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 ${\displaystyle X}$ with action of a topological group ${\displaystyle G}$ is defined as the ordinary cohomology ring with coefficient ring ${\displaystyle \Lambda }$ of the homotopy quotient ${\displaystyle EG\times _{G}X}$:

${\displaystyle H_{G}^{*}(X;\Lambda )=H^{*}(EG\times _{G}X;\Lambda ).}$

If ${\displaystyle G}$ is the trivial group, this is the ordinary cohomology ring of ${\displaystyle X}$, whereas if ${\displaystyle X}$ is contractible, it reduces to the cohomology ring of the classifying space ${\displaystyle BG}$ (that is, the group cohomology of ${\displaystyle G}$ when G is finite.) If G acts freely on X, then the canonical map ${\displaystyle EG\times _{G}X\to X/G}$ is a homotopy equivalence and so one gets: ${\displaystyle H_{G}^{*}(X;\Lambda )=H^{*}(X/G;\Lambda ).}$