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 $$X$$ 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, 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"$\Omega^n_G(X) = \bigoplus_{2k+i = n}(\text{Sym}^k(\mathfrak{g}^\vee)\otimes \Omega^i(X))^G$"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, $$\begin{align} H^*_{dR}(BU(1)) &= \bigoplus_{k=0}\text{Sym}^{2k}(\mathbb{R}^\vee) \\ &\cong \mathbb{R}[t] \\ &\text{ where } \deg(t) = 2 \end{align}$$ since the $$U(1)$$-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 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 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 EG → BG. This bundle X → XG → 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 $$X(\mathbb{C})$$, 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 $$H^2_G(M; \mathbb{Z})$$.

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

Relation to stacks

 * PDF page 10 has the main result with examples.