Kirwan map

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

1. F. C. Kirwan, Cohomology of Quotients in Complex and Algebraic Geometry, Mathematical Notes 31, Princeton University Press, Princeton N. J., 1984.
2. M. Harada, G. Landweber. Surjectivity for Hamiltonian G-spaces in K-theory. Trans. Amer. Math. Soc. 359 (2007), 6001--6025.