Centralizers of abelian Hamiltonian actions on rational ruled surfaces
Abstract
In this thesis, we compute the homotopy type of the group of equivariant symplectomorphisms of $S^2 \times S^2$ and $CP^2$ blown up once under the presence of Hamiltonian group actions of either $S^1$ or finite cyclic groups. For Hamiltonian circle actions, we prove that the centralizers 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. We can show that the same holds for the centralizers of most finite cyclic groups in the Hamiltonian group $\Ham(M)$. Our results rely on $J$-holomorphic techniques, on Delzant's classification of toric actions, on Karshon's classification of Hamiltonian circle actions on $4$-manifolds, and on the Chen-Wilczy\'nski smooth classification of $\Z_n$-actions on Hirzebruch surfaces.