View source for Equivariant cohomology

In [[mathematics]], '''equivariant cohomology''' (or ''Borel cohomology'') is a [[cohomology theory]] from [[algebraic topology]] which applies to [[topological space]]s with a ''[[Group action (mathematics)|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>X</math> with action of a topological group <math>G</math> is defined as the ordinary [[cohomology ring]] with coefficient ring <math>\Lambda</math> of the [[Universal bundle#Use_in_the_study_of_group_actions|homotopy quotient]] <math>EG \times_G X</math>:
:<math>H_G^*(X; \Lambda) = H^*(EG \times_G X; \Lambda).</math>
If <math>G</math> is the [[trivial group]], this is the ordinary [[cohomology ring]] of <math>X</math>, whereas if <math>X</math> is [[contractible]], it reduces to the cohomology ring of the [[classifying space]] <math>BG</math> (that is, the group cohomology of <math>G</math> when ''G'' is finite.) If ''G'' acts freely on ''X'', then the canonical map <math>EG \times_G X \to X/G</math> is a homotopy equivalence and so one gets: <math>H_G^*(X; \Lambda) = H^*(X/G; \Lambda).</math>

== Definitions ==
It is also possible to define the equivariant cohomology 
<math>H_G^*(X;A)</math> of <math>X</math> with coefficients in a 
<math>G</math>-module ''A''; these are [[abelian group]]s. 
This construction is the analogue of cohomology with local coefficients.

If ''X'' is a [[manifold]], ''G'' a [[compact Lie group]] and <math>\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 form]]s.)

The construction should not be confused with other cohomology theories,
such as [[Bredon cohomology]] or the cohomology of [[invariant differential form]]s: if ''G'' is a compact Lie group, then, by the averaging argument{{Citation needed|reason=Previous link was incorrect.|date=October 2017}}, 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>\mathfrak{X} = [X_1 \rightrightarrows X_0]</math> equivariant cohomology of a smooth manifold<ref>{{harvnb|Behrend|2004}}</ref> is a special example of the groupoid cohomology of a Lie groupoid. This is because given a <math>G</math>-space <math>X</math> for a compact Lie group <math>G</math>, there is an associated groupoid<blockquote><math>\mathfrak{X}_G = [G\times X \rightrightarrows X] </math></blockquote>whose equivariant cohomology groups can be computed using the Cartan complex <math>\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<blockquote><math>\Omega^n_G(X) = \bigoplus_{2k+i = n}(\text{Sym}^k(\mathfrak{g}^\vee)\otimes \Omega^i(X))^G</math></blockquote>where <math>\text{Sym}^\bullet(\mathfrak{g}^\vee)</math> is the symmetric algebra of the dual Lie algebra from the Lie group <math>G</math>, and <math>(-)^G</math> corresponds to the <math>G</math>-invariant forms. This is a particularly useful tool for computing the cohomology of <math>BG</math> for a compact Lie group <math>G</math> since this can be computed as the cohomology of<blockquote><math>[G \rightrightarrows *]</math></blockquote>where the action is trivial on a point. Then,<blockquote><math>H^*_{dR}(BG) = \bigoplus_{k\geq 0 }\text{Sym}^{2k}(\mathfrak{g}^\vee)^G</math></blockquote>For example,<blockquote><math>\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></blockquote>since the <math>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>X</math> by its <math>G</math>-action) in which <math>X</math> is first replaced by a larger but [[homotopy equivalent]] space so that the action is guaranteed to be [[Group action (mathematics)|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<sup>−1</sup>x''): moreover, this diagonal action is free since it is free on ''EG''.  So we define the homotopy quotient ''X''<sub>''G''</sub> to be the orbit space (''EG'' × ''X'')/''G'' of this free ''G''-action.

In other words, the homotopy quotient is the [[associated bundle|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''<sub>''G''</sub> → ''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 <math>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>BG</math> is [[n-connected|2-connected]] and ''X'' has real dimension 2. Fix some smooth ''G''-bundle <math>P_\text{sm}</math> on ''X''. Then any principal ''G''-bundle on <math>X</math> is isomorphic to <math>P_\text{sm}</math>. In other words, the set <math>\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>P_\text{sm}</math> or equivalently the set of holomorphic connections on ''X'' (since connections are integrable for dimension reason). <math>\Omega</math> is an infinite-dimensional complex affine space and is therefore contractible.

