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

$\displaystyle{ H_G^*(X; \Lambda) = H^*(EG \times_G X; \Lambda). }$

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

## Definitions

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

### Relation with groupoid cohomology

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

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

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

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

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

$\displaystyle{ [G \rightrightarrows *] }$

where the action is trivial on a point. Then,

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

For example,

\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} }

since the $\displaystyle{ U(1) }$-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 $\displaystyle{ X }$ by its $\displaystyle{ G }$-action) in which $\displaystyle{ X }$ 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 EGBG 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 EGBG. This bundle XXGBG 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 $\displaystyle{ X(\mathbb{C}) }$, 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 $\displaystyle{ BG }$ is 2-connected and X has real dimension 2. Fix some smooth G-bundle $\displaystyle{ P_\text{sm} }$ on X. Then any principal G-bundle on $\displaystyle{ X }$ is isomorphic to $\displaystyle{ P_\text{sm} }$. In other words, the set $\displaystyle{ \Omega }$ 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 $\displaystyle{ P_\text{sm} }$ or equivalently the set of holomorphic connections on X (since connections are integrable for dimension reason). $\displaystyle{ \Omega }$ is an infinite-dimensional complex affine space and is therefore contractible.

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

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

## Equivariant characteristic classes

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

## Localization theorem

Main page: Localization formula for equivariant cohomology

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

2. using Čech cohomology and the isomorphism $\displaystyle{ H^1(M; \mathbb{C}^*) \simeq H^2(M; \mathbb{Z}) }$ given by the exponential map.