equivariant cohomology

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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#### Representation theory
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[[!include representation theory - contents]]
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#Contents#
* table of contents
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## Idea
 {#Idea}

_Equivariant cohomology_ is [[cohomology]] in the presence of and taking into account [[group]]-[[actions]] (and generally [[∞-group]] [[∞-actions]]) both on the domain space and on the [[coefficients]].

Exactly what this comes down to depends on the choice of ambient [[(∞,1)-topos]] $\mathbf{H}$ and of the way that $G$ is regarded as an [[∞-group]] object of $\mathbf{H}$. Some important choices are the following:

* **"coarse" equivariance.** For $\mathbf{H} = $ [[∞Grpd]] $\simeq L_{whe}$ [[Top]], and $G$ a [[discrete group]], regarded via its [[delooping]] groupoid/[[classifying space]] $\mathbf{B}G \in \mathbf{H}$, then $\mathbf{H}_{/\mathbf{B}G}$ is [[presentable (∞,1)-category|presented]] by the [[Borel model structure]] on the category of [[simplicial sets]] equipped with $G$-action. (This is also called the _coarse_ [[equivariant homotopy theory]], in view of the next examples). This theory only knows [[homotopy quotients]] and [[homotopy fixed points]] of $G$ (in particular cofibrant replacement in the [[Borel model structure]] is indeed given by the [[Borel construction]] and so Borel equivariant cohomology theory appears here whenever the [[coefficients]] have trivial $G$-action). In the case that the domain itself is the points with trivial $G$-action then the equivariant cohomology here is precisely the [[group cohomology]] of $G$.

* **"fine" Bredon equivariance.** In order to bring in more geometric information one may equip [[G-spaces]] with information about the actual $G$-[[fixed points]], not just their [[homotopy fixed points]]. By general lore of [[topos theory]] this means to have all spaces be probe-able by fixed points, hence to have them be [[(∞,1)-presheaves]] on the [[global equivariant indexing category]] $Glob$, or if desired just on the [[global orbit category]] $Orb$, hence to set $(\mathbf{H} \to \mathbf{B}) =  (PSh_\infty(Glob) \to PSh_\infty(Orb))$, where the [[base (∞,1)-topos]] is that of [[orbispaces]] and $\mathbf{H}$ sitting [[cohesion|cohesively]] over it is the "[[global equivariant homotopy theory]]" proper (see there).

  Now we have $\mathbf{B}G \in \mathbf{H}$ naturally via the [[(∞,1)-Yoneda embedding]] and the [[slice (∞,1)-topos]] $\mathbf{B}_{/\mathbf{B}G} \simeq L_{fpwe} G Top $ is the traditional [[equivariant homotopy theory]] presented by the "fine" model structure on [[G-spaces]] whose weak equivalences are the $H$-fixed point wise [[weak homotopy equivalences]] for all suitable subgroups $H \hookrightarrow G$. The [[spectrum objects]] here are what are called [[spectra with G-action]] or "[[naive G-spectra]]".
See at _[[Elmendorf's theorem]]_ for details. By the discussion there every object in the fine model structure if fibrant and cofibrant replacement here is given by passage to [[G-CW complexes]], so that the [[derived hom spaces]] computing cohomology are the ordinary $G$-[[fixed points]] of the [[mapping spectra]] from such as [[G-CW complex]] into the coefficient spectrum (this is traditionally motivated via detour through [[genuine G-spectra]], see e.f. [Greenlees-May, equation (3.7)](#GreenleesMay)).

  Cohomology with [[Eilenberg-MacLane object]]-[[coefficients]] in $PSh_\infty(Orb)_{/\mathbf{B}G}$ is what [[Glen Bredon]] originally considered as what is now called _[[Bredon cohomology]]_.

## Presentations

> under construction

(...) [[Elmendorf theorem]] (...) [[Borel model structure]] (...)

## Borel equivariant cohomology
 {#Borel}

We first state the [[category theory|general abstract]] definition of _[[Borel equivariant cohomology]]_ and then derive from it the more concrete formulations that are traditionally given in the literature.

+-- {: .standout}

[[Borel equivariant cohomology]] is the [[cohomology]] of [[action groupoids]] ([[homotopy quotients]]/[[Borel constructions]]).

=--

For standard [[cohomology]] in the [[(∞,1)-topos]] $\mathbf{H} =$ [[Top]] these [[action groupoid]]s of a [[group]] $G$ acting on a [[topological space]] $X$ are traditionally known as the [[Borel construction]] $\mathcal{E}G \times_G X$.

Recall from the discussion at [[cohomology]] that in full generality we have a notion of cohomology of an object $X$ with coefficients in an object $A$ whenever $X$ and $A$ are objects of some [[(∞,1)-topos]] $\mathbf{H}$. The cohomology set $H(X,A)$ is the set of connected components in the [[hom-object]] [[∞-groupoid]] of maps from $X$ to $A$: $H(X,A) = \pi_0 \mathbf{H}(X,A)$.

Recall moreover from the discussion at [[space and quantity]] that objects of an [[(infinity,1)-category of (infinity,1)-sheaves|(∞,1)-topos of (∞,1)-sheaves]] have the interpretation of [[∞-groupoid]]s with extra structure. For instance for $(\infty,1)$-sheaves on a [[site]] of  smooth test spaces such as [[Diff]] these objects have the interpretation of [[Lie ∞-groupoid]]s.

In this case, for $X$ some such [[∞-groupoid]] with structure, let $X_0  \hookrightarrow X$ be its 0-truncation, which is the [[space]] of [[object]]s of $X$, the [[discrete groupoid|categorically discrete groupoid]] underlying $X$. We think of the morphisms in $X$ as determining which points of $X_0$ are related under some kind of action on $X_0$, the 2-morphisms as relating these relations on some higher action, and so on. **Equivariance** means, roughly: functorial transformation behaviour of objects on $X_0$ with respect to this "action" encoded in the morphisms in $X$. This is the intuition that is made precise in the following

In the simple special case that one should keep in mind, $X$ is for instance the [[action groupoid]] $X = X_0//G$ of the [[action]], in the ordinary sense, of a [[group]] $G$ on $X_0$: its morphisms $x \to g(x)$ connect those objects of $X_0$ that are related by the action by some group element $g \in G$.

It is natural to consider the [[relative cohomology]] of the inclusion $X_0 \hookrightarrow X$. Equivariant cohomology is essentially just another term for relative cohomology with respect to an inclusion of a space into a ($\infty$-)groupoid.

+-- {: .un_defn}
###### Definition (equivariant cohomology)

In some [[(∞,1)-topos]] $\mathbf{H}$ the **equivariant cohomology** with coefficient in an object $A$ of a 0-truncated object $X_0$ with respect to an action encoded in an inclusion $X_0 \hookrightarrow X$ is simply the $A$-valued cohomology $H(X,A)$ of $X$.

More specifically, an **equivariant structure** on an $A$-cocycle $c : X_0 \to A$ on $X_0$ is a choice of extension $\hat c$

$$
  \array{
    X_0 &\to& A
    \\
    \downarrow & \nearrow_{\hat c}
    \\
    X
  }
  \,.
$$

i.e. a lift of $c$ through the projection $\mathbf{H}(X,A) \to \mathbf{H}(X_0,A)$.

=--

### Examples

#### Group cohomology

By comparing the definition of equivariant cohomology with that of [[group cohomology]] one sees that group cohomology can be equivalently thought of as being **equivariant cohomology of the point**.

#### Equivariant cohomotopy

[[!include flavours of cohomotopy -- table]]

#### Equivariant bundles

For $G$ some [[group]] let $G Bund$ be the [[stack]] of $G$-[[principal bundle]]s. Let $K$ be some finite group (just for the sake of simplicity of the example) and let $K \to Aut(X_0)$ be an action of $K$ on a space $X_0$. Let $X = X_0 // K$ be the corresponding [[action groupoid]].

Then a cocycle in the $K$-equivariant cohomology $H(X_0//K, G Bund)$ is

* a $G$-[[principal bundle]] $P \to X$ on $X$;

* for each $k \in K$ an [[isomorphism]] of $G$-principal bundles $\lambda_k : P \to k^* P$

* such that for all $k_1, k_2 \in K$ we have $\lambda_{k_2}\circ \lambda_{k_1} = \lambda_{k_2\cdot k_1}$.

#### Local systems -- flat connections

For $X_0$ a [[space]] and $X := P_n(X_0)$ a version of its [[path n-groupoid]] we have a canonical inclusion $X_0 \hookrightarrow P_n(X_0)$ of $X_0$ as the collection of constant paths in $X_0$.

Consider for definiteness $\Pi(X_0) := \Pi_\infty(X_0)$, the [[path ∞-groupoid]] of $X_0$. (All other cases are in principle obtaind from this by truncation and/or strictification).

Then for  $A$ some coefficient $\infty$-groupoid, a morphism $g : X_0 \to A$ can be thought of as classifying a $A$-[[principal ∞-bundle]] on the space $X_0$.

On the other hand, a morphism out of $P_n(X_0)$ is something like a flat [[connection on a bundle|connection]] (see there for more details) on this principal $\infty$-bundle, also called an $A$-[[local system]]. (More details on this are at [[schreiber:differential cohomology]]).

Accordingly, an extension of $g : X_0 \to A$ through the inclusion $X_0 \hookrightarrow \Pi(X)$ is the process of equipping a principal $\infty$-bundle with a flat connection.

Comparing with the above definition of eqivariant cohomology, we see that flat connections on bundles may be regarded as **path-equivariant structures** on these bundles.

This is therefore an example of equivariance which is not with respect to a global [[group]] action, but genuinely a [[groupoid]]al one.

#### Equivariant de Rham cohomology

* [[equivariant de Rham cohomology]]

### Remarks

When pairing equivariant cohomology with other variants of cohomology such as [[twisted cohomology]] or [[differential cohomology]] one has to exercise a bit of care as to what it really is that one wants to consider. A discussion of this is (beginning to appear) at [[schreiber:differential equivariant cohomology]].

## Bredon equivariant cohomology {#Bredon}

See also

* [[Bredon cohomology]]

### Preliminary remarks

According to the [[nPOV]] on [[cohomology]], if $X$ and $A$ are objects in an [[(∞,1)-topos]], the 0th cohomology $H^0(X;A)$ is $\pi_0(Map(X,A))$, while if $A$ is a [[groupoid object in an (∞,1)-category|group object]], then $H^1(X;A)= \pi_0(Map(X,B A))$.  More generally, if $A$ is $n$ times [[delooping|deloopable]], then $H^n(X;A) = \pi_0(Map(X, B^n A)$.  In [[Top]], this gives you the usual notions if $A$ is a (discrete) group, and in general, $H^1(X;A)$ classifies [[principal ∞-bundle]]s in whatever [[(∞,1)-topos]].

Now consider the $(\infty,1)$-topos $G Top$ of $G$-equivariant spaces, which can also be described as the [[(∞,1)-presheaf|(∞,1)-presheaves]] on the [[orbit category]] of $G$.  For any other group $\Pi$ there is a notion of a principal $(G,\Pi)$-bundle (where $G$ is the group of equivariance, and $\Pi$ is the structure group of the bundle), and these are classified by maps into a classifying $G$-space $B_G \Pi$.  So the principal $(G,\Pi)$-bundles over $X$ can be called $H^0(X;B_G \Pi)$.  If we had something of which $B_G \Pi$ was a [[delooping]], we could call the principal $(G,\Pi)$-bundles "$H^1(X;?)$", but there does not seem to be such a thing.  It seems that $B_G \Pi$ is not connected, in the sense that ${*}\to B_G \Pi$ is not an [[effective epimorphism]] and thus $B_G \Pi$ is not the quotient of a [[groupoid object in an (∞,1)-category|group object]] in $G Top$.

### $G$-equivariant spectra

If we have an object $A$ of our $(\infty,1)$-topos that can be delooped infinitely many times, then we can define $H^n(X;A)$ for any integer $n$ by looking at all the spaces $\Omega^{-n} A = B^n A$.  These integer-graded [[cohomology group]]s are closely connected to each other, e.g. they often have [[cup product]]s or [[Steenrod square]]s or [[Poincare duality]], so it makes sense to consider them all together as a _[[cohomology theory]]_ .  We then are motivated to put together all of the objects $\{B^n A\}$ into a [[spectrum object]], a single object which encodes all of the cohomology groups of the theory.  A general spectrum is a sequence of objects $\{E_n\}$ such that $E_n \simeq \Omega E_{n+1}$; the stronger requirement that $E_{n+1} \simeq B E_n$ restricts us to "connective" spectra, those that can be produced by successively delooping a single object of the $(\infty,1)$-topos.  In [[Top]], the most "basic" spectra are the [[Eilenberg-MacLane spectrum|Eilenberg-MacLane spectra]] produced from the input of an ordinary abelian group.

Now we can do all of this in $G Top$, and the resulting notion of spectrum is called a **[[naive G-spectrum]]**: a sequence of $G$-spaces $\{E_n\}$ with $E_n \simeq \Omega E_{n+1}$.  Any naive $G$-spectrum represents a cohomology theory on $G$-spaces.  The most "basic" of these are "Eilenberg-Mac Lane $G$-spectra" produced from **coefficient systems**, i.e. abelian-group-valued presheaves on the  [[orbit category]].  The cohomology theory represented by such an Eilenberg-Mac Lane $G$-spectrum is called an (integer-graded) [[Bredon cohomology]] theory.

