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. Specifically, the equivariant cohomology ring of a space with action of a topological group is defined as the ordinary cohomology ring with coefficient ring of the homotopy quotient :
If is the trivial group, this is the ordinary cohomology ring of , whereas if is contractible, it reduces to the cohomology ring of the classifying space (that is, the group cohomology of when G is finite.) If G acts freely on X, then the canonical map is a homotopy equivalence and so one gets:
It is also possible to define the equivariant cohomology of with coefficients in a -module A; these are abelian groups. This construction is the analogue of cohomology with local coefficients.
If X is a manifold, G a compact Lie group and is the field of real numbers or the field of complex numbers (the most typical situation), then the above cohomology may be computed using the so-called Cartan model (see equivariant differential forms.)
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, 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.
For a Lie groupoid equivariant cohomology of a smooth manifold is a special example of the groupoid cohomology of a Lie groupoid. This is because given a -space for a compact Lie group , there is an associated groupoid
whose equivariant cohomology groups can be computed using the Cartan complex which is the totalization of the de-Rham double complex of the groupoid. The terms in the Cartan complex are
where is the symmetric algebra of the dual Lie algebra from the Lie group , and corresponds to the -invariant forms. This is a particularly useful tool for computing the cohomology of for a compact Lie group since this can be computed as the cohomology of
where the action is trivial on a point. Then,
since the -action on the dual Lie algebra is trivial.
The homotopy quotient, also called homotopy orbit space or Borel construction, is a “homotopically correct” version of the orbit space (the quotient of by its -action) in which is first replaced by a larger but homotopy equivalent space so that the action is guaranteed to be free.
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−1x): moreover, this diagonal action is free since it is free on EG. So we define the homotopy quotient XG 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 → XG → BG is called the Borel fibration.
The following example is Proposition 1 of .
Let X be a complex projective algebraic curve. We identify X as a topological space with the set of the complex points , which is a compact Riemann surface. Let G be a complex simply connected semisimple Lie group. Then any principal G-bundle on X is isomorphic to a trivial bundle, since the classifying space is 2-connected and X has real dimension 2. Fix some smooth G-bundle on X. Then any principal G-bundle on is isomorphic to . In other words, the set of all isomorphism classes of pairs consisting of a principal G-bundle on X and a complex-analytic structure on it can be identified with the set of complex-analytic structures on or equivalently the set of holomorphic connections on X (since connections are integrable for dimension reason). is an infinite-dimensional complex affine space and is therefore contractible.
Let be the group of all automorphisms of (i.e., gauge group.) Then the homotopy quotient of by classifies complex-analytic (or equivalently algebraic) principal G-bundles on X; i.e., it is precisely the classifying space of the discrete group .
Let E be an equivariant vector bundle on a G-manifold M. It gives rise to a vector bundle on the homotopy quotient so that it pulls-back to the bundle over . An equivariant characteristic class of E is then an ordinary characteristic class of , which is an element of the completion of the cohomology ring . (In order to apply Chern–Weil theory, one uses a finite-dimensional approximation of EG.)
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.)
In the non-equivariant case, the first Chern class can be viewed as a bijection between the set of all isomorphism classes of complex line bundles on a manifold M and  In the equivariant case, this translates to: the equivariant first Chern gives a bijection between the set of all isomorphism classes of equivariant complex line bundles and .
The localization theorem is one of the most powerful tools in equivariant cohomology.