arXiv subjects
Publications and source records attributed to Andrew Cotton-Clay.
The symplectic Floer homology HF_*(f) of a symplectomorphism f:S->S encodes data about the fixed points of f using counts of holomorphic cylinders in R x M_f, where M_f is the mapping torus of f. We give an algorithm to compute HF_*(f) for f 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, 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-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.