It turns out, though, that the cohomology theories arising in this way are kind of weird.  For instance, when one calculates with them, one sees [[torsion]] popping up in odd places where one wouldn't expect it.  It would also be nice to have a [[Poincare duality]] theorem for $G$-manifolds, but that fails with these theories.  The solution people have come up with is to widen the notion of "[[loop space object|looping]]" and "[[delooping]]" and thereby the grading:

instead of just looking at $\Omega^n = Map(S^n, -)$, we look at $\Omega^V = Map(S^V,-)$, where $V$ is a finite-dimensional [[representation]] of $G$ and $S^V$ is its [[one-point compactification]].  Now if $A$ is a $G$-space that can be [[delooping|delooped]] "$V$ times," we can define $H^V(X;A) = \pi_0(Map(X,\Omega^{-V} A)$.  If $A$ can be delooped $V$ times for all representations $V$, then our integer-graded cohomology theory can be expanded to an **[[RO(G)-grading|RO(G)-graded]] cohomology theory**, with cohomology groups $H^\alpha(X;A)$ for all formal differences of representations $\alpha = V - W$.  The corresponding notion of spectrum is a **[[genuine G-spectrum]]**, which consists of spaces $E_V$ for all representations $V$ such that $E_V \simeq \Omega^{W-V} E_W$.  A naive Eilenberg-Mac Lane $G$-spectrum can be extended to a genuine one precisely when the coefficient system it came from can be extended to a [[Mackey functor]], and in this case we get an **$RO(G)$-graded Bredon cohomology theory** .

$RO(G)$-graded Bredon cohomology has lots of formal advantages over the integer-graded theory.  For instance, the torsion that popped up in odd places before can now be seen as arising by "shifting" of something in the cohomology of a point in an "off-integer dimension," which was invisible to the integer-graded theory.  Also there is a [[Poincare duality]] for $G$-manifolds: if $M$ is a $G$-manifold, then we can embed it in a representation $V$ (generally not a trivial one!) and by [[Thom space]] arguments, obtain a Poincare duality theorem involving a dimension shift of $\alpha$, where $\alpha$ is generally not an integer (and, apparently, not even uniquely determined by $M$!).  Unfortunately, however, $RO(G)$-graded Bredon cohomology is kind of hard to compute.

For more see at _[[equivariant stable homotopy theory]]_ and _[[global equivariant stable homotopy theory]]_.

### Examples

* $\mathbb{Z}_2$-equivariant cohomology theories: [[KR-theory]], [[MR-theory]]

* [[modular group]]-equivariance: [[modular equivariant elliptic cohomology]]

## Multiplicative equivariant cohomology

For [[multiplicative cohomology theories]] there is a further refinement of equivariance where the equivariant cohomology groups are built from global sections on a [[sheaf]] over cerain systems of [[moduli spaces]]. For more on this see at

* [Equivariant multiplicative cohomology](http://ncatlab.org/nlab/show/A+Survey+of+Elliptic+Cohomology+-+A-equivariant+cohomology#equivariant)

* [[equivariant elliptic cohomology]]

## Examples

[[!include Segal completion -- table]]

## Related concepts

* [[equivariance]], [[equivariant structure]]

* [[equivariant bundle]], [[equivariant connection]]

* [[equivariant differential topology]]

* [[equivariant stable homotopy theory]], [[global equivariant stable homotopy theory]]

* [[equivariant rational homotopy theory]], [[rational equivariant stable homotopy theory]]

  * [[model structure on equivariant dgc-algebras]]

* [[equivariant cobordism theory]]

* [[equivariant localization]]

* [[Segal conjecture]]

* [[equivariant K-theory]], [[equivariant operator K-theory]], [[equivariant KK-theory]]

  * [[Baum-Connes conjecture]], [[Green-Julg theorem]], [[Atiyah-Segal completion theorem]]

  * [[quantization commutes with reduction]]

* [[equivariant elliptic cohomology]]

* [[equivariant complex oriented cohomology theory]]

* [[orbifold cohomology]]

* [[Chen-Ruan cohomology]], [[delocalized equivariant cohomology]]

* [[twisted equivariant cohomology]]

[[!include homotopy type representation theory -- table]]

## References

### General

Introduction to [[Borel equivariant cohomology]]:

* {#Tu11} [[Loring Tu]], _What is... Equivariant Cohomology?_, Notices of the AMS, Volume 85, Number 3, March 2011 ([pdf](https://www.ams.org/notices/201103/rtx110300423p.pdf), [[TuEquivariantCohomology.pdf:file]])

* [[Loring Tu]], *Introductory Lectures on Equivariant Cohomology*, Annals of Mathematics Studies **204**, AMS 2020 ([ISBN:9780691191744](https://press.princeton.edu/books/hardcover/9780691191744/introductory-lectures-on-equivariant-cohomology))

Introduction to [[Bredon equivariant cohomology]]:

* {#Blumberg17} [[Andrew Blumberg]], section 1.4 of _Equivariant homotopy theory_, 2017 ([pdf](https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf), [GitHub](https://github.com/adebray/equivariant_homotopy_theory))

* {#GreenleesMay} [[John Greenlees]], [[Peter May]], section 3 of _Equivariant stable homotopy theory_ ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))

* {#Guillou06} [[Bert Guillou]], *Equivariant Homotopy and Cohomology*, lecture notes, 2020 ([pdf](http://www.ms.uky.edu/~guillou/F20/751Notes.pdf), [[GuillouEquivariantHomotopyAndCohomology.pdf:file]])

Textbooks and lecture notes:

* {#tomDieck79} [[Tammo tom Dieck]], section 7 of _[[Transformation Groups and Representation Theory]]_, Lecture Notes in Mathematics 766, Springer 1979

* {#May96} [[Peter May]] et al., _Equivariant homotopy and cohomology theory_, CBMS Regional Conference Series in Mathematics Volume: 91; 1996 ([ISBN:978-0-8218-0319-6](https://bookstore.ams.org/cbms-91), [pdf](http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf), [pdf](https://ncatlab.org/nlab/files/MayEtAlEquivariant96.pdf))

* Matvei Libine, _Lecture Notes on Equivariant Cohomology_ ([arXiv](http://arxiv.org/abs/0709.3615))

* Sébastien Racanière, _Lecture on Equivariant Cohomology_, 2004 ([[RacaniereEquivariant04.pdf:file]])

For a brief modern survey see also the first three sections of

* [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], _The Arf-Kervaire problem in algebraic topology: Sketch of the proof_ ([[HHRKervaire.pdf:file]])

  (with an eye towards application to the [[Arf-Kervaire invariant problem]])

* blog on [Localization techniques in Equivariant Cohomology](http://www.aimath.org/wiki/localization/index.php/Main_Page)

Discussion of equivariant versions of [[differential cohomology]] is in

* Andreas Kübel, [[Andreas Thom]], _Equivariant Differential Cohomology_, Transactions of the American Mathematical Society (2018) ([arXiv:1510.06392](https://arxiv.org/abs/1510.06392))

See also at _[[equivariant de Rham cohomology]]_.

### In complex oriented generalized cohomology theory
 {#InComplexOrientedGeneralizedCohomologyTheory}

Equivariant [[complex oriented cohomology theory]] is discussed in the following articles.

* [[Michael Hopkins]], [[Nicholas Kuhn]], [[Douglas Ravenel]], _Generalized group characters and complex oriented cohomology theories_, J. Amer. Math. Soc. 13 (2000), 553-594 ([publisher](http://www.ams.org/journals/jams/2000-13-03/S0894-0347-00-00332-5/), [pdf](http://www.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf))

  (This deals with "naive" Borel-equivariant complex oriented cohomology, but discusses general [[character]] expressions and explicit formulas for equivariant [[Morava K-theory|K(n)]]-cohomology.)

Specifically equivariant [[complex cobordism cohomology]] is discussed in

* {#tomDiek70} [[Tammo tom Dieck]], _Bordism of $G$-manifolds and integrability theorems, Topology 9 (1970) 345-358

* {#Abrams13a} [[William Abram]], _Equivariant complex cobordism_, 2013 ([web](http://deepblue.lib.umich.edu/handle/2027.42/99796), [pdf](http://deepblue.lib.umich.edu/bitstream/handle/2027.42/99796/abramwc_1.pdf?sequence=1))

* [[William Abram]], [[Igor Kriz]], _The equivariant complex cobordism ring of a finite abelian group_ ([pdf](http://www.math.lsa.umich.edu/~ikriz/cobordism13022-1.pdf))

The following articles discuss [[equivariant formal group laws]]:

* [[John Greenlees]], _Equivariant formal group laws and complex oriented cohomology theories_, Homology Homotopy Appl. Volume 3, Number 2 (2001), ii-451 ([EUCLID](http://projecteuclid.org/euclid.hha/1139840255))

* [[William Abram]], _On the equivariant formal group law of the equivariant complex cobordism ring_, ([arXiv:1309.0722](http://arxiv.org/abs/1309.0722))

  (also [Abrams 13a, section III](#Abrams13a)).

See also the references at _[[equivariant elliptic cohomology]]_.

[[!include traditional orbifold cohomology -- references]]

[[!redirects Borel cohomology]]

[[!redirects equivariant cohomology theory]]
[[!redirects equivariant cohomology theories]]


equivariant cohomology -- table

**[[cohomology]] in the presence of [[∞-group]] $G$ [[∞-action]]**:

| [[Borel equivariant cohomology]] |  $\phantom{AAA}\leftarrow\phantom{AAA}$ | general ([[Bredon cohomology|Bredon]]) [[equivariant cohomology]] | $\phantom{AAA}\rightarrow\phantom{AAA}$ | non-equivariant [[cohomology]] with [[homotopy fixed point]] [[coefficients]]  |
|---|---|---|---|---|
| $\phantom{AA}\mathbf{H}(X_G, A)\phantom{AA}$ |  trivial [[∞-action|action]] on [[coefficients]] $A$ | $\phantom{AA}[X,A]^G\phantom{AA}$ | trivial [[∞-action|action]] on [[domain]] space $X$ |  $\phantom{AA}\mathbf{H}(X, A^G)\phantom{AA}$ |


Equivariant cohomology theories

This entry collects material related to the book

* [[Glen Bredon]],

  _Equivariant cohomology theories_,

  Lecture Notes in Mathematics Vol. 34

  Springer 1967

  ([doi:10.1007/BFb0082690](https://link.springer.com/book/10.1007/BFb0082690))

announced in

* [[Glen Bredon]], _Equivariant cohomology theories_, Bull. Amer. Math. Soc. Volume 73, Number 2 (1967), 266-268. ([Euclid](https://projecteuclid.org/euclid.bams/1183528794))

on [[equivariant cohomology]] ([[Bredon cohomology]]).

category: reference


equivariant cohomotopy

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
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[[!include homotopy - contents]]
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#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
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#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
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#### Manifolds and cobordisms
+--{: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[Bredon cohomology]]-[[equivariant cohomology|equivariant]] enhancement of [[cohomotopy]]-theory is _equivariant cohomotopy_:

For $G$ a [[group]], $V$ a [[finite dimensional vector space|finite-dimensional]] [[real vector space|real]]  $G$-[[representation]] $G \to O(V)$ and writing $S^V$ for the corresponding [[representation sphere]], the _equivariant cohomotopy_ in [[RO(G)-degree]] $V$ of a [[G-space]] $X$ is the set of $G$-[[equivariant homotopy theory|equivariant]] [[homotopy classes]] of maps from $X$ to $S^V$:

$$
  \pi_G^V\big( X \big)
  \;\coloneqq\;
  \left[
    X, S^V
  \right]^G
  \,.
$$

For $V = \mathbb{R}^n$ the [[trivial representation]] of [[dimension]] $n$, this reduces to the definition of plain [[cohomotopy]]-sets

$$
  \pi^{\mathbb{R}^n}(X)
  \;=\;
  \pi^n(X)
  \;=\;
  \left[ X, S^n\right]
  \,.
$$

The [[stabilization]] of this construction, in the sense of [[equivariant stable homotopy theory]], yields the concept of _[[equivariant stable cohomotopy]]_.

