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

- $H_{G}^{*}(M)\to H^{*}(M/\!/_{p}G)$

where

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

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