Symplectic Floer homology of area-preserving surface diffeomorphisms

Abstract

The symplectic Floer homology ${HF}_{\ast }\left(\varphi \right)$ of a symplectomorphism $\varphi :\Sigma \to \Sigma$ encodes data about the fixed points of $\varphi$ using counts of holomorphic cylinders in $ℝ\times M_{\varphi }$, where $M_{\varphi }$ is the mapping torus of $\varphi$. We give an algorithm to compute ${HF}_{\ast }\left(\varphi \right)$ for $\varphi$ a surface symplectomorphism in a pseudo-Anosov or reducible mapping class, completing the computation of Seidel’s ${HF}_{\ast }\left(h\right)$ for $h$ any orientation-preserving mapping class.