## Properties

### Equivariant Hopf degree theorem

* [[equivariant Hopf degree theorem]]

## Examples

### Equivariant Cohomotopy of $S^V$ in RO-degree $V$

As a special case of the [[equivariant Hopf degree theorem]] , we obtain the following:

+-- {: .num_example #EquivariantHomotopyOfSVInRODegreeV}
###### Proposition
**([[equivariant cohomotopy]] of [[representation sphere]] $S^V$ in [[RO(G)-degree]] $V$)**

  Let $G \in \mathrm{Grp}_{\mathrm{fin}}$
  and $V \in \mathrm{RO}(G)$ with $V^G = 0$.
  Then the bipointed [[equivariant cohomotopy]]
  of the [[representation sphere]] $S^V$
  in [[RO(G)-degree]] $V$ is
  the [[Cartesian product]] of one copy of the [[integers]] for
  each [[isotropy group|isotropy]] subgroup (eq:IsotropySubgroups) of $G$ in $S^V$
  except the full subgroup $G \subset G$

$$
  \array{
      \pi^V\left( S^V\right)^{\{0,\infty\}/}
      &
        \overset{\simeq}{\longrightarrow}
      &
      \underset{
        {
          {
              {H \in \mathrm{Isotr}_{S^V}(G)}
              \atop
              {H \neq G}
          }
        }
      }{\prod}
      \;\;
      {\vert W_G(H)\vert } \cdot \mathbb{Z}
      \\
      \big[
        S^V
          \overset{c}{\longrightarrow}
        S^V
      \big]
      &\mapsto&
      \Big(
        H
          \mapsto
        \mathrm{deg}
        \big(
          c^H
        \big)
        -
        \mathrm{offs}(c,H)
      \Big)
  }
$$
  where on the right
  $$
    \mathrm{deg}
    \Big(
      \big(
        S^V
      \big)^H
      \overset{
        c^H
      }{\longrightarrow}
      \big(
        S^V
      \big)^H
    \Big)
    \in \mathbb{Z}
  $$
  is the [[integer]] [[winding number]] of the underlying [[continuous function]]
  of $c$ ([[corestriction|co]])[[restriction|restricted]] to $H$-[[fixed points]], and part of the claim is  that this is an integer multiple of the order of the [[Weyl group]] $W_G(H)$ up to an offset

$$
  \mathrm{offs}(f,H)
  \;\in\;
  \big\{
    0,1, \cdots,  \left\vert W_G(H)\right\vert
  \big\}
  \;\subset\;
  \mathbb{Z}
$$

which depends in a definite way on the degrees of $c^K$ for  all isotropy groups $K \gt H$.

=--

For **[[proof]]** see [here](Hopf+degree+theorem#EquivariantHomotopyOfSVInRODegreeV) at _[[equivariant Hopf degree theorem]]_.

+-- {: .num_example #EquivariantCohomotopyOfRepresentationSphereOfSignRepresentationInThatDegree}
###### Example
**([[equivariant cohomotopy]] of $S^{\mathbb{R}_{sgn}}$ in [[RO(G)-degree]] the [[sign representation]] $\mathbb{R}_{sgn}$)**

Let $G = \mathbb{Z}_2$ the [[cyclic group of order 2]] and $\mathbb{R}_{sgn} \in RO(\mathbb{Z}_2)$ its 1-dimensional [[sign representation]].

Under equivariant [[stereographic projection]] ([here](representation\+sphere#Construction)) the corresponding [[representation sphere]] $S^{\mathbb{R}_{sgn}}$ is equivalently the [[unit circle]]

$$
  S^1
  \simeq
  S(\mathbb{R}^2)
$$

equipped with the $\mathbb{Z}_2$-[[action]] whose [[involution]] element $\sigma$ [[reflection|reflects]] one of the two [[coordinate functions|coordinates]] of the ambient [[Cartesian space]]

$$
  \sigma \;\colon\; (x_1,x_2) \mapsto (x_1, -x_2)
  \,.
$$

Equivalently, if we identify

\[
  \label{CircleAsQuaotientOfRByZ}
  S^1
  \;\simeq\;
  \mathbb{R}/\mathbb{Z}
\]

then the involution action is

$$
  \begin{aligned}
    \sigma
    \;\colon\;
    t
    \mapsto
    &
    \phantom{\sim}
    1 - t
    \\
    & \sim
    \phantom{1} - t
  \end{aligned}
  \,.
$$

This means that the [[fixed point space]] is the [[0-sphere]]

$$
  \big( S^1\big)^{\mathbb{Z}_2}
  \;\simeq\;
  S^0
$$

being two antipodal points on the circle, which in the presentation (eq:CircleAsQuaotientOfRByZ) are labeled $\{0,1/2\} \simeq S^0$.

Notice that the map

\[
  \label{ConstantParameterFunctionFromSignRepresentationSphereToItself}
  \array{
    S^1 &\overset{n}{\longrightarrow}& S^1
    \\
    t &\mapsto& n\cdot t
  }
\]

of constant parameter speed and [[winding number]] $n \in \mathbb{N}$ is equivariant for this $\mathbb{Z}_2$-[[action]] on both sides:

\begin{center}
\begin{xymatrix}
  t
    \ar@{|->}[r]
    \ar@{|->}[d]_{\sigma}
    &
  n\cdot t
    \ar@{|->}[d]^{\sigma}
  \\
  -t
    \ar@{|->}[r]
    &
  -n \cdot t
\end{xymatrix}
\end{center}

Now the restriction of the map $n \cdot (-)\in \mathbb{Z}$ from (eq:ConstantParameterFunctionFromSignRepresentationSphereToItself) to the [[fixed points]]

$$
  \array{
    S^0 = \left( S^{\mathbb{R}_{sgn}}\right)
    &\hookrightarrow&
    S^{\mathbb{R}_{sgn}}
    \\
    {}^{ \mathllap{  \left( \cdot n\right)^{\mathbb{Z}_2} } }
    \big\downarrow
      &&
    \big\downarrow^{\mathrlap{\cdot n}}
    \\
    S^0 = \left( S^{\mathbb{R}_{sgn}}\right)
    &\hookrightarrow&
    S^{\mathbb{R}_{sgn}}
  }
$$

sends (0 to 0 and) $1/2$ to either $1/2$ or to $0$, depending on whether the [[winding number]] is [[odd number|odd]] or [[even number|even]]:

$$
  \array{
     S^0
       &\overset{ \left(\cdot n\right)^{\mathbb{Z}_2} }{\longrightarrow}&
     S^0
     \\
     1/2 &\mapsto&
     \left\{
       \array{
         1/2 &\vert& n \;\text{is odd}
         \\
         0 &\vert& n \text{is even}
       }
     \right.
  }
$$

Hence if the restriction to the [[fixed locus]] is taken to be the [[identity function|identity]] (bipointed [[equivariant cohomotopy]]) then, in accord with Prop. \ref{EquivariantHomotopyOfSVInRODegreeV} there remains the [[integers]] worth of equivariant [[homotopy classes]], where each integer $k \in \mathbb{Z}$ corresponds to the odd winding integer $1 + 2k$

$$
  \array{
    \pi^{\mathbb{R}_{sgn}}
    \left(
      S^{\mathbb{R}_{sgn}}
    \right)^{\{0,\infty\}/}
    &\simeq&
    2 \cdot \mathbb{Z} + 1
    &\simeq&
   \mathbb{Z}
    \\
    \left[
      \mathbb{R}/\mathbb{Z} \overset{c}{\to} \mathbb{R}/\mathbb{Z}
    \right]_{{0 \mapsto 0} \atop {1/2 \mapsto 1/2}}
    &\mapsto&
    deg(c)
    &\mapsto&
    \big( deg(c) - 1\big)/2
  }
$$

=--

+-- {: .num_example #EquivariantCohomotopyOfRepresentationSphereOfQuaternionsInThatDegree}
###### Example
**([[equivariant cohomotopy]] of $S^{\mathbb{H}}$ in [[RO(G)-degree]] the [[quaternions]] $\mathbb{H}$)**

Let $G \subset SU(2) \simeq S(\mathbb{H})$ be a non-[[trivial group|trivial]] [[finite subgroup of SU(2)]] and let $\mathbb{H} \in RO(G)$ be the [[real vector space]] of [[quaternions]] regarded as a [[linear representation]] of $G$ by left multiplication with unit [[quaternions]].

Then the bi-pointed [[equivariant cohomotopy]] of the [[representation sphere]] $S^{\mathbb{H}}$ in [[RO(G)-degree]] $\mathbb{H}$ is

$$
  \array{
    \pi^{\mathbb{H}}
    \left(
      S^{\mathbb{H}}
    \right)^{\{0,\infty\}/}
    &\simeq&
    {\left\vert G\right\vert} \cdot \mathbb{Z} + 1
    &\simeq&
    {\left\vert G\right\vert} \cdot \mathbb{Z}
    &\simeq&
    \mathbb{Z}
    \\
    \left[
      S^{\mathbb{H}}
      \overset{c}{\longrightarrow}
       S^{\mathbb{H}}
    \right]
    &\mapsto&
    deg\left( c^{ \{e\}  }\right)
    &\mapsto&
    deg\left( c^{ \{e\}  }\right) - 1
    &\mapsto&
    \big(
      deg\left( c^{ \{e\}  }\right) - 1
    \big)/ {\left\vert G\right\vert}
  }
$$

=--

+-- {: .proof}
###### Proof

The only [[isotropy groups|isotropy]] [[subgroups]] of the left action of $G$ on $\mathbb{H}$ are the two extreme cases $Isotr_{\mathbb{H}}(G) = \{1, G\} \in Sub(G)$. Hence the only multiplicity that appears in Prop. \ref{EquivariantHomotopyOfSVInRODegreeV} is

$$
  \left\vert W_G(1)\right\vert
  \;=\;
  \left\vert G \right\vert
  \,.
$$

and all degrees  must differ from that of the class of the [[identity function]] by a multiple of this multiplicity. Finally, the offset of the identity function is clearly $offs\left(id_{S^{\mathbb{H}}},1\right) = deg\left( id_{S^{\mathbb{H}}}\right) = 1$.

=--

## Related concepts

[[!include flavours of cohomotopy -- table]]

* [[equivariant Hopf degree theorem]]

* [[equivariant Pontrjagin theorem]]

## References

### General

* {#Wasserman69} [[Arthur Wasserman]], section 3 of _Equivariant differential topology_, Topology Vol. 8, pp. 127-150, 1969 ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/wasserman.pdf))

* {#tomDieck79} [[Tammo tom Dieck]], section 8.4 of _[[Transformation Groups and Representation Theory]]_, Lecture Notes in Mathematics 766 Springer 1979

* {#Peschke94} [[George Peschke]], _Degree of certain equivariant maps into a representation sphere_, Topology and its Applications Volume 59, Issue 2, 30 September 1994, Pages 137-156 (doi:10.1016/0166-8641(94)90091-4)

* Zalman Balanov, _Equivariant hopf theorem_, Nonlinear Analysis: Theory, Methods & Applications Volume 30, Issue 6, December 1997, Pages 3463-3474 (doi:10.1016/S0362-546X(97)00020-5)

* {#Cruickshank03} [[James Cruickshank]], _Twisted homotopy theory and the geometric equivariant 1-stem_, Topology and its Applications Volume 129, Issue 3, 1 April 2003, Pages 251-271 (doi:10.1016/S0166-8641(02)00183-9)

* Davide L. Ferrario, _On the equivariant Hopf theorem_, _Topology
Volume 42, Issue 2, March 2003, Pages 447-465_ (doi:10.1016/S0040-9383(02)00015-0)

### Cocycle spaces

Discussion of [[cocycle spaces]] in equivariant Cohomotopy:

* [[Victor Vassiliev]], _Twisted homology of configuration spaces, homology of spaces of equivariant maps, and stable homology of spaces of non-resultant systems of real homogeneous polynomials_ ([arXiv:1809.05632](https://arxiv.org/abs/1809.05632))

* [[Victor Vassiliev]], *Cohomology of spaces of Hopf equivariant maps of spheres* ([arXiv:2102.07157](https://arxiv.org/abs/2102.07157))

### For M-brane charge quantization

Discussion of [[M-brane]] [[charge quantization]] in [[equivariant cohomotopy]]:

* [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_, Comm. Math. Phys. **371** (2019) 425-524 [[arXiv:1805.05987](https://arxiv.org/abs/1805.05987), [doi:10.1007/s00220-019-03442-3](https://doi.org/10.1007/s00220-019-03442-3)]

  ([[equivariant rational homotopy theory|equivariant]] [[rational Cohomotopy]])

* {#SatiSchreiber19} [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], _[[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation]]_ ([arXiv:1909.12277](https://arxiv.org/abs/1909.12277))

  (implying [[RR-field tadpole cancellation]])

[[!redirects equivariant Cohomotopy]]

[[!redirects equivariant cohomotopy theory]]
[[!redirects equivariant Cohomotopy theory]]


equivariant complex cobordism cohomology theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cobordism theory
+--{: .hide}
[[!include cobordism theory -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The lift of [[complex cobordism cohomology theory]] to [[equivariant stable homotopy theory]].

[[!include Segal completion -- table]]

## Properties

### Universal orientation

For an [[abelian group|abelian]] [[compact Lie group]] $G$,
[[equivariant complex cobordism theory]] $MU_G$ is an [[equivariant complex oriented cohomology theory]] ([Greenlees 01, Sec. 13](#Greenlees01)).

Much as in the non-equivariant case (see at _[[universal complex orientation on MU]]_), $MU_G$ is universal in that there is a [[bijection]] between equivariant complex orientations (in degree 2) on some cohomology theory $E_G$ and homotopy ring homomorphisms of $G$-spectra $MU_G \to E_G$ ([Cole-Greenlees-Kriz 02, Theorem 1.2](#ColeGreenleesKriz02)).

For the analogous statement on the equivariant [[Lazard ring]] see [Greenlees 01a](#Greenlees01a), [Greenlees 01, Theorem 13.1](#Greenlees01), [Cole-Greenlees-Kriz 02, Theorem 1.3](#ColeGreenleesKriz02).

