In differential geometry, the Kirwan map, introduced by British mathematician Frances Kirwan, is the homomorphism
H G ∗ ( M ) → 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 μ : M → g ∗ . H G ∗ ( M ) is the equivariant cohomology ring of M; i.e.. the cohomology ring of the homotopy quotient E G × G M of M by G. M / / p G = μ − 1 ( p ) / G is the symplectic quotient of M by G at a regular central value p ∈ Z ( g ∗ ) of μ .It is defined as the map of equivariant cohomology induced by the inclusion μ − 1 ( p ) ↪ M followed by the canonical isomorphism H G ∗ ( μ − 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.
