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 $S^2\times S^2$ and $\mathbb{C}{P}^2\#\overline{\mathbb{C}{P}^2}$ 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 ${\mathcal{J}}_{\omega }^{S^1}$. In particular, we show that this action preserves a decomposition of ${\mathcal{J}}_{\omega }^{S^1}$ 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.

Modified proof of Proposition A.11. Fixed minor typos. Published version

References in corpus (4)

• Symplectomorphism groups and embeddings of balls into rational ruled 4-manifolds
• Symplectic $(-2)$-spheres and the symplectomorphism group of small rational 4-manifolds, II
• Cyclic actions on rational ruled symplectic four-manifolds
• Centralizers of Hamiltonian finite cyclic group actions on rational ruled surfaces