Centralizers of Hamiltonian circle actions on rational ruled surfaces

TL;DR

This paper determines the homotopy type of equivariant symplectomorphism groups on certain rational ruled surfaces with Hamiltonian circle actions, revealing their structure depends on the extension to toric actions.

Contribution

It computes the homotopy type of equivariant symplectomorphism groups for specific 4-manifolds under Hamiltonian circle actions, connecting to toric extensions.

Findings

Groups are homotopy equivalent to a torus or a pushout of tori.

Action preserves a stratification of compatible almost complex structures.

Results depend on classification of toric and circle actions.

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 $\mathcal{J}^{S^1}_{\omega}$. 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.