## Related concepts

* [[equivariant bordism homology theory]]

* [[equivariant Morava K-theory]]

* [[equivariant complex oriented cohomology theory]]

\linebreak

[[!include flavours of cobordism cohomology theories -- table]]

## References

* [[Peter May]], _Equivariant and non-equivariant module spectra_, Journal of Pure and Applied Algebra Volume 127, Issue 1, 1 May 1998, Pages 83–97 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/modfinal.pdf))

* {#Greenlees01a} [[John Greenlees]], _The coefficient ring of equivariant homotopical bordism classifies equivariant formal group laws over Noetherian rings_, 2001 ([preprint](https://homepages.abdn.ac.uk/d.j.benson/pages/html/archive/greenlees.html))

* {#Greenlees01b} [[John Greenlees]], _Equivariant formal group laws and complex oriented cohomology theories_, Homology Homotopy Appl. Volume 3, Number 2 (2001), 225-263 ([euclid:hha/1139840255](https://projecteuclid.org/euclid.hha/1139840255))

* {#ColeGreenleesKriz02} [[Michael Cole]], [[John Greenlees]], [[Igor Kriz]], _The universality of equivariant complex bordism_, Math Z 239, 455–475 (2002) ([doi:10.1007/s002090100315](https://doi.org/10.1007/s002090100315))

* [[William Abram]], _A note on the equivariant formal group law of the equivariant complex cobordism ring_ ([arXiv:1309.0722](http://arxiv.org/abs/1309.0722))

* [[William Abram]], [[Igor Kriz]], _The equivariant complex cobordism ring of a finite abelian group_ ([arXiv:1509.08540](http://arxiv.org/abs/1509.08540))

Generalization of the [[Conner-Floyd isomorphism]] to [[equivariant cohomology theory]], relating [[equivariant cobordism cohomology]] to [[equivariant K-theory]]:

* [[Steven Costenoble]], *Equivariant Conner-Floyd isomorphism*, Trans. Amer. Math. Soc. **304** 2 (1987) 801--818, [[jstor:2000742](http://www.jstor.org/stable/2000742)]

[[!redirects equivariant complex cobordism theory]]

[[!redirects equivariant MU]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Equivariant complex oriented cohomology theory_ is the generalization of [[complex oriented cohomology theory]] to [[equivariant cohomology]].

In generalization of [[topological K-theory]] as the prototypical example of a [[complex oriented cohomology theory]], its generalization to [[equivariant K-theory]] is equivariantly complex oriented.

## Examples

### Equivariant complex K-theory

[[equivariant complex K-theory]] is an [[equivariant complex oriented cohomology theory]] ([Greenlees 01, Section 10](#Greenlees01)):

+-- {: .num_prop #EquivariantKTheoryOfProjectiveGSpace}
###### Proposition
**([[equivariant K-theory of projective G-space]])**

For $G$ an [[abelian group|abelian]] [[compact Lie group]], let

$$
  \underset{i}{\oplus}
  \,
  \mathbf{1}_{V_i}
  \;\;
  \in
  \;\;
  G Representations_{\mathbb{C}}^{fin}
$$

be a [[finite dimensional vector space|finite-dimensional]] [[direct sum]] of [[complex numbers|complex]] 1-[[dimension|dimensional]] [[linear representations]].

The $G$-[[equivariant K-theory]] [[cohomology ring|ring]] $K_G(-)$ of the corresponding [[projective G-space]] $P(-)$ is the following [[quotient ring]] of the [[polynomial ring]] in one [[variable]] $L$ over the complex [[representation ring]] $R(G)$ of $G$:

\[
  \label{EquivariantKTheoryRingOfProjectiveGSpace}
  K_G
  \Big(
    P
    \big(
      \underset{i}{\oplus}
      \,
      \mathbf{1}_{V_i}
    \big)
  \Big)
  \;\;
    \simeq
  \;\;
  R(G)
  \big[
    L
  \big]
  \big/
  \underset{i}{\prod}
  \big(
    1 - 1_{{}_{V_i}} L
  \big)
  \,,
\]

where

* $L \,=\, \big[ \mathcal{L}_{ \underset{i}{\oplus} \mathbf{1}_{V_i} }\big]$ is the K-theory class of the [[tautological equivariant line bundle]] on the given [[projective G-space]];

* $ 1_{{}_{V_i}}  L \;=\; \big[  \mathbf{1}_{V_i} \boxtimes \mathcal{L}_{ \underset{i}{\oplus} \mathbf{1}_{V_i} } \big]$ is the class of its [[external tensor product]] of [[equivariant vector bundles]] with the given [[linear representation]].

=--

+-- {: .num_cor #EquivariantComplexOrientationOfEquivariantComplexKTheory}
###### Corollary
**([[equivariant complex oriented cohomology|equivariant complex orientation]] of [[equivariant K-theory]])**

For $G$ an [[abelian group|abelian]] [[compact Lie group]] and $\mathbf{1}_V \,\in\, G Representations_{\mathbb{C}}^{fin}$ a complex 1-[[dimension|dimensional]] [[linear representation]], the corresponding [[representation sphere]] is the [[projective G-space]] $S^{\mathbf{1}_V} \,\simeq\, P\big( \mathbf{1}_V \oplus \mathbf{1} \big)$ ([this Prop.](representation+sphere#OneDimensionalRepresentationSpheresAsProjectiveLine)) and so, by Prop. \ref{EquivariantKTheoryOfProjectiveGSpace},

$$
  \begin{aligned}
    \widetilde K_G
    \big(
      S^{\mathbf{1}_V}
    \big)
    & \simeq\,
    K_G
    \big(
      P( \mathbf{1}_V \oplus \mathbf{1} )
      ;
      \,
      \underset{
        \simeq \, pt
      }{
      \underbrace{
        P( \mathbf{1} )
      }
      }
    \big)
    \\
    & \simeq
    ker
    \Big(
       R(G)\big[L\big]
       \big/
       (1 - 1_{{}_{V}} L)
       (1 - L)
      \longrightarrow
      \underset{
        \simeq \, R(G)
      }{
       \underbrace{
         R(G)\big[L\big]
         \big/
         (1 - L)
       }
       }
    \Big)
    \\
    & \simeq
     (1 - L)
     \cdot
       R(G)\big[L\big]
       \big/
       (1 - 1_{{}_{V}} L)
       (1 - L)
  \end{aligned}
$$

is generated by the [[Bott element]] $(1 - L)$ over $P\big( \mathbf{1}_V \oplus \mathbf{1} \big)$. By the nature of the [[tautological equivariant line bundle]], this Bott element is the restriction of that on [[infinite complex projective G-space]] $P\big(\mathcal{U}_G\big)$. The latter is thereby exhibited as an [[equivariant complex oriented cohomology theory|equivariant complex orientation]] in [[equivariant complex K-theory]].

=--

([Greenlees 01, p. 248 (24 of 39)](#Greenlees01))

### Equivariant complex cobordism
  {#ExampleEquivariantComplexCobordism}

For an [[abelian group|abelian]] [[compact Lie group]] $G$,
[[equivariant complex cobordism theory]] $MU_G$ is an [[equivariant complex oriented cohomology theory]] ([Greenlees 01, Sec. 13](#Greenlees01)).

Much as in the non-equivariant case (see at _[[universal complex orientation on MU]]_), $MU_G$ is universal in that there is a [[bijection]] between equivariant complex orientations (in degree 2) on some cohomology theory $E_G$ and homotopy ring homomorphisms of $G$-spectra $MU_G \to E_G$ ([Cole-Greenlees-Kriz 02, Theorem 1.2](#ColeGreenleesKriz02)).

For the analogous statement on the equivariant [[Lazard ring]] see [Greenlees 01a](equivariant+complex+cobordism+cohomology+theory#Greenlees01a), [Greenlees 01, Theorem 13.1](#Greenlees01), [Cole-Greenlees-Kriz 02, Theorem 1.3](#ColeGreenleesKriz02).

## Related concepts

* [[equivariant formal group]]

* [[projective G-space]]

* [[tautological equivariant line bundle]]

* [[real oriented cohomology theory]], [[quaternionic oriented cohomology theory]]

## References

* {#Greenlees01} [[John Greenlees]], _Equivariant formal group laws and complex oriented cohomology theories_, Homology Homotopy Appl. Volume 3, Number 2 (2001), 225-263 ([euclid:hha/1139840255](https://projecteuclid.org/euclid.hha/1139840255))

* [[Michael Cole]], [[John Greenlees]], [[Igor Kriz]], _Equivariant Formal Group Laws_, Proceedings of the LMS, Volume 81, Issue 2 2000 ([doi:10.1112/S0024611500012466]( https://doi.org/10.1112/S0024611500012466))

* {#ColeGreenleesKriz02} [[Michael Cole]], [[John Greenlees]], [[Igor Kriz]], _The universality of equivariant complex bordism_, Math Z 239, 455–475 (2002) ([doi:10.1007/s002090100315](https://doi.org/10.1007/s002090100315))

See also:

* [[Steven Costenoble]], [[Peter May]], [[Stefan Waner]], _Equivariant orientation theory_, Homology Homotopy Appl.
Volume 3, Number 2 (2001), 265-339 ([euclid:hha/1139840256](https://projecteuclid.org/euclid.hha/1139840256))

[[!redirects equivariant complex oriented cohomology theories]]

[[!redirects equivariant complex-oriented cohomology theory]]
[[!redirects equivariant complex-oriented cohomology theories]]

[[!redirects equivariant complex oriented cohomology]]
[[!redirects equivariant complex-oriented cohomology]]


equivariant connection

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _equivariant connection_ is a [[connection on a bundle]] $\nabla$ over a [[space]] $X$ with [[action]] by a group $H$ which is equipped with $H$-[[equivariant structure]], hence equivalently -- in the language of [[higher differential geometry]] of [[smooth groupoids]] -- an extension of a connection $\nabla \;\colon\; X \longrightarrow \mathbf{B}G_{conn}$ to a connection $\nabla_{equ}$ on the [[action groupoid]] $X//H$:

$$
  \array{
    X &\stackrel{\nabla}{\longrightarrow}& \mathbf{B}G_{conn}
    \\
    \downarrow & \nearrow_{\mathrlap{\nabla_{equ}}}
    \\
    X//H
  }
  \,.
$$

## Examples

* [[reduced phase space]]

## Related concepts

* [[equivariant bundle]]

* [[equivariant cohomology]]

[[!redirects equivariant connections]]


equivariant connective topological K-theory

#Contents#
* table of contents
{:toc}

## Idea

The [[equivariant stable homotopy theory|equivariant]] version of [[connective]] [[topological K-theory]] $ko$ and $ku$.

## Related concepts

* [[equivariant K-theory]]

[[!include Segal completion -- table]]

## References

* [[John Greenlees]], _Equivariant version of real and complex connective K-theory_, Homology Homotopy Appl. Volume 7, Number 3 (2005), 63-82. ([Euclid](http://projecteuclid.org/euclid.hha/1139839291))


equivariant de Rham cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[equivariant cohomology]]-generalization of [[de Rham cohomology]].

| [[cohomology]] | [[Borel equivariant cohomology|Borel]]-[[equivariant cohomology]] |
|---|---|
| [[real cohomology|real]] [[ordinary cohomology]] | real [[equivariant ordinary cohomology]] |
| [[de Rham cohomology]] | [[equivariant de Rham cohomology]]

## Properties

Throughout we consider the following setup:

+-- {: .num_defn #SmoothGManifold}
###### Definition
**([[smooth manifold]] with [[smooth function|smooth]] [[action]] of a [[Lie group]])**

Let

1. $X$ be a [[smooth manifold]] with [[de Rham algebra]] denoted $\big( \Omega^\bullet(X), d_{dR} \big)$,

1. $G$ a [[Lie group]] with [[Lie algebra]] denoted $\big(\mathfrak{g}, [-,-]\big)$,

1. $X \times G \overset{\rho}{\longrightarrow} X$ a [[smooth function|smooth]] [[action]] of $G$ on $X$.

Often this data is called a _smooth $G$-manifold $X$_, or similar.

