CENTRALIZERS OF HAMILTONIAN FINITE CYCLIC GROUP ACTIONS ON RATIONAL RULED SURFACES

Let M = (M, omega) be either S-2 x S-2 or CP2#(CP2) over bar endowed with any symplectic form omega. Suppose a finite cyclic group Z(n) is acting effectively on (M, omega) through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism Z(n) -> Ham(M, omega). In this paper, we investigate the homotopy type of the group Symp(Zn) (M, omega) of equivariant symplectomorphisms. We prove that for some infinite families of Z(n) actions satisfying certain inequalities involving the order n and the symplectic cohomology class [omega], the actions extend to either one or two toric actions, and accordingly, that the centralizers are homotopically equivalent to either a finite dimensional Lie group, or to the homotopy pushout of two tori along a circle. Our results rely on J-holomorphic techniques, on Delzant's classification of toric actions, on Karshon's classification of Hamiltonian circle actions on 4-manifolds, and on the Chen-Wilczynski classification of smooth Z(n)-actions on Hirzebruch surfaces.