I ask whether every homologically trivial cyclic action on a symplectic
four-manifold extends to a Hamiltonian circle action. By a cyclic action I
mean an action of a cyclic group of finite order; it is homologically
trivial if it induces the identity map on homology. 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 it does
not extend to a 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 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.