# Equivariant cohomology

The construction should not be confused with other cohomology theories,
such as Bredon cohomology or the cohomology of invariant differential forms: if *G* is a compact Lie group, then, by the averaging argument^{[citation needed]}, any form may be made invariant; thus, cohomology of invariant differential forms does not yield new information.

Koszul duality is known to hold between equivariant cohomology and ordinary cohomology.

To this end, construct the universal bundle *EG* → *BG* for *G* and recall that *EG* admits a free *G*-action. Then the product *EG* × *X* —which is homotopy equivalent to *X* since *EG* is contractible—admits a “diagonal” *G*-action defined by (*e*,*x*).*g* = (*eg*,*g ^{−1}x*): moreover, this diagonal action is free since it is free on

*EG*. So we define the homotopy quotient

*X*

_{G}to be the orbit space (

*EG*×

*X*)/

*G*of this free

*G*-action.

In other words, the homotopy quotient is the associated *X*-bundle over *BG* obtained from the action of *G* on a space *X* and the principal bundle *EG* → *BG*. This bundle *X* → *X*_{G} → *BG* is called the **Borel fibration**.

Alternatively, one can first define an equivariant Chern class and then define other characteristic classes as invariant polynomials of Chern classes as in the ordinary case; for example, the equivariant Todd class of an equivariant line bundle is the Todd function evaluated at the equivariant first Chern class of the bundle. (An equivariant Todd class of a line bundle is a power series (not a polynomial as in the non-equivariant case) in the equivariant first Chern class; hence, it belongs to the completion of the equivariant cohomology ring.)

The localization theorem is one of the most powerful tools in equivariant cohomology.