Centralizers of Hamiltonian circle actions on rational ruled surfaces

Abstract

In this paper, we compute the homotopy type of the group of equivariant symplectomorphisms of S 2 Ŝ2 and CP 2 #CP 2 under the presence of Hamiltonian group actions of the circle S 1 . We prove that the group of equivariant symplectomorphisms are homotopy equivalent to either a torus, or to the homotopy pushout of two tori depending on whether the circle action extends to a single toric action or to exactly two non-equivalent toric actions. This follows from the analysis of the action of equivariant symplectomorphisms on the space of compatible and invariant almost complex structures J S 1 ω . In particular, we show that this action preserves a decomposition of J S 1 ω into strata which are in bijection with toric extensions of the circle action. Our results rely on J-holomorphic techniques, on Delzant's classification of toric actions and on Karshon's classification of Hamiltonian circle actions on 4-manifolds.