Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces

Let M = ( M , ω ) M=(M,ω ) be either S 2 × S 2 S^2 × S^2 or C P 2 # C P 2 ¯ CP^2# CP^2̄ endowed with any symplectic form ω ω . Suppose a finite cyclic group Z n Z_n is acting effectively on ( M , ω ) (M,ω ) through Hamiltonian diffeomorphisms, that is, there is an injective homomorphism Z n ↪ H a m ( M , ω ) Z_n Ham(M,ω ) . In this paper, we investigate the homotopy type of the group S y m p Z n ( M , ω ) SympZ_n(M,ω ) of equivariant symplectomorphisms. We prove that for some infinite families of Z n Z_n actions satisfying certain inequalities involving the order n n and the symplectic cohomology class [ ω ] [ω ] , 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 J -holomorphic techniques, on Delzant’s classification of toric actions, on Karshon’s classification of Hamiltonian circle actions on 4 4 -manifolds, and on the Chen-Wilczyński classification of smooth Z n Z_n -actions on Hirzebruch surfaces.