Homotopy theory | Symplectic topology | Group actions (mathematics) | Algebraic topology

Equivariant cohomology

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 ordinary cohomology theory. Specifically, the equivariant cohomology ring of a space with action of a topological group is defined as the ordinary cohomology ring with coefficient ring of the homotopy quotient : If is the trivial group, this is the ordinary cohomology ring of , whereas if is contractible, it reduces to the cohomology ring of the classifying space (that is, the group cohomology of when G is finite.) If G acts freely on X, then the canonical map is a homotopy equivalence and so one gets: (Wikipedia).

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Related pages

Cohomology ring | Topological space | Moduli stack of principal bundles | Group cohomology | Associated bundle | Localization formula for equivariant cohomology | Algebraic topology | Trivial group | Classifying space | Bredon cohomology | Koszul duality | Riemann surface | GKM variety | Mathematics | Algebraic curve | Lie groupoid | Equivariant differential form | Universal bundle | Kirwan map | Abelian group | Quotient stack