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Given a closed, oriented surface, possibly with boundary, and a mappingclass, we obtain sharp lower bounds on the number of fixed points of a surfacesymplectomorphism (i.e. area-preserving map) in the given mapping class, bothwith and without nondegeneracy assumptions on the fixed points. Thisgeneralizes the Poincar\'e-Birkhoff fixed point theorem to arbitrary surfacesand mapping classes. These bounds often exceed those for non-area-preservingmaps. We obtain these bounds from Floer homology computations with certaintwisted coefficients plus a method for obtaining fixed point bounds on entiresymplectic mapping classes on monotone symplectic manifolds from suchcomputations. For the case of possibly degenerate fixed points, we usequantum-cup-length-type arguments for certain cohomology operations we defineon summands of the Floer homology.

A sharp bound on fixed points of surface symplectomorphisms in each mapping class