 Equivariant cohomology

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.

Equivariant cohomology is a variant of cohomology theory that is designed to study the topological properties of spaces with a group action. It generalizes classical cohomology theories by incorporating the symmetry of a group acting on a topological space and allows for the analysis of spaces that are equipped with a continuous group action, which is particularly useful in various fields such as algebraic topology, algebraic geometry, and mathematical physics.