Kirwan map

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

$H_G^{\ast }(M)\to H^{\ast }(M/{/}_{p}G)$

where

• $M$ is a Hamiltonian G-space; i.e., a symplectic manifold acted by a Lie group G with a moment map $\mu :M\to {𝔤}^{\ast }$.
• $H_G^{\ast }(M)$ 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 $p\in Z({𝔤}^{\ast })$ 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 $H_G^{\ast }({\mu }^{-1}(p))=H^{\ast }(M/{/}_{p}G)$.

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 $M$.

References

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