Kirwan map explained

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

H ^{*} _{G(M)} \to H ^{*(M} // _{p} G) whereM is a Hamiltonian G-space; i.e., a symplectic manifold acted by a Lie group G with a moment map \mu: M \to { ak g} ^{*} .H ^{*} _{G(M)} is the equivariant cohomology ring of M ; i.e.. the cohomology ring of the homotopy quotient EG x _{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({ ak 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 ^{*(\mu} _{G} ^{-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]

References

1. Book: Kirwan, F.C. . Frances Kirwan . [{{GBurl|4wfZBnlSaJ0C|pg=PP5}} Cohomology of Quotients in Complex and Algebraic Geometry ]. Mathematical Notes . 31 . Princeton University Press . 1984 . 978-0-691-21456-6 .
2. M. . Harada . G. . Landweber . Surjectivity for Hamiltonian G-spaces in K-theory . Trans. Amer. Math. Soc. . 359 . 2007 . 12 . 6001–25 . 10.1090/S0002-9947-07-04164-5 . 20161853 . math/0503609 . 17690407 .