Equivariant cohomology explained

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 Λ of the homotopy quotient EG x _{G} X :H ^{*(X;} _{G} Λ ) = H ^{*(EG} x _{G} X; Λ ). 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 x _{G} X \to X/G is a homotopy equivalence and so one gets: H ^{*(X;} _{G} Λ ) = H ^{*(X/G;} Λ ).

Definitions

It is also possible to define the equivariant cohomology

H ^{*(X;A)} _{G} 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

Λ 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

ak{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

ak{X} _{G} = [G x X \rightrightarrows X]

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

\Omega ^{n} _{G(X)} = o plus _{2k+i} ( Sym ^{k(ak{g}} ^{\vee) ⊗} \Omega ^{i(X))} ^{G}

where Sym ^{\bullet(ak{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) = o plus _{k\geq} Sym ^{2k} ( ak{g} ^{\vee)} ^{G}

For example,

\begin{align} H ^{*} _{dR} (BU( 1 )) & = o plus _{k=0} Sym ^{2k} ( R ^{\vee)} \\ &\cong R [t] \\ & 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 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 http://www.math.harvard.edu/~lurie/282ynotes/LectureIV-Approaches.pdf.

Let X be a complex projective algebraic curve. We identify X as a topological space with the set of the complex points

X( 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 _{sm} on X. Then any principal G-bundle on X is isomorphic to P _{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 _{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

l{G} be the group of all automorphisms of P _{sm} (i.e., gauge group.) Then the homotopy quotient of \Omega by l{G} classifies complex-analytic (or equivalently algebraic) principal G-bundles on X; i.e., it is precisely the classifying space B l{G} of the discrete group l{G} .\operatorname{Bun} _{G(X)} as the quotient stack [\Omega/ l{G}] and then the homotopy quotient B l{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 x _{G} M so that it pulls-back to the bundle \widetilde{E} = EG x E over EG x 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} x _{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;} 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;} Z ) .

Localization theorem

See main article: Localization formula for equivariant cohomology. 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

References

• • Book: Brion, M. . Equivariant cohomology and equivariant intersection theory . http://www-fourier.ujf-grenoble.fr/~mbrion/notesmontreal.pdf . Representation Theories and Algebraic Geometry . Springer . Nato ASI Series . 514 . 1998 . 1–37 . 978-94-015-9131-7 . 10.1007/978-94-015-9131-7_1 . math/9802063 . 14961018 .
  • Book: Hsiang, Wu-Yi . Cohomology Theory of Topological Transformation Groups. Springer . 1975 . 10.1007/978-3-642-66052-8 . 978-3-642-66052-8 .
• Tu . Loring W. . What Is . . . Equivariant Cohomology? . . 58 . 3 . 423–6 . March 2011 . 1305.4293 .

Relation to stacks

• Book: Behrend, K. . Cohomology of stacks . https://www.math.ubc.ca/~behrend/CohSta-1.pdf . Intersection theory and moduli . ICTP Lecture Notes . 19 . 2004 . 9789295003286 . 249–294 . PDF page 10 has the main result with examples.

Further reading

• Book: V.W. . Guillemin . S. . Sternberg . Supersymmetry and equivariant de Rham theory . Springer . 1999 . 978-3-662-03992-2 . 10.1007/978-3-662-03992-2 .
• Web site: Vergne . M. . Paycha . S. . Cohomologie équivariante et théoreme de Stokes . 1998 . Département de Mathématiques, Université Blaise Pascal .

External links

• — Excellent survey article describing the basics of the theory and the main important theorems
  • Web site: Young-Hoon Kiem . Introduction to equivariant cohomology theory . 2008 . Seoul National University .
• What is the equivariant cohomology of a group acting on itself by conjugation?

Notes and References

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