On cohomology of invariant submanifolds of Hamiltonian actions

Abstract

In this note we prove the following theorem: Let $G$ be a compact Lie group acting on a compact symplectic manifold $M$ in a Hamiltonian fashion. If $L$ is an $l$-dimensional closed invariant submanifold of $M$, on which the $G$-action is locally free then the fundamental class $[L]$ is trivial in $H_l(M,{\mathbb Q})$. We also prove similar results for lower homology groups of $L$, in case the group $G$ is a finite product of copies of $S^1$ and SU(2). The key ingredients of the proofs are Kirwan's theorem that Hamiltonian spaces are equivariantly formal and symplectic reduction.