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.

If ${\displaystyle G}$ acts freely, then the equivariant cohomology ring is just the singular cohomology ring of the quotient space ${\displaystyle X/G}$. In particular, if ${\displaystyle G}$ is the trivial group, then the equivariant cohomology ring is the ordinary cohomology ring of ${\displaystyle X}$. 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 ${\displaystyle G}$ is finite.)