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

One can define the [[moduli stack of principal bundles]] <math>\operatorname{Bun}_G(X)</math> as the [[quotient stack]] <math>[\Omega/\mathcal{G}]</math> and then the homotopy quotient <math>B\mathcal{G}</math> is, by definition, the [[homotopy type]] of <math>\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>\widetilde{E}</math> on the homotopy quotient <math>EG \times_G M</math> so that it pulls-back to the bundle <math>\widetilde{E}=EG \times E</math> over <math>EG \times M</math>. An equivariant characteristic class of ''E'' is then an ordinary characteristic class of <math>\widetilde{E}</math>, which is an element of the completion of the cohomology ring <math>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>H^2(M; \mathbb{Z}).</math><ref>using [[Čech cohomology]] and the isomorphism <math>H^1(M; \mathbb{C}^*) \simeq H^2(M; \mathbb{Z})</math> given by the [[exponential function|exponential map]].</ref> 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>H^2_G(M; \mathbb{Z})</math>.

== Localization theorem ==
{{expand-section|date=April 2014}}
{{main|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]]

==Notes==
{{reflist}}

==References==
* {{citation|first1=Michael|last1= Atiyah|author1-link=Michael Atiyah| first2= Raoul|last2= Bott|author2-link=Raoul Bott| title=The moment map and equivariant cohomology|journal=[[Topology (journal)|Topology]]|volume= 23 |pages= 1–28|year=1984|doi= 10.1016/0040-9383(84)90021-1|doi-access=free}}
*{{cite book |first=M. |last=Brion |chapter=Equivariant cohomology and equivariant intersection theory |chapter-url=http://www-fourier.ujf-grenoble.fr/~mbrion/notesmontreal.pdf |title=Representation Theories and Algebraic Geometry |publisher=Springer |series=Nato ASI Series |volume=514 |date=1998 |pages=1–37 |isbn=978-94-015-9131-7 |doi=10.1007/978-94-015-9131-7_1 |arxiv=math/9802063 |s2cid=14961018 }}
*{{citation|first1=Mark|last1= Goresky|author1-link=Mark Goresky| first2=Robert |last2=Kottwitz|author2-link=Robert Kottwitz|first3= Robert|last3= MacPherson|author3-link=Robert MacPherson (mathematician)| title=Equivariant cohomology, Koszul duality, and the localization theorem|journal= [[Inventiones Mathematicae]]|volume= 131 |year=1998|pages=25–83 |doi=10.1007/s002220050197 |citeseerx= 10.1.1.42.6450|s2cid= 6006856}}
*{{cite book|first=Wu-Yi |last=Hsiang|title=Cohomology Theory of Topological Transformation Groups|publisher=Springer |year=1975 |doi=10.1007/978-3-642-66052-8  |isbn=978-3-642-66052-8 }}
*{{Cite journal
    | last = Tu
    | first = Loring W.
    | title = What Is . . . Equivariant Cohomology?
    | journal = [[Notices of the American Mathematical Society]]
    | volume = 58
    | issue = 3
    | pages = 423–6
    | date = March 2011 |arxiv=1305.4293
    | url = https://www.ams.org/notices/201103/rtx110300423p.pdf }}

=== Relation to stacks ===

*{{cite book |first=K. |last=Behrend |chapter=Cohomology of stacks |chapter-url=https://www.math.ubc.ca/~behrend/CohSta-1.pdf |title=Intersection theory and moduli |series=ICTP Lecture Notes |volume=19 |date=2004 |isbn=9789295003286 |pages=249–294 }} PDF page 10 has the main result with examples.

==Further reading==
*{{cite book |first1=V.W. |last1=Guillemin |first2=S. |last2=Sternberg |title=Supersymmetry and equivariant de Rham theory  |publisher=Springer |location= |date=1999 |isbn=978-3-662-03992-2 |doi=10.1007/978-3-662-03992-2 }}
*{{cite web |last1=Vergne |first1=M. |last2=Paycha |first2=S. |title=Cohomologie équivariante et théoreme de Stokes |date=1998 |publisher=Département de Mathématiques, Université Blaise Pascal |url=https://www.emis.de/journals/SC/2003/7/pdf/smf_sem-cong_7_1-43.pdf}}

== External links ==
*{{citation |first=E. |last=Meinrenken |chapter=Equivariant cohomology and the Cartan model |chapter-url=http://www.math.toronto.edu/mein/research/enc.pdf |title=Encyclopedia of mathematical physics |publisher= |location= |date=2006 |isbn=978-0-12-512666-3 |pages=242–250 |url=}} — Excellent survey article describing the basics of the theory and the main important theorems
*{{Springer|id=e/e036090|title=Equivariant cohomology}}
*{{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 }}
*[https://mathoverflow.net/q/20671 What is the equivariant cohomology of a group acting on itself by conjugation?]

[[Category:Algebraic topology]]
[[Category:Homotopy theory]]
[[Category:Symplectic topology]]
[[Category:Group actions (mathematics)]]