=--

### Models
 {#Models}

Given a smooth $G$-manifold $X$ (Def. \ref{SmoothGManifold}) various [[dg-algebras]] are used to model the corresponding $G$-equivariant de Rham cohomology $X$, known as

* _[the Weil model](#TheWeilModel)_

* _[the Cartan model](#TheCartanModel)_

* _the BRST/Kalkman model_

#### The Weil model
 {#TheWeilModel}

Let $\big( W(\mathfrak{g}), d_W \big)$ denote the [[Weil algebra]] of $\mathfrak{g}$. If $\{t_a\}$ is a [[linear basis]] for $\mathfrak{g}$

\[
  \label{LinearBasisForG}
  \mathfrak{g}
  \;=\;
  span\big(  \{t_a\} \big)
\]

with induced structure constants for the [[Lie bracket]] being $\big\{f_{b c}^a\big\}_{a,b,c}$

\[
  \label{StructureConstants}
  [t_a, t_b] \;=\; f_{a b}^c t^b \wedge t^c
\]

(using the [[Einstein summation convention]] throughout)

then the [[Weil algebra]] is the [[dgc-algebra]] explicitly given by [[generators and relations]] as follows:

\[
  \label{ExplicitWeilAlgebra}
  W(\mathfrak{g})
  \;\coloneqq\;
  \mathbb{R}\big[
    \{
      \underset
        { deg = 1 }
        {
          \underbrace{ t^a }
        }
      \}_a,
    \{
      \underset
        { deg = 2 }
        {
          \underbrace{ r^a }
        }
      \}_a
  \big]
  \Big/
  \left(
    \begin{aligned}
      d_W \, t^a & =  - \tfrac{1}{2}f^a_{b c} t^b \wedge t^c + r^a
      \\
      d_W \, r^a & = f^a_{b c} t^b \wedge r^c
    \end{aligned}
  \right)
\]

Now consider the [[tensor product of algebras|tensor product of]] [[dgc-algebras]] of the [[de Rham algebra]] of $X$ with the [[Weil algebra]] of $\mathfrak{g}$

\[
  \label{TensorProductOfdeRhamAlgebraWithWeilAlgebra}
  \Big(
    \Omega^\bullet
    \big(
      X
    \big)
    \otimes
    W(\mathfrak{g}),
    \,
    d_{dR} + d_W
  \Big)
  \,.
\]

On this consider the following joint [[Cartan calculus]] operations: for each basis element $t_a$ (eq:LinearBasisForG) a graded [[derivation]] of degree -1 (contraction)

\[ \label{WeilModelContractionOperation}
  \iota_a
  \;\colon\;
  \Big(
    \Omega
    \big(
      X
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)^{\bullet}
  \longrightarrow
  \Big(
    \Omega
    \big(
      X
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)^{\bullet - 1 }
\]

and a graded derivation of degree 0 (generalized [[Lie derivative]])

\[
  \label{WeilModelLieDerivative}
  \mathcal{L}_a
  \;\colon\;
  \Big(
    \Omega
    \big(
      X
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)^\bullet
  \longrightarrow
  \Big(
    \Omega
    \big(
      X
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)^{\bullet}
\]

defined on $\omega \in \Omega^\bullet(X)$ any [[differential form]] and $t^a, r^a$ as in (eq:ExplicitWeilAlgebra) as follows

\[
  \label{WeilModelContractionOnGenerators}
  \iota_a \;\colon\;
  \left\{
    \begin{aligned}
      \omega & \mapsto \iota_{v^a} \omega
      \\
      t^b &\mapsto \delta^a_b
      \\
      r^b & \mapsto 0
    \end{aligned}
  \right.
\]

and

\[
  \label{WeilModelLieDerivativeOnGenerators}
  \mathcal{L}_a
  \;\colon\;
  \left\{
    \begin{aligned}
      \omega & \mapsto \mathcal{L}_{v^a} \omega
      \\
      t^b &\mapsto f_{a c}^b t^c
      \\
      r^b & \mapsto f_{a c}^b r^c
    \end{aligned}
  \right.
\]

where

$$
  v^a
  \;\colon\;
  X
  \overset{
    \big( (e,t_a), 0 \big)
  }{\hookrightarrow}
  T G \times T X
  \simeq
  T ( G \times X )
  \overset{ d \rho } {\longrightarrow}
  T X
$$

is the [[vector field]] on $X$ which is the [[derivative]] of the [[action]] $\rho$ of $G$ along the [[Lie algebra]]-element $t_a \in \mathfrak{g} \simeq T_e G$,

and where $\iota_{v^a}$ is ordinary contraction of [[vector fields]] into [[differential forms]] and $\mathcal{L}_{v^a} = [d_{dR}, \iota-{v^a}]$ is [[Lie derivative]] of differential forms.

With this one defines the sub-[[chain complex]] of _[[horizontal differential forms]]_ as the joint [[kernel]] of the contraction operators (eq:WeilModelContractionOperation)

\[
  \label{WeilModelHorizontalForms}
  \Big(
    \Omega^\bullet
    \big(
      X
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)_{hor}
  \overset{
    ker\big( \{\iota_a\}_a \big)
  }{\hookrightarrow}
  \Big(
    \Omega^\bullet
    \big(
      X
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)
\]

(this subspace need not be preserved by the differential, but the following further subspace is)

and the further sub-[[dgc-algebra]] of _[[basic differential forms]]_, which are in addition in the [[kernel]] of the [[Lie derivatives]] (eq:WeilModelLieDerivative)

\[
  \label{WeilModelBasicForms}
  \Big(
    \Omega^\bullet
    \big(
      X
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)_{basic}
  \overset{
    ker\big( \{\mathcal{L}_a\}_a \big)
  }{\hookrightarrow}
  \Big(
    \Omega^\bullet
    \big(
      X
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)_{hor}
  \overset{
    ker\big( \{\iota_a\}_a \big)
  }{\hookrightarrow}
  \Big(
    \Omega^\bullet
    \big(
      X
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)
\]

Since the [[differential]] $d_{dR} + d_W$ (eq:TensorProductOfdeRhamAlgebraWithWeilAlgebra) graded-commutes with $\iota_a$ to $\mathcal{L}_a$ (by definition and by [[Cartan's magic formula]]) and hence graded-commutes with the [[Lie derivative]] $\mathcal{L}_a$ itself, it restricts to this joint [[kernel]], thus defining a sub-[[dgc-algebra]] (just no longer [[semi-free dg-algebra|semi-free]], in generaL)

\[
  \label{WeilModel}
  \Big(
  \Big(
    \Omega^\bullet
    \big(
      X
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)_{basic}
  ,
  d_{dR} + d_W
  \Big)
\]

This [[dgc-algebra]] is called the _Weil model_ for $G$-equivariant de Rham cohomology of $X$.

(see [Atiyah-Bott 84](#AtiyahBott84), [Mathai-Quillen 86, Sec. 5](#MathaiQuillen86) [Kalkman 93, Sec.2.1](#Kalkman93), [Miettinen 96, Sec. 2](#Miettinen96)).

#### The Cartan model
 {#TheCartanModel}

The Cartan model follows from the Weil model [above](#TheWeilModel) by algebraically solving the horizontality constraint (eq:WeilModelHorizontalForms). This we discuss first [below](#LabelCartanViaHorizontalProjection). Then we describe the resulting dgc-algebra further [below](#CartanModelDirectDefinition).

Reviews include ([Mathai-Quillen 86, Sec. 5](#MathaiQuillen86), [Kalkman 93, section 2.2](#Kalkman93))

##### Via horizontal projection of the Weil model
  {#LabelCartanViaHorizontalProjection}

The _Cartan model_ arises form the Weil model [above](#TheWeilModel) by the observation that the first of the two constraints defining [[basic differential forms]] (eq:WeilModelBasicForms), namely the constrain for [[horizontal differential forms]] (eq:WeilModelHorizontalForms), may be uniformly solved:

+-- {: .num_lemma #ProjectionOntoHorizontalDifferentialForms}
###### Lemma
**([[projection operator]] onto [[horizontal differential forms]])**

Consider the [[normal ordering|normal ordered]] [[exponential]] of minus the sum of the contraction [[derivations]] (eq:WeilModelContractionOperation) followed by wedge product with the corresponding degree-1 generator (eq:ExplicitWeilAlgebra)

\[
  \label{ExponentialProjectionOperator}
  :
    \exp
    \big(
      - \theta^a \iota_a
    \big)
  :
  \;=\;
  1
  - \underset{a}{\sum} \theta^a \iota_a
  + \tfrac{1}{2} \underset{a,b}{\sum} \theta^a \theta^b \iota_b \iota_a
  - \cdots
  \;\;\colon\;\;
  \Omega^\bullet
  \big(
    X
  \big)
  \otimes
  W(\mathfrak{g})
  \longrightarrow
  \Omega^\bullet
  \big(
    X
  \big)
  \otimes
  W(\mathfrak{g})
\]

We have:

1. This is the [[projection operator]] onto the sub-space of [[horizontal differential forms]] (eq:WeilModelHorizontalForms).

1. The restriction of this projector to $\Omega^\bullet\big(X \big)$ is a [[graded algebra]]-[[isomorphism]] onto the horizontal forms in $CE(\mathfrak{g}) \otimes \Omega^\bullet\big(X \big)$

   $$
     \Omega^\bullet\big(X \big)
     \underoverset{\simeq}{ :\exp\big( - \theta^a \iota_a \big):  }{\longrightarrow}
     \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{hor}
   $$

1. Hence the further [[tensor product]] with $\mathbb{R}\big[ \{r^a\}_a \big]$ is an algebra isomorphism onto the full subspace of [[horizontal differential forms]] (eq:WeilModelHorizontalForms)

   $$
     \mathbb{R}\big[ \{r^a\}_a \big]
     \otimes
     \Omega^\bullet\big(X \big)
     \underoverset{\simeq}{
       :\exp\big( - \theta^a \iota_a \big):
     }{
       \longrightarrow
     }
     \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{hor}
   $$

1. The operator commutes with the [[Lie derivative]] (eq:WeilModelLieDerivative) and hence restricts to an isomorphism onto the sub-[[dgc-algebra]] of [[basic differential forms]] (eq:WeilModelBasicForms)

   \[
     \label{IsomorphisamCartanToWeilModel}
     \big(
       \mathbb{R}\big[ \{r^a\}_a \big]
       \otimes
       \Omega^\bullet\big(X \big)
     \big)^G
     \underoverset{\simeq}{
       \;\;\;
       :\exp\big( - \theta^a \iota_a \big):
       \;\;\;
     }{
       \longrightarrow
     }
     \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{bas}
   \]

1. The [[inverse function|inverse]] of (eq:IsomorphisamCartanToWeilModel)

   \[
     \label{IsomorphismWeilToCartanModel}
     \big(
       \mathbb{R}\big[ \{r^a\}_a \big]
       \otimes
       \Omega^\bullet\big(X \big)
     \big)^G
     \underoverset{\simeq}{
       \;\;\;
       \epsilon
       \;\;\;
     }{
       \longleftarrow
     }
     \big( CE(\mathfrak{g}) \otimes \Omega^\bullet(X) \big)_{bas}
   \]

   is the [[algebra homomorphism]] given setting all generaotors $\theta^a$ in (eq:ExplicitWeilAlgebra)  to zero

   \[
     \label{WeilAugmentationOverCartan}
     \epsilon \;\colon\;
     \left\{
       \array{
          \omega & \mapsto \omega
          \\
          \theta^a & \mapsto 0
          \\
          r^a & \mapsto r^a
       }
     \right.
   \]

1. The induced [[differential]] on the left, which hence makes $: \exp\big( - \theta^a \iota_a\big) :$ a [[dgc-algebra]]-[[isomorphism]] and hence in particular a [[quasi-isomorphism]] is

   \[
     \label{InducedDifferentialOnCartanModel}
     \epsilon
     \circ
     \big( d_{dR} + d_W\big)
     \circ
     : \exp\big(  -\theta^a \iota_a \big) :
     \;=\;
     d_{dR}
     +
     r^a \iota_{v^a}
     \,.
   \]

=--

This is the _Mathai-Quillen isomorphism_ ([Mathai-Quillen 86, around (5.9)](#MathaiQuillen86)).

+-- {: .proof}
###### Proof

Observe that the operator (eq:ExponentialProjectionOperator) is equal to the product

$$
  :
    \exp
    \big(
      - \theta^a \iota_a
    \big)
  :
  \;=\;
  \big(
    id - \theta^1 \iota_1
  \big)
  \big(
    id - \theta^2 \iota_2
  \big)
  \cdots
  \big(
    id - \theta^{dim(\mathfrak{g})} \iota_{dim(\mathfrak{g})}
  \big)
  \,.
$$

Here all factors commute with each other, and each factor is itself a [[projection operator]], with [[image]] the [[kernel]] of the corresponding single contraction operator, e.g.

$$
  im
  \big(
    1 - \theta^1 \iota_1
  \big)
  \;\simeq\;
  ker\big( \iota_1\big)
$$

etc.

Hence the joint image is the joint kernel of the contraction operators.

It is clear by inspection that $\epsilon$ in (eq:IsomorphismWeilToCartanModel) is a [[linear map|linear]] [[inverse function|inverse]] to $: \exp\big( - \theta^a \iota_a\big) :$. Therefore, since $\epsilon$ is manifestly an [[algebra homomorphism]], so is $: \exp\big( - \theta^a \iota_a\big) :$.

This implies that the induced differential (eq:InducedDifferentialOnCartanModel) is a graded [[derivation]] and hence that it may be identified by its action on generators. Direct inspection indeed yields

for all generators $r^a$

$$
  \begin{aligned}
     \epsilon
     \circ
     \big( d_{dR} + d_W\big)
     \circ
     : \exp\big(  -\theta^a \iota_a \big) :
     \big(
       r^a
     \big)
     & = 0
     \\
     & =
     \big( d_{dR} + r^a \iota_{v^a} \big) ( r^a )
  \end{aligned}
$$

and for all differential forms $\omega \in \Omega^\bullet\big( X \big)$:

$$
  \begin{aligned}
     \epsilon
     \circ
     \big( d_{dR} + d_W\big)
     \circ
     : \exp\big(  -\theta^a \iota_a \big) :
     \big( \omega \big)
     & =
     \epsilon
     \circ
     \big( d_{dR} + d_W\big)
     \big(
       \omega
       -
       \theta^a \iota_{v^a} \omega
       +
       \cdots
     \big)
     \\
     & =
     \epsilon
     \circ
     \big(
       d_{dR} \omega
       +
       \underset{a}{\sum}
       \theta^a
       d_{dR} \iota_{v^a} \omega
       +
       \big( r^a - \tfrac{1}{2}f^a_{b c} t^b \wedge t^c \big)
       \iota_{v^a} \omega
       +
       \cdots
     \big)
     \\
     & =
       d_{dR} \iota_{v^a} ( \omega)
       +
       r^a \iota_{v^a} (\omega)
  \end{aligned}
$$

because $\epsilon$ annihilates, by (eq:WeilAugmentationOverCartan), all summands containing a $\theta^a$-factor.

