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# Kirwan map

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 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.[1]

References

M. Harada, G. Landweber. Surjectivity for Hamiltonian G-spaces in K-theory. Trans. Amer. Math. Soc. 359 (2007), 6001--6025.

F. C. Kirwan, Cohomology of Quotients in Complex and Algebraic Geometry, Mathematical Notes 31, Princeton University Press, Princeton N. J., 1984.

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