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

Abstract

Given a closed, oriented surface Σ, 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 Poincaré-Birkhoff fixed point theorem to arbitrary surfaces and mapping classes. These bounds often exceed those for non-areapreserving 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.1 Our results extend to the nonnegative Euler characteristic case but these exceptional cases would be cumbersome to carry around. All of these cases are already understood.2 That is, the fixed points of φ are cut out transversally in the sense that det(1 − dφx) = 0 for fixed points x.

The growth rate of Floer homology and symplectic zeta function

“…Corollary 2.10. As an application, A. Cotton-Clay gave recently [5] a sharp lower bound on the number of fixed points of area-preserving map in any prescribed mapping class(rel boundary), generalising the Poincare-Birkhoff fixed point theorem.…”