Equivariant cohomology ($H_G^{\ast }(X;R)=H^{\ast }(EG{\times }_{G}X;R)$)

The Borel version of equivariant cohomology is the ordinary cohomology of the Borel construction. It retains information about both the space and its group action. For a point it is the cohomology of the classifying space; for a free action with the usual bundle hypotheses it is the cohomology of the orbit space.

For a group action on $X$ and a contractible free $G$-space $EG$, the diagonal orbit space $EG{\times }_{G}X$ maps to the classifying space $BG$ with fibre $X$. It defines equivariant cohomology. When $X\to X/G$ is a principal $G$-bundle, the map $EG{\times }_{G}X\to X/G$ has contractible fibre $EG$ and gives a homotopy equivalence for spaces of CW type. Freeness without the bundle hypotheses should not be substituted for this assertion for arbitrary topological groups.