=--

The left hand side [[graded algebra]] of the isomorphism (eq:IsomorphisamCartanToWeilModel) equipped with the induced differential (eq:InducedDifferentialOnCartanModel) is called the _Cartan model_, and that isomorphism exhibits it as equivalent to the Weil model:

\[
  \label{QuasiIsoFromCartanToWeilModel}
  \array{
  \Big(
     \Big(
       \Omega^\bullet\big(X \big)
       \otimes
       \mathbb{R}\big[ \{r^a\}_a \big]
     \Big)^G
     \,,\,
     d_{dR} + r^a \iota_{v^a}
  \Big)
  &
  \underoverset{\simeq}{
     : \exp\big( - \theta^a \iota_a \big) :
  }{\longrightarrow}
  &
  \Big(
  \Big(
    \Omega^\bullet
    \big(
      X
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)_{basic}
  ,
  d_{dR} + d_W
  \Big)
  \\
  \text{Cartan model}
  &&
  \text{Weil model}
  }
\]

This statement is originally due to [Cartan 50, Sec. 6](#Cartan50).

\linebreak

In summary, the Cartan model is explicitly the following dgc-algebra:

##### Direct definition
  {#CartanModelDirectDefinition}

Write

$$
  (\Omega^\bullet(G, \mathfrak{g}^\ast[1])^G
  \hookrightarrow
  \Omega(G, \mathfrak{g}^\ast[1])
$$

for the $G$-[[invariant differential forms]] on $G$ with [[coefficients]] in the [[linear dual]] of the Lie algebra $\mathfrak{g}$, shifted up in degree. So for $\{F^a\}$ a dual [[basis]], a general element of this space in degree $2 p + q$ is of the form

$$
   \omega
   \;=\;
   F^{a_1} \wedge \cdots F^{a_p}
     \wedge
    \omega_{a_1,\cdots ,a_q}
  \,,
$$

where $\omega_{\cdots}$ are [[differential n-form|differential q-forms]], such that for each $t_a \in \mathfrak{g}$ the [[Lie derivative]] of these forms satisfies

$$
  \mathcal{L}_{v^a} \omega_{a_1, a_2 \cdots , a_p}
  =
  f_{a a_1}{}^b \omega_{b , a_2, \cdots , a_p}
  +
  f_{a a_2}^{}^b \omega_{a_1 ,  b,  \cdots , a_p}
  +
  \cdots
  \,,
$$

where $\{f_{a b}{}^b\}$ are the structure constants of $\mathfrak{g}$ (eq:StructureConstants).

Equip this [[graded vector space]] $\Omega^\bullet(G, \mathfrak{h}^\ast[1])^G$ with a [[differential]] $d$ by

$$
  d
    \colon
  \omega
    \mapsto
  d_{dR}\omega - F^a \iota_{v^a} \omega
$$

(e.g. [Kalkman 93 (1.15)](#Kalkman93)).

The resulting [[dgc-algebra]] $(\Omega^\bullet(G,\mathfrak{g}^\ast[1])^G, d)$ is the _Cartan model_ for $G$-equivariant de Rham cohomology on $X$.

\linebreak

### Equivariant de Rham theorem
 {#EquDeRhamTheorem}

The point of the [above](#Models) dgc-algebra models is that, under suitable conditions, their [[cochain cohomology]] computes the [[real cohomology]] of the [[homotopy type]] of the [[homotopy quotient]] $X \sslash H$, which, as an actual [[topological space]], may be presented by the [[Borel construction]] $X \times_G E G$, hence the [[Borel equivariant cohomology|Borel equivariant]] [[de Rham cohomology]] of $X$.

This is the [[equivariant cohomology]]-generalization of the plain [[de Rham theorem]]:

+-- {: .num_prop #EquivariantDeRhamTheorem}
###### Proposition
**([[equivariant de Rham theorem]])**

Let

1. $G$ be a [[Lie group]] which is

   1. [[compact Lie group|compact]];

   1. [[connected topological space|connected]];

1. $X$ be a smooth $G$-manifold (Def. \ref{SmoothGManifold}).

Then the [[cochain cohomology]] of (the [[cochain complex]] underlying) the Weil model [[dgc-algebra]] (eq:WeilModel), and hence, by Lemma \ref{ProjectionOntoHorizontalDifferentialForms}, also of the Cartan model [[dgc-algebra]] (eq:QuasiIsoFromCartanToWeilModel). is [[isomorphism|isomorphic]] to the [[real cohomology]] of the [[homotopy quotient]] $X \!\sslash\! G$ of the action on (the [[topological space]] underlying) $X$ by the ([[topological group]] underlying) $G$, hence in particular of the [[Borel construction]] $X \times_G E G \simeq  X \!\sslash\! G $:

\[
  \array{
  \text{Cartan model cohomology}
  \\
  H^\bullet
  \Big(
     \Big(
       \Omega^\bullet\big(X \big)
       \otimes
       \mathbb{R}\big[ \{r^a\}_a \big]
     \Big)^G
     \,,\,
     d_{dR} + r^a \iota_{v^a}
  \Big)
  \\
  {}^{\simeq}
  \Big\downarrow
  {}^{
     H^\bullet\big( : \exp\big( - \theta^a \iota_a \big) : \big)
  }
  \\
  H^\bullet
  \Big(
  \Big(
    \Omega^\bullet
    \big(
      X
    \big)
    \otimes
    W(\mathfrak{g})
  \Big)_{basic}
  ,
  d_{dR} + d_W
  \Big)
  &\underoverset{\simeq}{\;\;\;\;\;\;\;\;\;\;\;\;\;}{\longrightarrow}&
  H^\bullet
  \big(
    X \!\!\sslash\!\! G
    \,,\,
    \mathbb{R}
  \big)
  \\
  \text{Weil model cohomology}
  &&
  \mathclap{
    \text{equivariant real cohomology}
  }
  }
\]

=--

(e.g [Meinrenken 06, Theorem 6.1](#Meinrenken06))

+-- {: .proof #ProofOfEquivariantDeRhamTheorem}
###### Proof idea

Recall that the [[product topological space]] $X \times E G$ of $X$ with the total space $E G$ of the [[universal principal bundle]], equipped with the [[diagonal action]] by the group $G$, constitutes a [[resolution]] of $X$ as a [[topological G-space]], in that the [[projection]]

$$
  X \times  E G
  \underoverset{\simeq_{whe}}{ \;\;\; pr_1 \;\;\; }{\longrightarrow}
  X
$$

is a $G$-[[equivariant function]] which is a [[weak homotopy equivalence]] (since $E G$ is a [[weakly contractible topological space]]) and the [[diagonal action|diagonal]] $G$-[[action]] on $X \times E G$ is [[free action|free]] (since the action on $E G$ is). Therefore the [[homotopy quotient]] of $X$ by $G$ is presented by the ordinary [[quotient space]] of $X \times E G$ by $G$, which is what is called the [[Borel construction]]

$$
  X \times_G E G
    \;\coloneqq\;
  \big(
    X \times E G
  \big)/G
  \;\simeq_{whe}\;
  X \sslash G
$$

The point now is that the Weil model (eq:WeilModel) for equivariant cohomology is exactly the analog of the Borel construction in terms of dgc-algebraic [[rational homotopy theory]]-type models in [[real cohomology]]:

By the ordinary [[de Rham theorem]] the image of the smooth manifold $X$ in dgc-algebra [[rational homotopy theory]] (with [[real number]]-[[coefficients]]) is given by the [[de Rham algebra]] $\Omega^\bullet(X)$, and the image of $E G$ is the [[Weil algebra]] $W(\mathfrak{g})$: The [[contractible topological space|contractability]] of $E G$ corresponds to the free propery ([here](Weil+algebra#FreeProperty)) of the Weil algebra, and the $G$-[[action]] on $E G$ corresponds to the canonical $\mathfrak{g}$-[[Cartan calculus]] on $W(\mathfrak{g})$.

Since for a [[free action]] the invariant forms are the [[basic differential forms]], this shows that/how the Weil model is the image of the [[Borel construction]] in dgc-algebraic [[rational homotopy theory]]:

\begin{xymatrix}
  &
  X \times E G \ar@(ul,ur)^G
  \ar[d]^-{ q }
  &
  \Omega^\bullet( X )
  \otimes
  W(g)
  \ar@(ur,ul)_{ g }
  \ar@{<-^{)}}[d]^-{ q^\ast }
  \\
  \;
  X // G
  \ar@{}[r]|-{
    \simeq_{\mathrm{whe}}
  }
  &
  \big(
    X \times E G
  \big)/G
  &
  \big(
    \Omega^\bullet( X )
    \otimes
    W(g)
  \big)_{\mathrm{basic}}
  \ar@{<-}[r]^-{ :\exp\big( t^a \iota_a\big): }_-{\simeq}
  &
  \big(
    \Omega^\bullet( X )\big[ r^a \big]
  \big)^G
  \\
  &
  \mbox{Borel construction}
  &
  \mbox{Weil model}
  &
  \mbox{Cartan model}
\end{xymatrix}

$\,$

=--

+-- {: .num_remark}
###### Remark

A generalization of the [[equivariant de Rham theorem]] to non-[[compact Lie group|compact]] Lie groups exists ([Getzler 94](#Getzler94)) but this uses the [[simplicial de Rham complex]] of the [[action groupoid]] $X \sslash G$ ([Bott-Shulman-Stasheff 76](#BottShulmanStasheff76)) and is thus a fair bit more complicated, computationally.

=--

### Cartan's map
  {#CartanMap}

If the [[G-space|G-manifold]] $X$ has a [[free action]], hence is the total space $X = P$ of a $G$-[[principal bundle]] $P \to B$, then the _Cartan map_ (or _[[Cartan's map]]_ or similar) is a [[quasi-isomorphism]] from the [[Cartan model]] for the [[equivariant de Rham cohomology]] of $X = P$ to the ordinary [[de Rham complex]] model for the ordinary [[de Rham cohomology]] of the base manifold $B$

$$
  \array{
    \bigg(
       \Big(
         \Omega^\bullet\big(P \big)
         \otimes
         \mathbb{R}\big[ \{r^a\}_a \big]
       \Big)^G
       \,,\,
       d_{dR} + r^a \iota_{v^a}
    \bigg)
    &
    \overset{
       \simeq_{qi}
    }{\longrightarrow}
    &
    \Big(
      \Omega^\bullet_{dR}(B),
      \,
      d_{dR}
    \Big)
    \\
    \omega \otimes \langle - \rangle
    &
    \mapsto
    &
    \omega_{hor} \otimes \langle F_\nabla\rangle
  }
$$

given by choosing an [[Ehresmann connection]] $\nabla$ on $P \to B$ and inserting its [[curvature form]] into the [[invariant polynomials]] $\langle-\rangle$ (essentially the [[Chern-Weil homomorphism]]).

