TRIVIALITY OF SYMPLECTIC SU(2)-ACTIONS ON HOMOLOGY

YILDIRAY OZAN

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, ω) on the rational homology groups H_{∗}(M, ℚ) is trivial. In this note, given a symplectic action of SU(2), ϕ : SU(2) × M → M, we will construct a symplectic fiber bundle P_{ϕ} → ℂP^{2} with fiber (M, ω) 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 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 ϕ and do some explicit calculations.