We ask whether every homologically trivial cyclic action on a symplectic four-manifold extend to a Hamiltonian circle action. By a cyclic action we mean an action of a cyclic group of finite order; it is homologically trivial if it induces the identity map on homology. We assume that the manifold is closed and connected. In the talk, I will give an example of a homologically trivial symplectic cyclic action on a four-manifold that admits Hamiltonian circle actions, and show that is does not extend to a Hamiltonian circle action. I will also discuss symplectic four-manifolds on which every homologically trivial cyclic action extends to a Hamiltonian circle action. I will deduce corollaries on the existence of homologically trivial cyclic actions and on embedding finite-order cyclic subgroups of the group of Hamiltonian symplectomorphisms in circle subgroups. This work applies holomorphic methods to extend combinatorial tools developed for circle actions to study cyclic actions.
Tue, 12/09/2017 - 12:00 to 13:00
Ross Building Room 70A