# Equivariant cohomology

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In mathematics, equivariant 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: $H_G^*(X; \Lambda) = H^*(X/G; \Lambda).$

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 so-called Cartan model (see equivariant differential forms.)

The construction should not be confused as a more naive 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.

The Koszul duality is known to hold between equivariant cohomology and ordinary cohomology.

## 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 G. 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 $X(\mathbb{C})$, 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 $EG \times E \to EG \times M$. 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 the 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 $H^2(M; \mathbb{Z}).$[1] 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.

## Notes

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

## References

• M. Atiyah, R. Bott, The moment map and equivariant cohomology, Topology 23 (1984).
• Michel Brion, Equivariant cohomology and equivariant intersection theory [2]
• M. Goresky, R. Kottwitz, and R. MacPherson. Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131 (1998).
• Tu, Loring W. (March 2011). "What Is . . . Equivariant Cohomology?" (PDF). AMS Notices 58 (03): 423–426.