Symplectic Floer homology of area-preserving surface diffeomorphisms and sharp fixed point bounds

The symplectic Floer homology HF*(&phis;) of a symplectomorphism &phis; : Sigma → Sigma encodes data about the fixed points of &phis; using counts of holomorphic cylinders in R x M&phis;, where M &phis; is the mapping torus of &phis;. We give an algorithm to compute HF*(&phis;) for &phis; 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.;Given a closed, oriented surface Sigma, possibly with boundary, and a mapping class g ∈ pi0(Diff+(Sigma, ∂Sigma)), we obtain a sharp lower bound on the number of fixed points of a surface symplectomorphism (i.e. area-preserving map) in the given mapping class, generalizing the Poincare-Birkhoff fixed point theorem. This bound often exceeds that for non-area-preserving maps. These bounds come from Floer homology computations with certain twisted coefficients and a new method for obtaining fixed point bounds on entire symplectic mapping classes on monotone symplectic manifolds from such computations.