# 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 [math]\displaystyle{ X }[/math] with action of a topological group [math]\displaystyle{ G }[/math] is defined as the ordinary cohomology ring with coefficient ring [math]\displaystyle{ \Lambda }[/math] of the homotopy quotient [math]\displaystyle{ EG \times_G X }[/math]:

- [math]\displaystyle{ H_G^*(X; \Lambda) = H^*(EG \times_G X; \Lambda). }[/math]

If [math]\displaystyle{ G }[/math] is the trivial group, this is the ordinary cohomology ring of [math]\displaystyle{ X }[/math], whereas if [math]\displaystyle{ X }[/math] is contractible, it reduces to the cohomology ring of the classifying space [math]\displaystyle{ BG }[/math] (that is, the group cohomology of [math]\displaystyle{ G }[/math] when *G* is finite.) If *G* acts freely on *X*, then the canonical map [math]\displaystyle{ EG \times_G X \to X/G }[/math] is a homotopy equivalence and so one gets: [math]\displaystyle{ H_G^*(X; \Lambda) = H^*(X/G; \Lambda). }[/math]

## Definitions

It is also possible to define the equivariant cohomology
[math]\displaystyle{ H_G^*(X;A) }[/math] of [math]\displaystyle{ X }[/math] with coefficients in a
[math]\displaystyle{ G }[/math]-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 [math]\displaystyle{ \Lambda }[/math] 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.

### Relation with groupoid cohomology

For a Lie groupoid [math]\displaystyle{ \mathfrak{X} = [X_1 \rightrightarrows X_0] }[/math] equivariant cohomology of a smooth manifold^{[1]} is a special example of the groupoid cohomology of a Lie groupoid. This is because given a [math]\displaystyle{ G }[/math]-space [math]\displaystyle{ X }[/math] for a compact Lie group [math]\displaystyle{ G }[/math], there is an associated groupoid

[math]\displaystyle{ \mathfrak{X}_G = [G\times X \rightrightarrows X] }[/math]

whose equivariant cohomology groups can be computed using the Cartan complex [math]\displaystyle{ \Omega_G^\bullet(X) }[/math] which is the totalization of the de-Rham double complex of the groupoid. The terms in the Cartan complex are

[math]\displaystyle{ \Omega^n_G(X) = \bigoplus_{2k+i = n}(\text{Sym}^k(\mathfrak{g}^\vee)\otimes \Omega^i(X))^G }[/math]

where [math]\displaystyle{ \text{Sym}^\bullet(\mathfrak{g}^\vee) }[/math] is the symmetric algebra of the dual Lie algebra from the Lie group [math]\displaystyle{ G }[/math], and [math]\displaystyle{ (-)^G }[/math] corresponds to the [math]\displaystyle{ G }[/math]-invariant forms. This is a particularly useful tool for computing the cohomology of [math]\displaystyle{ BG }[/math] for a compact Lie group [math]\displaystyle{ G }[/math] since this can be computed as the cohomology of

[math]\displaystyle{ [G \rightrightarrows *] }[/math]

where the action is trivial on a point. Then,

[math]\displaystyle{ H^*_{dR}(BG) = \bigoplus_{k\geq 0 }\text{Sym}^{2k}(\mathfrak{g}^\vee)^G }[/math]

For example,

[math]\displaystyle{ \begin{align} H^*_{dR}(BU(1)) &= \bigoplus_{k=0}\text{Sym}^{2k}(\mathbb{R}^\vee) \\ &\cong \mathbb{R}[t] \\ &\text{ where } \deg(t) = 2 \end{align} }[/math]

since the [math]\displaystyle{ U(1) }[/math]-action on the dual Lie algebra is trivial.

## 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 [math]\displaystyle{ X }[/math] by its [math]\displaystyle{ G }[/math]-action) in which [math]\displaystyle{ X }[/math] 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

*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 [1].

Let *X* be a complex projective algebraic curve. We identify *X* as a topological space with the set of the complex points [math]\displaystyle{ X(\mathbb{C}) }[/math], 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 [math]\displaystyle{ BG }[/math] is 2-connected and *X* has real dimension 2. Fix some smooth *G*-bundle [math]\displaystyle{ P_\text{sm} }[/math] on *X*. Then any principal *G*-bundle on [math]\displaystyle{ X }[/math] is isomorphic to [math]\displaystyle{ P_\text{sm} }[/math]. In other words, the set [math]\displaystyle{ \Omega }[/math] 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 [math]\displaystyle{ P_\text{sm} }[/math] or equivalently the set of holomorphic connections on *X* (since connections are integrable for dimension reason). [math]\displaystyle{ \Omega }[/math] is an infinite-dimensional complex affine space and is therefore contractible.

