Triviality of symplectic SU(2)-actions on homology

Lalonde and McDuff showed that the natural action of the rational homology of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $(M, \omega)$ontherationalhomologygroups$H_*(M,{\mathbb Q})$ is trivial. In this note, given a symplectic action of SU(2), $\varphi :SU(2)\times M\to M$, we will construct a symplectic fiber bundle $P_{\varphi }\to {\mathbb{C}P}^2$ 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 ${\mathbb{C}P}^2$’s, the counter parts of holomorphic discs bounding circles in case of Hamiltonian circle actions. We will also define some invariants of the action $\varphi$ and do some explicit calculations.