Equivariant cohomology – Wikipedia

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

${displaystyle EGtimes _{G}X}$

:

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

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

${displaystyle EGtimes _{G}Xto X/G}$

is a homotopy equivalence and so one gets:

${displaystyle H_{G}^{*}(X;Lambda )=H^{*}(X/G;Lambda ).}$

Definitions[edit]

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[edit]

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

${displaystyle {mathfrak {X}}_{G}=[Gtimes Xrightrightarrows X]}$

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

${displaystyle [Grightrightarrows *]}$

where the action is trivial on a point. Then,

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

For example,

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

since the

${displaystyle U(1)}$

-action on the dual Lie algebra is trivial.

Homotopy quotient[edit]

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[edit]

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[edit]

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

${displaystyle EGtimes _{G}M}$

so that it pulls-back to the bundle

${displaystyle {widetilde {E}}=EGtimes E}$

over

${displaystyle EGtimes M}$

. 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

${displaystyle H^{*}(EGtimes _{G}M)=H_{G}^{*}(M)}$

. (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[edit]

This section needs expansion. You can help by adding to it. (April 2014)

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

See also[edit]

References[edit]

• 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[edit]

• 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[edit]

External links[edit]