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^[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. ^ Kirwan, F.C. (1984). Cohomology of Quotients in Complex and Algebraic Geometry. Vol. 31. Mathematical Notes. 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