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## Summary

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

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

It is also possible to define the equivariant cohomology $H_{G}^{*}(X;A)$  of $X$  with coefficients in a $G$ -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 $\Lambda$  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 ${\mathfrak {X}}=[X_{1}\rightrightarrows X_{0}]$  equivariant cohomology of a smooth manifold is a special example of the groupoid cohomology of a Lie groupoid. This is because given a $G$ -space $X$  for a compact Lie group $G$ , there is an associated groupoid

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

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

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

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

$[G\rightrightarrows *]$

where the action is trivial on a point. Then,

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

For example,

{\begin{aligned}H_{dR}^{*}(BU(1))&=\bigoplus _{k=0}{\text{Sym}}^{2k}(\mathbb {R} ^{\vee })\\&\cong \mathbb {R} [t]\\&{\text{ where }}\deg(t)=2\end{aligned}}

since the $U(1)$ -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 $X$  by its $G$ -action) in which $X$  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 .

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

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

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

## Equivariant characteristic classes

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

## Localization theorem

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