equivariant K-theory

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#### Cohomology
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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

_Equivariant K-theory_ is the [[equivariant cohomology]] version of the [[generalized cohomology theory]] [[K-theory]].

To the extent that [[K-theory]] is given by [[equivalence classes]] of [[virtual vector bundles]] ([[topological K-theory]], [[operator K-theory]]), equivariant K-theory is given by equivalence classes of virtual [[equivariant bundles]] or generalizations to [[noncommutative topology]] thereof, as in _[[equivariant operator K-theory]]_, _[[equivariant KK-theory]]_.

## Properties

### Bott periodicity
 {#BottPeriodicity}

The [[Bott periodicity]] of plain K-theory generalizes to equivariant K-theory:

Complex equivariant K-theory is invariant under smashing with [[representation spheres]] of complex representations ([Atiyah 68, Theorem 4.3](#Atiyah68)), while real equivariant K-theory is invariant under smashing with representation spheres of real 8d reps with spin structure ([Atiyah 68, Theorem 6.1](#Atiyah68)).

Review in [Karoubi 05, Section 5](#Karoubi05).

### Complex orientation

Equivariant complex K-theory is an [[equivariant complex oriented cohomology theory]] ([Greenlees 01, Sec. 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]].

=--

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

+-- {: .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))

### Relation to operator K-theory of crossed product algebras

The _[[Green-Julg theorem]]_ identifies, under some conditions, equivariant K-theory with [[operator K-theory]] of corresponding [[crossed product algebras]].

### Relation to representation theory
 {#RelationToRepresentationTheory}

#### Equivariant $KU$ and the complex representation ring
 {#EquivariantKUAndTheComplexRepresentationRing}

The [[representation ring]] of $G$ over the [[complex numbers]] is the $G$-[[equivariant K-theory]] of the point, or equivalently by the [[Green-Julg theorem]], if $G$ is a [[compact Lie group]], the [[operator K-theory]] of the [[group algebra]] (the [[groupoid convolution algebra]] of the [[delooping]] groupoid of $G$):

\[
  \label{RepresentationRingAsEquivariantKTheoryOfThePoint}
  R_{\mathbb{C}}(G)
    \simeq
  KU^0_G(\ast)
    \simeq
  KK(\mathbb{C}, C(\mathbf{B}G))
  \,.
\]

The first [[isomorphism]] here follows immediately from the elementary definition of equivariant [[topological K-theory]], since a $G$-[[equivariant vector bundle]] over the point is manifestly just a [[linear representation]] of $G$ on a [[complex vector space]].

(e.g. [Greenlees 05, section 3](#Greenlees05), [Wilson 16, example 1.6 p. 3](#Wilson16))

#### Chern classes of linear representations
 {#ChernClassesOfLinearRepresentations}

Under the identification (eq:RepresentationRingAsEquivariantKTheoryOfThePoint) and the [[Atiyah-Segal completion]] map

$$
  R_{\mathbb{C}}(G)
  \simeq
  KU_G^0(\ast)
    \overset{ \widehat{(-)} }{\longrightarrow}
  KU(BG)

$$

one may ask for the [[Chern character]] of the K-theory class $\widehat{V} \in KU(B G)$ expressed in terms of the actual [[character]] of the [[representation]] $V$. For more see at _[[Chern class of a linear representation]]_.

There is a closed formula at least for the [[first Chern class]] ([Atiyah 61, appendix](#Atiyah61)):

For 1-dimensional representations $V$ their [[first Chern class]]  $c_1(\widehat{V}) \in H^2(B G, \mathbb{Z})$ is their image under the canonical [[isomorphism]] from 1-[[dimension|dimensional]] characters in $Hom_{Grp}(G,U(1))$ to the [[group cohomology]] $H^2_{grp}(G, \mathbb{Z})$ and further to the [[ordinary cohomology]] $H^2(B G, \mathbb{Z})$ of the [[classifying space]] $B G$:

$$
  c_1\left(\widehat{(-)}\right)
  \;\colon\;
  Hom_{Grp}(G, U(1))
  \overset{\simeq}{\longrightarrow}
  H^2_{grp}(G,\mathbb{Z})
  \overset{\simeq}{\longrightarrow}
  H^2(B G, \mathbb{Z})
  \,.
$$

More generally, for $n$-[[dimension|dimensional]] [[linear representations]] $V$ their [[first Chern class]] $c_1(\widehat V)$ is the previously defined first Chern-class of the [[line bundle]] $\widehat{\wedge^n V}$ corresponding to the  $n$-th [[exterior power]] $\wedge^n V$ of $V$. The latter is a 1-dimensional representation, corresponding to the [[determinant line bundle]] $det(\widehat{V}) = \widehat{\wedge^n V}$:

$$
  c_1(\widehat{V})
    \;=\;
  c_1(det(\widehat{V}))
    \;=\;
  c_1( \widehat{\wedge^n V} )
  \,.
$$

([Atiyah 61, appendix, item (7)](#Atiyah61))

More explicitly, via the formula for the [[determinant]] as a [[polynomial]] in [[traces]] of powers (see [there](determinant#eq:DeterminantAsPolynomialInTracesOfPowers)) this means that the first Chern class of the $n$-dimensional representation $V$ is expressed in terms of its [[character]] $\chi_V$ as

\[
  \label{FirstChernClassOfRepresentationInTermsOfTheCharacter}
  c_1(V)
  =
  \chi_{\left(\wedge^n V\right)}
  \;\colon\;
  g
  \;\mapsto\;
  \underset{
     { k_1,\cdots, k_n \in \mathbb{N} }
     \atop
     { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n }
  }{\sum}
  \underoverset{ l = 1 }{ n }{\prod}
  \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l !  }
  \left(\chi_V(g^l)\right)^{k_l}
\]

