Topological properties of Hamiltonian circle actions

Abstract

This paper studies Hamiltonian circle actions, i.e. circle subgroups of the group Ham(M,ω) of Hamiltonian symplectomorphisms of a closed symplectic manifold (M,ω). Our main tool is the Seidel representation of π1(Ham(M,ω)) in the units of the quantum homology ring. We show that if the weights of the action at the points at which the moment map is a maximum are sufficiently small then the circle represents a nonzero element of π1(Ham(M,ω)). Further, if the isotropy has order at most two and the circle contracts in Ham(M,ω) then various symmetry properties hold. For example, the image of the normalized moment map is a symmetric interval [−a, a]. If the action is semifree (i.e. the isotropy weights are 0 or ±1) then we calculate the leading order term in the Seidel representation, an important technical tool in understanding the quantum cohomology of manifolds that admit semifree Hamiltonian circle actions. If the manifold is toric, we use our results about this representation to describe the basic multiplicative structure of the quantum cohomology ring of an arbitrary toric manifold. There are two important technical ingredients; one relates the equivariant cohomology of M to the Morse flow of the moment map, and the other is a version of the localization principle for calculating Gromov–Witten invariants on symplectic manifolds with S1-actions.