Kirwan map

Last updated October 23, 2022

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

H_{G}^{*}(M)\to H^{*}(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 $\mu :M\to {\mathfrak {g}}^{*}$ .
$H_{G}^{*}(M)$ 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 $p\in Z({\mathfrak {g}}^{*})$ 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 $H_{G}^{*}(\mu ^{-1}(p))=H^{*}(M/\!/_{p}G)$ .

A theorem of Kirwan^[1] 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 $M$ .^[2]

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References

↑ Kirwan, F.C. (1984). Cohomology of Quotients in Complex and Algebraic Geometry. Mathematical Notes. Vol. 31. Princeton University Press. ISBN 978-0-691-21456-6.
↑ Harada, M.; Landweber, G. (2007). "Surjectivity for Hamiltonian G-spaces in K-theory". Trans. Amer. Math. Soc. 359 (12): 6001–25. arXiv: math/0503609 . doi:10.1090/S0002-9947-07-04164-5. JSTOR 20161853. S2CID 17690407.

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[1] Kirwan, F.C. (1984). Cohomology of Quotients in Complex and Algebraic Geometry. Mathematical Notes. Vol. 31. Princeton University Press. ISBN 978-0-691-21456-6.

[2] Harada, M.; Landweber, G. (2007). "Surjectivity for Hamiltonian G-spaces in K-theory". Trans. Amer. Math. Soc. 359 (12): 6001–25. arXiv: math/0503609 . doi:10.1090/S0002-9947-07-04164-5. JSTOR 20161853. S2CID 17690407.

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