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

${\displaystyle H_{G}^{*}(X;\Lambda )=H^{*}(EG\times _{G}X;\Lambda ).}$

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

Definitions

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

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

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

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

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

${\displaystyle [G\rightrightarrows *]}$

where the action is trivial on a point. Then,

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

For example,

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

since the ${\displaystyle U(1)}$-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 ${\displaystyle X}$ by its ${\displaystyle G}$-action) in which ${\displaystyle X}$ 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 [1].

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

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

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

Equivariant characteristic classes

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

Localization theorem

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

See also

• Equivariant differential form
• Kirwan map
• Localization formula for equivariant cohomology
• GKM variety
• Bredon cohomology

Notes

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.

Further reading

• Guillemin, V.W.; Sternberg, S. (1999). Supersymmetry and equivariant de Rham theory. Springer. doi:10.1007/978-3-662-03992-2. ISBN 978-3-662-03992-2.
• Vergne, M.; Paycha, S. (1998). "Cohomologie équivariante et théoreme de Stokes" (PDF). Département de Mathématiques, Université Blaise Pascal.

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?