  Given a closed, oriented surface, possibly with boundary, and a mapping
class, we obtain sharp lower bounds on the number of fixed points of a surface
symplectomorphism (i.e. area-preserving map) in the given mapping class, both
with and without nondegeneracy assumptions on the fixed points. This
generalizes the Poincar\'e-Birkhoff fixed point theorem to arbitrary surfaces
and mapping classes. These bounds often exceed those for non-area-preserving
maps. We obtain these bounds from Floer homology computations with certain
twisted coefficients plus a method for obtaining fixed point bounds on entire
symplectic mapping classes on monotone symplectic manifolds from such
computations. For the case of possibly degenerate fixed points, we use
quantum-cup-length-type arguments for certain cohomology operations we define
on summands of the Floer homology.
