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^[1] 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 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 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 .

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 $H^{2}(M;\mathbb {Z} ).$ ^[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 $H_{G}^{2}(M;\mathbb {Z} )$ .

Localization theorem

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

Notes

↑ Behrend 2004
↑ using Čech cohomology and the isomorphism $H^{1}(M;\mathbb {C} ^{*})\simeq H^{2}(M;\mathbb {Z} )$ 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
Brion, M. (1998). "Equivariant cohomology and equivariant intersection theory" (PDF). Representation Theories and Algebraic Geometry. Nato ASI Series. Vol. 514. Springer. pp. 1–37. arXiv:math/9802063. doi:10.1007/978-94-015-9131-7_1. ISBN 978-94-015-9131-7. S2CID 14961018.
Goresky, Mark; Kottwitz, Robert; MacPherson, Robert (1998), "Equivariant cohomology, Koszul duality, and the localization theorem", Inventiones Mathematicae, 131: 25–83, CiteSeerX 10.1.1.42.6450, doi:10.1007/s002220050197, S2CID 6006856
Hsiang, Wu-Yi (1975). Cohomology Theory of Topological Transformation Groups. Springer. doi:10.1007/978-3-642-66052-8. ISBN 978-3-642-66052-8.
Tu, Loring W. (March 2011). "What Is . . . Equivariant Cohomology?" (PDF). Notices of the American Mathematical Society. 58 (3): 423–6. arXiv:1305.4293.

Relation to stacks

Behrend, K. (2004). "Cohomology of stacks" (PDF). Intersection theory and moduli. ICTP Lecture Notes. Vol. 19. pp. 249–294. ISBN 9789295003286. PDF page 10 has the main result with examples.

External links

Meinrenken, E. (2006), "Equivariant cohomology and the Cartan model" (PDF), Encyclopedia of mathematical physics, pp. 242–250, ISBN 978-0-12-512666-3 — Excellent survey article describing the basics of the theory and the main important theorems
"Equivariant cohomology", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Young-Hoon Kiem (2008). "Introduction to equivariant cohomology theory" (PDF). Seoul National University.
What is the equivariant cohomology of a group acting on itself by conjugation?

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[1] Behrend 2004

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

[1]

[2]