Equivariant Homology Theory

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 = _1 \rightrightarrows X_0/math> 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

\mathfrak_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

$\Omega^n_G(X) = \bigoplus_(\text^k(\mathfrak^\vee)\otimes \Omega^i(X))^G$

where $\text^\bullet(\mathfrak^\vee)$ 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

\rightrightarrows */math>

where the action is trivial on a point. Then,

$H^*_(BG) = \bigoplus_\text^(\mathfrak^\vee)^G$

For example,

\begin
H^*_(BU(1)) &= \bigoplus_\text^(\mathbb^\vee) \\
&\cong \mathbb \\
&\text \deg(t) = 2
\end

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 o

Let ''X'' be a complex projective algebraic curve. We identify ''X'' as a topological space with the set of the complex points $X(\mathbb)$, 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 $P_\text$ on ''X''. Then any principal ''G''-bundle on $X$ is isomorphic to $P_\text$. 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 $P_\text$ 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 $\mathcal$ be the group of all automorphisms of $P_\text$ (i.e., gauge group.) Then the homotopy quotient of $\Omega$ by $\mathcal$ classifies complex-analytic (or equivalently algebraic) principal ''G''-bundles on ''X''; i.e., it is precisely the classifying space $B\mathcal$ of the discrete group $\mathcal$.

One can define the moduli stack of principal bundles $\operatorname_G(X)$ as the quotient stack Omega/\mathcal/math> and then the homotopy quotient $B\mathcal$ is, by definition, the homotopy type of $\operatorname_G(X)$.

Equivariant characteristic classes

Let ''E'' be an equivariant vector bundle on a ''G''-manifold ''M''. It gives rise to a vector bundle $\widetilde$ on the homotopy quotient $EG \times_G M$ so that it pulls-back to the bundle $\widetilde=EG \times E$ over $EG \times M$. An equivariant characteristic class of ''E'' is then an ordinary characteristic class of $\widetilde$, 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 $H^2(M; \mathbb).$using Čech cohomology and the isomorphism $H^1(M; \mathbb^*) \simeq H^2(M; \mathbb)$ given by the exponential map. 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 $H^2_G(M; \mathbb)$.

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

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*

Relation to stacks

* PDF page 10 has the main result with examples.

Further reading

*
*

External links

* — Excellent survey article describing the basics of the theory and the main important theorems
*
*{{cite web , author=Young-Hoon Kiem , url=http://www.math.snu.ac.kr/~kiem/mylecture-equivcoh.pdf , title=Introduction to equivariant cohomology theory , date=2008 , publisher=Seoul National University
What is the equivariant cohomology of a group acting on itself by conjugation?

Algebraic topology
Homotopy theory
Symplectic topology
Group actions (mathematics)