Let [math]\displaystyle{ \mathcal{G} }[/math] be the group of all automorphisms of [math]\displaystyle{ P_\text{sm} }[/math] (i.e., gauge group.) Then the homotopy quotient of [math]\displaystyle{ \Omega }[/math] by [math]\displaystyle{ \mathcal{G} }[/math] classifies complex-analytic (or equivalently algebraic) principal *G*-bundles on *X*; i.e., it is precisely the classifying space [math]\displaystyle{ B\mathcal{G} }[/math] of the discrete group [math]\displaystyle{ \mathcal{G} }[/math].

One can define the moduli stack of principal bundles [math]\displaystyle{ \operatorname{Bun}_G(X) }[/math] as the quotient stack [math]\displaystyle{ [\Omega/\mathcal{G}] }[/math] and then the homotopy quotient [math]\displaystyle{ B\mathcal{G} }[/math] is, by definition, the homotopy type of [math]\displaystyle{ \operatorname{Bun}_G(X) }[/math].

## Equivariant characteristic classes

Let *E* be an equivariant vector bundle on a *G*-manifold *M*. It gives rise to a vector bundle [math]\displaystyle{ \widetilde{E} }[/math] on the homotopy quotient [math]\displaystyle{ EG \times_G M }[/math] so that it pulls-back to the bundle [math]\displaystyle{ \widetilde{E}=EG \times E }[/math] over [math]\displaystyle{ EG \times M }[/math]. An equivariant characteristic class of *E* is then an ordinary characteristic class of [math]\displaystyle{ \widetilde{E} }[/math], which is an element of the completion of the cohomology ring [math]\displaystyle{ H^*(EG \times_G M) = H^*_G(M) }[/math]. (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 [math]\displaystyle{ H^2(M; \mathbb{Z}). }[/math]^{[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 [math]\displaystyle{ H^2_G(M; \mathbb{Z}) }[/math].

## Localization theorem

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

## See also

- Equivariant differential form
- Kirwan map
- Localization formula for equivariant cohomology
- GKM variety
- Bredon cohomology

## Notes

- ↑ Behrend. "Cohomology of Stacks". https://secure.math.ubc.ca/~behrend/CohSta-1.pdf.
- ↑ using Čech cohomology and the isomorphism [math]\displaystyle{ H^1(M; \mathbb{C}^*) \simeq H^2(M; \mathbb{Z}) }[/math] given by the exponential map.

## References

- Atiyah, Michael; Bott, Raoul (1984), "The moment map and equivariant cohomology",
*Topology***23**: 1–28, doi:10.1016/0040-9383(84)90021-1 - Michel Brion, "Equivariant cohomology and equivariant intersection theory" [2]
- Goresky, Mark; Kottwitz, Robert; MacPherson, Robert (1998), "Equivariant cohomology, Koszul duality, and the localization theorem",
*Inventiones Mathematicae***131**: 25–83, doi:10.1007/s002220050197 - Hsiang, Wu-Yi (1975).
*Cohomology Theory of Topological Transformation Groups*. New York: Springer. - Tu, Loring W. (March 2011). "What Is . . . Equivariant Cohomology?".
*Notices of the American Mathematical Society***58**(3): 423–426. https://www.ams.org/notices/201103/rtx110300423p.pdf.

### Relation to stacks

- Cohomology of Stacks - page 10 has the main result with examples

## Further reading

- V. W. Guillemin and S. Sternberg. Supersymmetry and equivariant de Rham theory . Springer-V erlag, Berlin, 1999
- CM Vergne, Cohomologie équivariante et théorème de Stokes

## External links

- Equivariant cohomology and the Cartan Model - Excellent survey article describing the basics of the theory and the main important theorems
- Hazewinkel, Michiel, ed. (2001), "Equivariant cohomology",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=e/e036090 - Young-Hoon Kiem, Introduction to equivariant cohomology theory
- What is the equivariant cohomology of a group acting on itself by conjugation?

Original source: https://en.wikipedia.org/wiki/Equivariant cohomology.
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