Kirwan map

In differential geometry, the Kirwan map, introduced by British mathematician Frances Kirwan, is the homomorphism

[math]\displaystyle{ H^*_G(M) \to H^*(M /\!/_p G) }[/math]

where

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

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

A theorem of Kirwan says that if [math]\displaystyle{ M }[/math] 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 [math]\displaystyle{ M }[/math].