Homotopy type of equivariant symplectomorphisms of rational ruled surfaces

Abstract: In this talk, we present results on the homotopy type of the
group of equivariant symplectomorphisms of $S^2 \times S^2$ and
$\mathbb{C}P^2$ blown up once under the presence of a Hamiltonian circle
actions. We prove that the group of equivariant symplectomorphisms is
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 nonequivalent toric actions. 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. Time permitting we will explain results of a similar flavour
on the homotopy type of $\mathbb{Z}_n$ equivariant symplectomorphisms for
a large family of finite cyclic groups in the Hamiltonian group. This is
based on joint work with Martin Pinsonnault.