The cosupport axiom, equivariant cohomology and the intersection cohomology of certain symplectic quotients

Let $M$ be a proper Hamiltonian $K$-space with proper moment map $\mu$. The symplectic quotient $X=\mu^{-1}(0)/K$ is in general a singular stratified space. In this paper we first generalize the Kirwan map to this symplectic setting which maps the $K$ equivariant cohomology of $\mu^{-1}(0)$ to the middle perversity intersection cohomology $IH^*(X)$. Next, we show that there is a natural right inverse of the Kirwan map, under an assumption coming from the cosupport axiom for the intersection homology sheaf. Furthermore, we can identify the subspace of $H^*_K(\mu^{-1}(0))$ which corresponds to $IH^*(X)$. The naturality of the splitting ensures that the intersection pairing corresponds to the cup product on equivariant cohomology. Finally, we show that the same set of ideas can be applied to any circle action and prove similar results without any assumption.