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 formula_1 with action of a topological group formula_2 is defined as the ordinary cohomology ring with coefficient ring formula_3 of the homotopy quotient formula_4:
If formula_2 is the trivial group, this is the ordinary cohomology ring of formula_1, whereas if formula_1 is contractible, it reduces to the cohomology ring of the classifying space formula_9 (that is, the group cohomology of formula_2 when "G" is finite.) If "G" acts freely on "X", then the canonical map formula_11 is a homotopy equivalence and so one gets: formula_12
It is also possible to define the equivariant cohomology 
formula_13 of formula_1 with coefficients in a 
formula_2-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 formula_3 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.
Homotopy quotient.
The homotopy quotient, also called homotopy orbit space or Borel construction, is a “homotopically correct” version of the orbit space (the quotient of formula_1 by its formula_2-action) in which formula_1 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 "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.
An example of a homotopy quotient.
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 formula_20, 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 formula_9 is 2-connected and "X" has real dimension 2. Fix some smooth "G"-bundle formula_22 on "X". Then any principal "G"-bundle on formula_1 is isomorphic to formula_22. In other words, the set formula_25 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 formula_22 or equivalently the set of holomorphic connections on "X" (since connections are integrable for dimension reason). formula_25 is an infinite-dimensional complex affine space and is therefore contractible.
Let formula_28 be the group of all automorphisms of formula_22 (i.e., gauge group.) Then the homotopy quotient of formula_25 by formula_28 classifies complex-analytic (or equivalently algebraic) principal "G"-bundles on "X"; i.e., it is precisely the classifying space formula_32 of the discrete group formula_28.
One can define the moduli stack of principal bundles formula_34 as the quotient stack formula_35 and then the homotopy quotient formula_32 is, by definition, the homotopy type of formula_34.
Equivariant characteristic classes.
Let "E" be an equivariant vector bundle on a "G"-manifold "M". It gives rise to a vector bundle formula_38 on the homotopy quotient formula_39 so that it pulls-back to the bundle formula_40 over formula_41. An equivariant characteristic class of "E" is then an ordinary characteristic class of formula_38, which is an element of the completion of the cohomology ring formula_43. (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 formula_44 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 formula_45.
Localization theorem.
The localization theorem is one of the most powerful tools in equivariant cohomology.