Symplectic Floer homology of area-preserving surface diffeomorphisms

Cotton-Clay, Andrew

doi:10.2140/gt.2009.13.2619

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Symplectic Floer homology of area-preserving surface diffeomorphisms

Andrew Cotton-Clay

Geometry & Topology 13 (2009) 2619–2674

DOI: 10.2140/gt.2009.13.2619

Abstract

The symplectic Floer homology ${H F}_{*} (ϕ)$ of a symplectomorphism $ϕ : Σ \to Σ$ encodes data about the fixed points of $ϕ$ using counts of holomorphic cylinders in $ℝ \times M_{ϕ}$ , where $M_{ϕ}$ is the mapping torus of $ϕ$ . We give an algorithm to compute ${H F}_{*} (ϕ)$ for $ϕ$ a surface symplectomorphism in a pseudo-Anosov or reducible mapping class, completing the computation of Seidel’s ${H F}_{*} (h)$ 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

References

Publication

Received: 18 July 2008

Revised: 29 April 2009

Accepted: 5 February 2009

Published: 30 July 2009

Proposed: Peter Ozsváth

Seconded: Yasha Eliashberg, Leonid Polterovich

Authors

Andrew Cotton-Clay
Department of Mathematics Harvard University One Oxford St Cambridge, MA 01238 USA
http://math.harvard.edu/~acotton

Volume 13, issue 5 (2009)