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In differential geometry, the Kirwan map, introduced by British mathematician Frances Kirwan, is the homomorphism
where
It is defined as the map of equivariant cohomology induced by the inclusion μ − 1 ↪ M {\displaystyle \mu ^{-1}\hookrightarrow M} followed by the canonical isomorphism H G ∗ ] = H ∗ {\displaystyle H_{G}^{*}]=H^{*}}.
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}.