Equivariant cohomology

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

[math]\displaystyle{ H_G^*(X; \Lambda) = H^*(EG \times_G X; \Lambda). }[/math]

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

Definitions

It is also possible to define the equivariant cohomology [math]\displaystyle{ H_G^*(X;A) }[/math] of [math]\displaystyle{ X }[/math] with coefficients in a [math]\displaystyle{ G }[/math]-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 [math]\displaystyle{ \Lambda }[/math] 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 [math]\displaystyle{ \mathfrak{X} = [X_1 \rightrightarrows X_0] }[/math] equivariant cohomology of a smooth manifold[1] is a special example of the groupoid cohomology of a Lie groupoid. This is because given a [math]\displaystyle{ G }[/math]-space [math]\displaystyle{ X }[/math] for a compact Lie group [math]\displaystyle{ G }[/math], there is an associated groupoid

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

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

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

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

[math]\displaystyle{ [G \rightrightarrows *] }[/math]

where the action is trivial on a point. Then,

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

For example,

[math]\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} }[/math]

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

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

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

Equivariant characteristic classes

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

Localization theorem

Main page: Localization formula for equivariant cohomology

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

See also

Notes

  1. Behrend. "Cohomology of Stacks". https://secure.math.ubc.ca/~behrend/CohSta-1.pdf. 
  2. using Čech cohomology and the isomorphism [math]\displaystyle{ H^1(M; \mathbb{C}^*) \simeq H^2(M; \mathbb{Z}) }[/math] given by the exponential map.

References

Relation to stacks

Further reading

External links