Volume 13, issue 5 (2009)

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Symplectic Floer homology of area-preserving surface diffeomorphisms

Andrew Cotton-Clay

Geometry & Topology 13 (2009) 2619–2674
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 $ℝ×{M}_{\varphi }$, 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.

Keywords
Floer homology, symplectomorphism, surface diffeomorphism, mapping class group, fixed point, Nielsen class
Mathematical Subject Classification 2000
Primary: 53D40, 37J10