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

${\displaystyle H_{G}^{*}(M)\to H^{*}(M/\!/_{p}G)}$

where

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

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

A theorem of Kirwan^[1] says that if ${\displaystyle M}$ 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 ${\displaystyle M}$.^[2]

References