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

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

where

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

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

A theorem of Kirwan says that if M {\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 M {\displaystyle M}.