In differential geometry, the Kirwan map, introduced by British mathematician

Frances Kirwan Dame Frances Clare Kirwan, (born 21 August 1959) is a British mathematician, currently Savilian Professor of Geometry at the University of Oxford. Her fields of specialisation are algebraic and symplectic geometry. Education Kirwan was ed ...

, is the

homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...

H^*_G(M) \to H^*(M /\!/_p G)

where *

M

is a Hamiltonian G-space; i.e., a

symplectic manifold In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called s ...

acted by a Lie group ''G'' with a

moment map In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the actio ...

\mu: M \to ^*

. *

H^*_G(M)

is the

equivariant cohomology ring In mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from algebraic topology which applies to topological spaces with a '' group action''. It can be viewed as a common generalization of group cohomology and an o ...

M

; i.e.. the cohomology ring of the homotopy quotient

EG \times_G M

M

G

. *

M /\!/_p G = \mu^(p)/G

is the

symplectic quotient In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the actio ...

M

G

at a regular central value

p \in Z(^*)

\mu

. It is defined as the map of equivariant cohomology induced by the inclusion

\mu^(p) \hookrightarrow M

followed by the canonical isomorphism

H_G^*(\mu^(p)) = H^*(M /\!/_p G)

. A theorem of Kirwan says that if

M

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

, then the map is surjective in rational coefficients. The analogous result holds between the

K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geom ...

of the symplectic quotient and the equivariant topological K-theory of

M

References

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