Equivariant cohomology

Summary

Equivariant cohomology is a cohomology theory for spaces with an action of a topological group $G$. In the Borel construction, one replaces the space $X$ by the homotopy quotient $EG \times_G X$ and takes its ordinary cohomology. For a free action of a compact Lie group this is the cohomology of the orbit space. For the action on a point it is the cohomology of the classifying space $BG$. It is a main tool for localization theorems, such as the Atiyah–Bott formula, and for the study of fixed points.