Joint Topology Seminar

Abstract. We show that the rank of a certain twisted version of symplectic Floer homology gives a bound on the number of fixed points of any map with nondegenerate fixed points in a given symplectic mapping class on a monotone symplectic manifold. We apply this to the case of area-preserving surface diffeomorphisms to give a sharp lower bound on the number of fixed points in a given mapping class which frequently exceeds the classical sharp bound on non-area-preserving maps given by Nielsen theory. This generalizes the Poincaré-Birkhoff fixed point theorem for area-preserving twist maps on the cylinder to arbitrary surfaces and mapping classes.