Abstract

The symplectic Floer homology HF_*(f) of a symplectomorphism f:S->S encodes data about the fixed points of f using counts of holomorphic cylinders in R x M_f, where M_f is the mapping torus of f. We give an algorithm to compute HF_*(f) for f a surface symplectomorphism in a pseudo-Anosov or reducible mapping class, completing the computation of Seidel's HF_*(h) for h any orientation-preserving mapping class.