([Guillemin-Sternberg 99, Chapter 5](#GuilleminSternberg99), [Albin-Melrose 09, Theorem 11.1](#AlbinMelrose09))

\linebreak

## Related concepts

* [[equivariant localization]]

* [[equivariant ordinary differential cohomology]]

* [[equivariant PL de Rham complex]], [[equivariant PL de Rham theorem]]

* [[gauged WZW model]]

* [[action Lie algebroid]], [[BRST complex]]

## References
  {#References}

The Cartan model for equivariant de Rham cohomology is originally due to

* {#Cartan50} [[Henri Cartan]], _La transgression dans un groupe de Lie et dans un espace fibré principal_, Colloque de topologie (espaces fibrés). Bruxelles, 1950

Review:

* [[Nicole Berline]], [[Ezra Getzler]], [[Michèle Vergne]], Section 7.1 of: *Heat Kernels and Dirac Operators*, Grundlehren **298**, Springer 2004 ([ISBN:9783540200628](https://www.springer.com/gp/book/9783540200628))

* {#Meinrenken03} [[Eckhard Meinrenken]], Section 5 of: *Group actions on manifolds*, Lecture Notes 2003 ([pdf](http://www.math.toronto.edu/mein/teaching/LectureNotes/action.pdf), [[Meinrenken_GroupActionsOnManifolds.pdf:file]])

* {#Meinrenken06} [[Eckhard Meinrenken]], _Equivariant cohomology and the Cartan model_, in: _Encyclopedia of Mathematical Physics_, Pages 242-250 Academic Press 2006 ([pdf](http://www.math.toronto.edu/mein/research/enc.pdf), [doi:10.1016/B0-12-512666-2/00344-8](https://doi.org/10.1016/B0-12-512666-2/00344-8))

* [[Oliver Goertsches]], [[Leopold Zoller]], _Equivariant de Rham Cohomology: Theory and Applications_, São Paulo J. Math. Sci. (2019) ([arXiv:1812.09511](https://arxiv.org/abs/1812.09511), [doi:10.1007/s40863-019-00129-4](https://doi.org/10.1007/s40863-019-00129-4))

Comprehensive textbook account:

* {#Tu20} [[Loring Tu]], *Introductory Lectures on Equivariant Cohomology*, Annals of Mathematics Studies **204**, AMS 2020 ([ISBN:9780691191744](https://press.princeton.edu/books/hardcover/9780691191744/introductory-lectures-on-equivariant-cohomology))

See also

* Camilo Arias Abad, [[Marius Crainic]], Sec. 1 of: _The Weil algebra and the Van Est isomorphism_ ([arXiv:0901.0322](https://arxiv.org/abs/0901.0322))

Early discussion of the Weil model includes

* {#AtiyahBott84} [[Michael Atiyah]], [[Raoul Bott]], _The moment map and equivariant cohomology_, Topology 23, 1 (1984) (<a href="https://doi.org/10.1016/0040-9383(84)90021-1">doi:10.1016/0040-9383(84)90021-1</a>, [pdf](https://www.math.stonybrook.edu/~mmovshev/MAT570Spring2008/BOOKS/atiyahbott_moment.pdf))

The slick proof of the equivalence between the Weil model and the the Cartan model via the _Mathai-Quillen isomorphism_ (Lemma \ref{ProjectionOntoHorizontalDifferentialForms}) is due to

* {#MathaiQuillen86} [[Varghese Mathai]], [[Daniel Quillen]], Sec. 5 of _Superconnections, Thom classes and equivariant differential forms_, Topology 25, 85 (1986)  (<a href="https://doi.org/10.1016/0040-9383(86)90007-8">doi:10.1016/0040-9383(86)90007-8</a>)

A review of the Weil model and the Cartan model and the introduction of the "BRST model" (Kalkman model) is in

* {#Kalkman93} [[Jaap Kalkman]], _BRST model applied to symplectic geometry_, Ph.D. Thesis, Utrecht, 1993 ([arXiv:hep-th/9308132 (broken)](https://arxiv.org/abs/hep-th/9308132), [cds:9308132](http://cds.cern.ch/record/568522), [pdf](http://cds.cern.ch/record/568522/files/9308132.pdf))

* [[Jaap Kalkman]], _BRST Model for Equivariant Cohomology and  Representatives for the Equivariant Thom Class_, Comm. Math. Phys. Volume 153, Number 3 (1993), 447-463. ([euclid:1104252784](http://projecteuclid.org/euclid.cmp/1104252784))

Generalization of the [[equivariant de Rham theorem]] to non-compact Lie groups is due to

* {#Getzler94} [[Ezra Getzler]], _The Equivariant Chern Character for Non-compact Lie Groups_, Advances in Mathematics Volume 109, Issue 1, November 1994, Pages 88-107 ([doi:10.1006/aima.1994.1081](https://doi.org/10.1006/aima.1994.1081))

based on the [[simplicial de Rham complex]]

* {#BottShulmanStasheff76} [[Raoul Bott]], [[Herbert Shulman]], [[Jim Stasheff]], _On the de Rham theory of certain classifying spaces_, Advances in Mathematics, Volume 20, Issue 1, April 1976, Pages 43-56 (<a href="https://doi.org/10.1016/0001-8708(76)90169-9">doi:10.1016/0001-8708(76)90169-9</a>, [pdf](https://core.ac.uk/download/pdf/82496263.pdf))

Discussion of [[equivariant de Rham cohomology]] with emphasis on [[characteristic forms]] and [[ordinary equivariant differential cohomology]]:

* [[Andreas Kübel]], [[Andreas Thom]], _Equivariant characteristic forms in the Cartan model and Borel equivariant cohomology_ ([arXiv:1508.07847](https://arxiv.org/abs/1508.07847))

* [[Andreas Kübel]], [[Andreas Thom]], _Equivariant Differential Cohomology_, Trans. Amer. Math. Soc. 370 (2018), 8237-8283  ([arXiv:1510.06392](https://arxiv.org/abs/1510.06392), [doi:10.1090/tran/7315](https://doi.org/10.1090/tran/7315))

Review with emphasis on [[equivariant localization]] formulas:

* [[Vasily Pestun]], _Review of localization in geometry_ ([arXiv:1608.02954](https://arxiv.org/abs/1608.02954)), chapter in: [[Vasily Pestun]], [[Maxim Zabzine]] (eds.) _Localization techniques in quantum field theories_,  J. Phys. A: Math. Theor. 50 440301, 2017 ([doi:10.1088/1751-8121/aa63c1](https://iopscience.iop.org/article/10.1088/1751-8121/aa63c1), [arXiv:1608.02952](https://arxiv.org/abs/1608.02952), [full pdf](http://pestun.ihes.fr/pages/LocalizationReview/LocQFT.pdf))

Discussion in relation to [[resolution of singularities]]:

* {#AlbinMelrose09} Pierre Albin, [[Richard Melrose]], _Equivariant cohomology and resolution_ ([arXiv:0907.3211](https://arxiv.org/abs/0907.3211))

See also

* Hugo Garcia-Compean, Pablo Paniagua, [[Bernardo Uribe]], _Equivariant extensions of differential forms for non-compact Lie groups_ ([arXiv:1304.3226](https://arxiv.org/abs/1304.3226))

Some related discussion for equivariant [[Riemannian geometry]] in

* [[Peter Michor]], _Basic Differential Forms for Actions of Lie Groups_, Proceedings of the American Mathematical Society Vol. 124, No. 5 (May, 1996), pp. 1633-1642 ([jstor:](ttps://www.jstor.org/stable/2161476))

Discussion in the broader context of [[equivariant ordinary differential cohomology]] is in

* Andreas Kübel, [[Andreas Thom]], _Equivariant Differential Cohomology_, Transactions of the American Mathematical Society (2018) ([doi:10.1090/tran/7315](https://doi.org/10.1090/tran/7315), [arXiv:1510.06392](https://arxiv.org/abs/1510.06392))

Discussion in the context of the [[gauged WZW model]] includes

* {#Witten92} [[Edward Witten]], appendix of _On holomorphic factorization of WZW and coset models_,  Comm. Math. Phys. Volume 144, Number 1 (1992), 189-212. ([EUCLID](http://projecteuclid.org/euclid.cmp/1104249222))

* {#FigueroaOFarrillStanciu94} [[José Figueroa-O'Farrill]], S Stanciu, _Gauged Wess-Zumino terms and Equivariant Cohomology_, Phys.Lett. B341 (1994) 153-159 ([arXiv:hep-th/9407196](http://arxiv.org/abs/hep-th/9407196))

* [[José de Azcárraga]],  J. C. Perez Bueno, _On the general structure of gauged Wess-Zumino-Witten terms_ ([arXiv:hep-th/9802192](http://arxiv.org/abs/hep-th/9802192))

Discussion in view of [[supersymmetry]]:

* {#GuilleminSternberg99} [[Victor Guillemin]], [[Shlomo Sternberg]], _Supersymmetry and equivariant de Rham theory_, Springer, (1999) ([doi:10.1007/978-3-662-03992-2](https://link.springer.com/book/10.1007/978-3-662-03992-2))

* {#Miettinen96} Mauri Miettinen, _Weil Algebras and Supersymmetry_ ([arXiv:hep-th/9612209](https://arxiv.org/abs/hep-th/9612209), [cds:317377](http://cds.cern.ch/record/317377), [spire:427720](http://inspirehep.net/record/427720))

Discussion in relation to [[equivariant K-theory]] and [[equivariant elliptic cohomology]]:

* [[Michel Duflo]], [[Michèle Vergne]], *Cohomologie équivariante et descente*, Astérisque, no. 215 (1993) ([numdam:AST_1993__215__5_0](http://www.numdam.org/book-part/AST_1993__215__5_0))

* [[Michèle Vergne]], *Bouquets revisited and equivariant elliptic cohomology*, International Journal of Mathematics 2021 ([arXiv:2005.00312](https://arxiv.org/abs/2005.00312), [doi:10.1142/S0129167X21400127](https://doi.org/10.1142/S0129167X21400127))

Generalization to equivariant-[[twisted de Rham cohomology]] (as the [[codomain]] for the [[twisted equivariant Chern character]]):

* [[Varghese Mathai]], [[Danny Stevenson]], p. 18 of: *Chern character in twisted K-theory: equivariant and holomorphic cases*, Commun. Math. Phys. **236** (2003) 161-186 ([arXiv:hep-th/0201010](https://arxiv.org/abs/hep-th/0201010), [doi:10.1007/s00220-003-0807-7](https://doi.org/10.1007/s00220-003-0807-7))

[[!redirects equivariant de Rham complex]]
[[!redirects equivariant de Rham complexes]]

[[!redirects Weil model]]
[[!redirects Weil models]]
[[!redirects Cartan model]]
[[!redirects Cartan models]]
[[!redirects Kalkman model]]
[[!redirects Kalkman models]]

[[!redirects Weil model for equivariant cohomology]]
[[!redirects Cartan model for equivariant cohomology]]
[[!redirects Kalkman model for equivariant cohomology]]

[[!redirects equivariant de Rham theorem]]


equivariant derived category

# Equivariant derived categories
* table of contents
{: toc}

## Idea

Given a [[locally compact space|locally compact]] [[topological group|group]] $G$ and a $G$-[[G-space|space]] $X$, one can consider the category of $G$-[[equivariant sheaves]] on $X$. It has been understood by [[Joseph Bernstein]] and [[Valery Lunts]] that the [[derived category]] of the [[abelian category]] of equivariant sheaves (or of any other abelian category) is not the most sensible choice, when the [[action]] is not [[free action|free]] and $G$ is not discrete. Instead one needs to use $G$-[[resolutions]], i.e. to effectively consider resolutions of the space $X$ itself while defining the derived category.

## References

* J. Bernstein, V. Lunts, Equivariant sheaves and functors, Springer Lecture Notes in Math. 1578 (1994). [MR95k:55012](http://www.ams.org/mathscinet-getitem?mr=95k:55012)

* notes by Zhiwei Yun, [pdf](http://stanford.edu/~zwyun/Bernstein-Lunts.pdf)

* [[Masaki Kashiwara]], _Equivariant derived category and representation of real semisimple Lie groups_, [pdf](http://www.kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/sd.pdf)

* Geoff Vooys, _Equivariant Functors and Sheaves_ ([arXiv:2107.01778](https://arxiv.org/abs/2110.01130))

[[!redirects equivariant derived category]]
[[!redirects equivariant derived categories]]


equivariant dgc-algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $G$ be a [[finite group]].

+-- {: .num_example #EquivariantPLDeRhamComplex}
###### Example

A $G$-equivariant dgc-algebra in the sense of [[equivariant rational homotopy theory]] is a [[functor]]

$$
  G Orbits
  \longrightarrow
  dgcAlgebras
$$

from the [[orbit category]] of $G$ to the [[category]] of [[dgc-algebras]].

=--

## Examples

+-- {: .num_example #EquivariantPLDeRhamComplex}
###### Example
**([[equivariant PL de Rham complex]])**

Let $S \in G SimplicialSets$ be a [[simplicial set]] equipped with $G$-[[action]], for instance the [[singular simplicial set]] of a [[topological G-space]].

The _equivariant PL de Rham complex_ of $S$ is the [[equivariant dgc-algebra]] given as the [[functor]] from the [[orbit category]] of $G$ to the category of [[dgc-algebras]]

$$
  \array{
   G Orbits
    &
    \overset{
      \Omega^\bullet_{PLdR}
      \big(
        Maps(-,X)^G
      \big)
    }{\longrightarrow}
    &
    dgcAlgebras
    \\
    G/H
    &\mapsto&
    \Omega^\bullet_{PLdR}
    \big(
      X^H
    \big)
  }
$$

which to a [[coset space]] $G/H$ assigns the [[PL de Rham complex]] of the $H$-[[fixed locus]] $X^H \subset X$.

=--

## Related concepts

* [[equivariant Sullivan models]]

* [[model structure on equivariant dgc-algebras]]

* [[equivariant rational homotopy theory]]

## References

* {#Triantafillou82} [[Georgia Triantafillou]], _Equivariant minimal models_, Trans. Amer. Math. Soc. vol 274 pp 509-532 (1982) ([jstor:1999119](http://www.jstor.org/stable/1999119))

* [[Marek Golasiński]], _Componentwise injective models of functors to DGAs_, Colloquium Mathematicum, Vol. 73, No. 1 (1997)  ([dml:21048](https://eudml.org/doc/210480), [[GolasinskiInjectiveModels.pdf:file]])

* [[Marek Golasiński]], _Injective models of $G$-disconnected simplicial sets_,  Annales de l'Institut Fourier, Volume 47 (1997) no. 5, p. 1491-1522 ([numdam:AIF_1997__47_5_1491_0](http://www.numdam.org/item/?id=AIF_1997__47_5_1491_0))

* {#Scull08} [[Laura Scull]], _A model category structure for equivariant algebraic models_, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 ([doi:10.1090/S0002-9947-07-04421-2](https://doi.org/10.1090/S0002-9947-07-04421-2))

[[!redirects equivariant dgc-algebras]]


equivariant differential K-theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Equivariant differential K-theory is an [[equivariant cohomology]]-enhancement of [[differential K-theory]], or conversely a [[differential cohomology]]-enhancement of [[equivariant K-theory]].

