Triviality of symplectic SU(2)-actions on homology

Abstract

Lalonde and McDuff showed that the natural action of the rational homology of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M, \omega)$ on the rational homology groups $H_*(M,{\mathbb Q})$ is trivial. In this note, given a symplectic action of SU(2), $\phi:SU(2)\times M \to M$, we will construct a symplectic fiber bundle $P_\phi\to {\mathbb CP}^2$ with fiber $(M,\omega)$ and use it to construct the chains, which bound the images of the homology cycles under the trace map given by the SU(2)-action. It turns out that the natural chains bounded by the SU(2)-orbits in $M$ are punctured ${\mathbb CP}^2$'s, the counter parts of holomorphic discs bounding circles in case of Hamiltonian circle actions. We will also define some invariants of the action $\phi$ and do some explicit calculations.