Kirwan map

In differential geometry, the Kirwan map, introduced by British mathematician Frances Kirwan, is the homomorphism
 * $$H^*_G(M) \to H^*(M /\!/_p G)$$

where
 * $$M$$ is a Hamiltonian G-space; i.e., a symplectic manifold acted by a Lie group G with a moment map $$\mu: M \to {\mathfrak g}^*$$.
 * $$H^*_G(M)$$ is the equivariant cohomology ring of $$M$$; i.e.. the cohomology ring of the homotopy quotient $$EG \times_G M$$ of $$M$$ by $$G$$.
 * $$M /\!/_p G = \mu^{-1}(p)/G$$ is the symplectic quotient of $$M$$ by $$G$$ at a regular central value $$ p \in Z({\mathfrak g}^*) $$ of $$\mu$$.

It is defined as the map of equivariant cohomology induced by the inclusion $$\mu^{-1}(p) \hookrightarrow M$$ followed by the canonical isomorphism $$H_G^*(\mu^{-1}(p)) = H^*(M /\!/_p G)$$.

A theorem of Kirwan says that if $$M$$ is compact, then the map is surjective in rational coefficients. The analogous result holds between the K-theory of the symplectic quotient and the equivariant topological K-theory of $$M$$.