Centralizers of Hamiltonian circle actions on rational ruled surfaces

In this paper, we compute the homotopy type of the group of equivariant symplectomorphisms of $S^2\times S^2$ and $\mathbf{C}P^2\sharp\overline{\mathbf{C}P^2}$ 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 $\mathcal{J}^{S^1}(\omega)$. In particular, we show that this action preserves a decomposition of $\mathcal{J}^{S^1}(\omega)$ 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.

To appear in Memoirs of the AMS.

arXiv version