This is supposed to coincide with [[orbifold differential K-theory]] on [[global quotient orbifolds]].

## Related concepts

* [[orbifold K-theory]], [[orbifold differential K-theory]]

* [[equivariant ordinary differential cohomology]]

## References

* [[Richard Szabo]], [[Alessandro Valentino]], _Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory_, Commun. Math. Phys. 294:647-702, 2010 ([arXiv:0710.2773](https://arxiv.org/abs/0710.2773))

* [[Ulrich Bunke]], [[Thomas Schick]], _Differential orbifold K-Theory_, J. Noncommut. Geom. 7 (2013), no. 4, 1027-1104 ([arXiv:0905.4181](https://arxiv.org/abs/0905.4181))

* {#Ortiz09} Michael L. Ortiz, _Differential Equivariant K-Theory_ ([arXiv:0905.0476](https://arxiv.org/abs/0905.0476))

* {#BunkeSchick10} [[Ulrich Bunke]], [[Thomas Schick]], Section 9 of: _Differential K-theory. A survey_, Global differential geometry, Springer, Heidelberg (2012), 303-358 ([arXiv:1011.6663](https://arxiv.org/abs/1011.6663), [doi:10.1088/1126-6708/1998/12/019](http://dx.doi.org/10.1088/1126-6708/1998/12/019))

[[!redirects differential equivariant K-theory]]


equivariant differential topology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
#### Differential geometry
+--{: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The subject of _equivariant differential topology_ is the enhancement of results of [[differential topology]] from plain [[manifolds]]/[[topological spaces]] to those equipped with [[actions]] of some [[group]] ([[G-spaces]]) -- the _[[equivariance group]]_.

## Properties

### Fixed submanifolds
 {#FixedSubmanifolds}

+-- {: .num_prop #FixedLociOfSmoothProperActionsAreSubmanifolds}
###### Proposition
**([[fixed loci]] of [[smooth function|smooth]] [[proper actions]] are [[submanifolds]])**

Let $X$ be a [[smooth  manifold]], $G$ a [[Lie group]] and $\rho \;\colon\; G \times X \to X$ a _[[proper action|proper]]_ [[action]] by [[diffeomorphisms]] (e.g. any smooth action if $G$ is [[compact Lie group|compact]], by [this Prop.](proper+action#SmoothActionOfCompactLieGroupOnSmoothManifoldIsProper)).

Then the $G$-[[fixed locus]] $X^G \hookrightarrow X$ is a [[smooth manifold|smooth]] [[submanifold]].

If in addition $X$ is equipped with a [[Riemannian metric]] and $G$ acts by [[isometries]], then the [[submanifold]] $X^G$ is a [[totally geodesic submanifold]].

=--

(e.g. [Ziller 13, theorem 3.5.2](#Ziller13), see also [this MO discussion](https://math.stackexchange.com/a/1739784/58526))

+-- {: .proof}
###### Proof

Let $x \in X^G \subset X$ be any [[fixed point]]. Since this is in particular a closed invariant [[submanifold]], Prop. \ref{ExistenceOfGInvariantTubularNeighbourhoods} applies and shows that an [[open neighbourhood]] of $x$ in $X$ is $G$-equivariantly [[diffeomorphism|diffeomorphic]] to a [[linear representation]] $V \in RO(G)$. The [[fixed locus]] $V^G \subset V$ of that is hence [[diffeomorphism|diffeomorphic]] to an [[open neighbourhood]] of $x$ in $\Sigma$.

=--

+-- {: .num_remark}
###### Remark

Without the assumption of [[proper action]] in Prop. \ref{FixedLociOfSmoothProperActionsAreSubmanifolds} the conclusion generally fails. See [this MO comment](https://math.stackexchange.com/a/1739768/58526) for a counter-example.

=--

### Invariant Riemannian metrices

+-- {: .num_prop #ExistenceOfGInvariantRiemannianMetrics}
###### Proposition
**(existence of $G$-[[invariant Riemannian metrics]] on [[G-manifolds]])**

Let $X$ be a [[smooth  manifold]], $G$ a [[compact Lie group]] and $\rho \;\colon\; G \times X \to X$ a _[[proper action|proper]]_ [[action]] by [[diffeomorphisms]].

Then there exists a [[Riemannian metric]] on $X$ which is [[invariant Riemannian metric|invariant]] with respect to the $G$-[[action]], hence such that all elements of $G$ act by [[isometries]].

=--

([Bredon 72, VI Theorem 2.1](#Bredon72), see also [Ziller 13, Theorem 3.0.2](#Ziller13))

### Equivariant tubular neighbourhoods

+-- {: .num_defn #GInvariantTubularNeighbourhood}
###### Definition
**($G$-[[equivariant tubular neighbourhood]])**

Let $X$ be a [[smooth  manifold]], $G$ a [[Lie group]] and $\rho \;\colon\; G \times X \to X$ a _[[proper action|proper]]_ [[action]] by [[diffeomorphisms]].

For $\Sigma \subset X^G \subset X$ a [[closed manifold|closed]] [[smooth manifold|smooth]] [[submanifold]] inside the [[fixed locus]], a _$G$-[[equivariant tubular neighbourhood]]_ $\mathcal{N}(\Sigma \subset X)$ of $\Sigma$ in $X$ is

1. a [[smooth vector bundle]] $E \to \Sigma$ equipped with a [[fiber]]-wise [[linear map|linear]] $G$-[[action]];

1. an equivariant [[diffeomorphism]] $E \overset{}{\longrightarrow} X$ onto an [[open neighbourhood]] of $\Sigma$ in $X$ which takes the [[zero section]] identically to $\Sigma$.

=--

+-- {: .num_prop #GActionOnNormalBundleToFixedLocus}
###### Proposition/Definition
**($G$-action on [[normal bundle]] to [[fixed locus]])**

Let $X$ be a [[smooth  manifold]], $G$ a [[Lie group]] and $\rho \;\colon\; G \times X \to X$ a _[[proper action|proper]]_ [[action]] by [[diffeomorphisms]].

Then linearization of the $G$-action aroujnd the [[fixed locus]] $X^G \subset X$ equips the [[normal bundle]] $N_X\left( X^G\right)$ with [[smooth function|smooth]] and [[fiber]]-wise [[linear map|linear]]  $G$-[[action]].

=--

(e.g. [Crainic-Struchiner 13, Example 1.7](#CrainicStruchiner13))

+-- {: .num_prop #ExistenceOfGInvariantTubularNeighbourhoods}
###### Proposition
**(existence of $G$-[[equivariant tubular neighbourhoods]])**

Let $X$ be a [[smooth  manifold]], $G$ a [[Lie group]] and $\rho \;\colon\; G \times X \to X$ a _[[proper action|proper]]_ [[action]] by [[diffeomorphisms]].

If $\Sigma \overset{\iota}{\hookrightarrow} X$ is a [[closed manifold|closed]] [[smooth manifold|smooth]] [[submanifold]] inside the $G$-[[fixed locus]]

\begin{center}
  \begin{xymatrix}
    \Sigma
    \ar@{^{(}->}[rr]^-{\iota^G}
    \ar@{^{(}->}[dr]_{\iota}
    &&
    X^G
    \ar@{^{(}->}[dl]
    \\
    &
    X
  \end{xymatrix}
\end{center}

then

1. $\Sigma$ admits a $G$-[[equivariant tubular neighbourhood]] $\Sigma \subset U \subset X$ (Def. \ref{GInvariantTubularNeighbourhood});

1. any two choices of such $G$-equivariant tubular neighbourhoods are $G$-[[equivariant function|equivariantly]] [[isotopy|isotopic]];

1. there always exists an $G$-equivariant tubular neighbourhood parametrized specifically by the [[normal bundle]] $N(\Sigma \subset X)$ of $Sigma$ in $X$, equipped with its induced $G$-[[action]] from Def. \ref{GActionOnNormalBundleToFixedLocus}, and such that the $G$-equivariant [[diffeomorphism]] is given by the [[exponential map]]

   $$
     \exp_\epsilon
       \;\colon\;
     N(\Sigma \subset X)
       \overset{\simeq}{\longrightarrow}
     \mathcal{N}(\Sigma \subset X)
   $$

with respect to a $G$-invariant [[Riemannian metric]] (which exists according to Prop. \ref{ExistenceOfGInvariantRiemannianMetrics}):

=--

The existence of the $G$-equivariant tubular neighbourhoods is for instance in [Bredon 72 VI Theorem 2.2](#Bredon72), [Kankaanrinta 07, theorem 4.4](#Kankaanrinta07). The uniqueness up to equivariant isotopy is in [Kankaanrinta 07, theorem 4.4, theorem 4.6](#Kankaanrinta07). The fact that one may always use the [[normal bundle]] appears at the end of the proof of [Bredon 72 VI Theorem 2.2](#Bredon72), and as a special case of a more general statement about invariant tubular neighbourhoods in [[Lie groupoids]] it follows from [Pflaum-Posthuma-Tang 11, Theorem 4.1](#PflaumPosthumaTang11) by applying the construction there to each point in $\Sigma$ for one and the same choice of background metric. See also for instance [Pflaum-Wilkin 17, Example 2.5](#PflaumWilkin17).

## Applications

* [[equivariant Hopf degree theorem]]

* [[equivariant Pontrjagin theorem]]

## Related concepts

* [[equivariant triangulation]], [[equivariant triangulation theorem]]

* [[equivariant bundle]]

* [[equivariant good open cover]]

* [[equivariant extension theorem]]

* [[equivariant homotopy theory]]

* [[equivariant stable homotopy theory]]

* [[equivariant rational homotopy theory]]

* [[equivariant cohomology]]

* [[equivariant cobordism theory]]

## References

* {#Wasserman69} [[Arthur Wasserman]], section 3 of: _Equivariant differential topology_, Topology Vol. 8, pp. 127-150, 1969 (doi:10.1016/0040-9383(69)90005-6[pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/wasserman.pdf))

* {#Bredon72} [[Glen Bredon]], _[[Introduction to compact transformation groups]]_, Academic Press 1972 ([pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf), [ISBN:9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1))

* {#Illman72} [[Sören Illman]], *Equivariant algebraic topology*, Princeton University 1972 ([[Illman_EquivariantAlgebraicTopology1972.pdf:file]])

* [[Tammo tom Dieck]], Section I.5 of: _[[Transformation Groups]]_, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

* {#Kankaanrinta07} [[Marja Kankaanrinta]], _Equivariant collaring, tubular neighbourhood and gluing theorems for proper Lie group actions_,     Algebr. Geom. Topol. Volume 7, Number 1 (2007), 1-27 ([euclid:agt/1513796653](https://projecteuclid.org/euclid.agt/1513796653))

* {#PflaumPosthumaTang11} [[Markus Pflaum]], [[Hessel Posthuma]], X. Tang, *Geometry of orbit spaces of proper Lie groupoids*, Journal für die reine und angewandte Mathematik (Crelles Journal) 2014.694 [[arXiv:1101.0180](https://arxiv.org/abs/1101.0180), [doi:10.1515/crelle-2012-0092](https://doi.org/10.1515/crelle-2012-0092)]

* {#Ziller13} [[Wolfgang Ziller]], _Group actions_, 2013 ([pdf](https://www.math.upenn.edu/~wziller/math661/LectureNotesLee.pdf), [[ZillerGroupActions.pdf:file]])

* {#CrainicStruchiner13} Marius Crainic, Ivan Struchiner, _On the linearization theorem for proper Lie groupoids_,  Annales scientifiques de l'École Normale Supérieure, Série 4, Volume 46 (2013) no. 5, p. 723-746 ([numdam:ASENS_2013_4_46_5_723_0](http://www.numdam.org/item/ASENS_2013_4_46_5_723_0/) [doi:10.24033/asens.2200](https://doi.org/10.24033/asens.2200))

* {#PflaumWilkin17} [[Markus Pflaum]], Graeme Wilkin, _Equivariant control data and neighborhood deformation retractions_, Methods and Applications of Analysis, 2019 ([arXiv:1706.09539](https://arxiv.org/abs/1706.09539))

See also:

* {#LashofRothenberg78} [[Richard Lashof]], [[Melvin Rothenberg]], Section 1 of: _$G$-Smoothing theory_, p. 211-266 in: _Algebraic and Geometric Topology_, Proceedings of Symposia in Pure Mathematics, Vol. XXXII, Part 1, American Mathematical Society 1978 ([doi:10.1090/pspum/032.1](https://doi.org/10.1090/pspum/032.1), [[LashofRothenbergGSmoothingTheory.pdf:file]])

In relation to [[Chern-Weil theory]] and [[equivariant de Rham cohomology]]:

* {#Meinrenken03} [[Eckhard Meinrenken]], *Group actions on manifolds*, Lecture Notes 2003 ([pdf](http://www.math.toronto.edu/mein/teaching/LectureNotes/action.pdf), [[Meinrenken_GroupActionsOnManifolds.pdf:file]])

[[!redirects differential equivariant topology]]