Equivariant cohomology

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:

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 groupoidwhose 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 arewhere 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 ofwhere the action is trivial on a point. Then,For example,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
One can define the moduli stack of principal bundles as the quotient stack and then the homotopy quotient is, by definition, the homotopy type of .

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.

PDF page 10 has the main result with examples.