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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^{[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.

For a Lie groupoid equivariant cohomology of a smooth manifold^{[1]} 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,

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 ^{−1}x*): moreover, this diagonal action is free since it is free on

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**.

The following example is Proposition 1 of [1].

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 ^{[2]} 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.

**^**Behrend 2004**^**using Čech cohomology and the isomorphism given by the exponential map.

- Atiyah, Michael; Bott, Raoul (1984), "The moment map and equivariant cohomology",
*Topology*,**23**: 1–28, doi:10.1016/0040-9383(84)90021-1 - Brion, M. (1998). "Equivariant cohomology and equivariant intersection theory" (PDF).
*Representation Theories and Algebraic Geometry*. Nato ASI Series. Vol. 514. Springer. pp. 1–37. arXiv:math/9802063. doi:10.1007/978-94-015-9131-7_1. ISBN 978-94-015-9131-7. S2CID 14961018. - Goresky, Mark; Kottwitz, Robert; MacPherson, Robert (1998), "Equivariant cohomology, Koszul duality, and the localization theorem",
*Inventiones Mathematicae*,**131**: 25–83, CiteSeerX 10.1.1.42.6450, doi:10.1007/s002220050197, S2CID 6006856 - Hsiang, Wu-Yi (1975).
*Cohomology Theory of Topological Transformation Groups*. Springer. doi:10.1007/978-3-642-66052-8. ISBN 978-3-642-66052-8. - Tu, Loring W. (March 2011). "What Is . . . Equivariant Cohomology?" (PDF).
*Notices of the American Mathematical Society*.**58**(3): 423–6. arXiv:1305.4293.

- Behrend, K. (2004). "Cohomology of stacks" (PDF).
*Intersection theory and moduli*. ICTP Lecture Notes. Vol. 19. pp. 249–294. ISBN 9789295003286. PDF page 10 has the main result with examples.

- Meinrenken, E. (2006), "Equivariant cohomology and the Cartan model" (PDF),
*Encyclopedia of mathematical physics*, pp. 242–250, ISBN 978-0-12-512666-3 — Excellent survey article describing the basics of the theory and the main important theorems - "Equivariant cohomology",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Young-Hoon Kiem (2008). "Introduction to equivariant cohomology theory" (PDF). Seoul National University.
- What is the equivariant cohomology of a group acting on itself by conjugation?