For example, for a representation of dimension $n = 2$ this reduces to

$$
  c_1(V)
  =
  \chi_{V \wedge V}
  \;\colon\;
  g
  \;\mapsto\;
  \frac{1}{2}
  \left(
    \left( \chi_V(g)\right)^2
    -
    \chi_V(g^2)
  \right)
$$

(see also e.g. [tom Dieck 09, p. 45](representation+theory#tomDieck09))

$\,$

#### Equivariant $KO$ and the real representation ring

An isomorphism analogous to (eq:RepresentationRingAsEquivariantKTheoryOfThePoint) identifies the $G$-representation ring over the [[real numbers]] with the equivariant orthogonal $K$-theory of the point in degree 0:

$$
  R_{\mathbb{R}}(G)
  \;\simeq\;
  KO_G^0(\ast)
  \,.
$$

But beware that equivariant [[KO]], even of the point, is much richer in higher degree ([Wilson 16, remark 3.34](#Wilson16)).

{#EquivariantKOOfThePoint} In fact, [[equivariant KO-theory]] of the point subsumes the [[representation rings]] over the [[real numbers]], the [[complex numbers]] and the [[quaternions]]:

$$
  KO_G^n(\ast)
  \;\simeq\;
  \left\{
  \array{
    0 &\vert& n = 7
    \\
    R_{\mathbb{C}}(G)/ R_{\mathbb{R}}(G) &\vert& n = 6
    \\
    R_{\mathbb{H}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 5
    \\
    R_{\mathbb{H}}(G) \phantom{/ R_{\mathbb{R}}(G) }  &\vert& n = 4
    \\
    0 &\vert& n = 3
    \\
    R_{\mathbb{C}}(G)/ R_{\mathbb{H}}(G) &\vert& n = 2
    \\
    R_{\mathbb{R}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 1
    \\
    R_{\mathbb{R}}(G) \phantom{/ R_{\mathbb{R}}(G)} &\vert& n =0
  }
  \right.
$$

([Greenlees 05, p. 3](#Greenlees05))

Accordingly the construction of an [[index]] ([[push-forward in generalized cohomology|push-forward]] to the point) in equivariant K-theory is a way of producing $G$-[[representations]] from [[equivariant vector bundles]]. This method is also called _[[Dirac induction]]_.

Specifically, applied to equivariant [[complex line bundles]] on [[coadjoint orbits]] of $G$, this is a K-theoretic formulation of the [[orbit method]].

### Relation to K-theory of homotopy quotient spaces (Borel constructions)

For $X$ a [[topological space]] equipped with a $G$-[[action]] for $G$ a [[topological group]], write $X//G$ for the [[homotopy type]] of the corresponding [[homotopy quotient]]. A standard model for this is the [[Borel construction]]

$$
  X//G \simeq (X \times EG)/G
  \,.
$$

The ordinary [[topological K-theory]] of $X//G$ is also called the _Borel-equivariant K-theory_ of $X$, denoted

$$
  K_G^{Bor}(X) \coloneqq K(X//G)
  \,.
$$

There is a canonical map

$$
  K_G(X) \to K_G^{Bor}(X)
$$

from the genuine equivariant K-theory to the Borel equivariant K-theory. In terms of the [[Borel construction]] this is given by the composite

$$
  K_G(X) \to K_G(X \times E G) \simeq K((X \times E G) / G ) \simeq K_G^{Bor}(X)
  \,,
$$

where the first map is pullback along the [[projection]] $X \times E G \to X$ and the first equivalence holds because the $G$-action on $X \times E G$ is free.

This map from genuine to Borel equivariant K-theory is not in general an isomorphism.

Specifically for $X$ the point, then $K_G(\ast) \simeq R(G)$ is the [[representation ring]] and $K_G^{Bor}(\ast) \simeq K(B G)$ is the [[topological K-theory]] of the [[classifying space]] $B G$ of $G$-[[principal bundles]]. In this case the above canonical map is of the form

$$
  R(G) \to K(B G)
  \,.
$$

This is never an [[isomorphism]], unless $G$ is the trivial group. But the [[Atiyah-Segal completion theorem]] says that the map identifies $K(B G)$ as the completion of $R(G)$ at the [[ideal]] of [[virtual representations]] of rank 0.

[[!include Segal completion -- table]]

### Rationalization
 {#Rationalization}

[[!include incarnations of rational equivariant topological K-theory -- table ]]

### Equivariant Chern-character

There is an [[equivariant Chern character]] map from equivariant K-theory to [[rational cohomology|rational]] [[equivariant ordinary cohomology]] [above](#Rationalization)

(e.g. [Stefanich](#Stefanich), [Sati-Schreiber 20, Sec. 3.4](#SatiSchreiber20))

## Related concepts

* [[rational equivariant K-theory]]

* [[equivariant Chern character]]

* [[Baum-Connes conjecture]], [[Green-Julg theorem]],

* [[Atiyah-Segal completion theorem]]

* [[orbifold K-theory]]

* [[twisted equivariant K-theory]]

  * [[twisted ad-equivariant K-theory]]

* [[equivariant elliptic cohomology]]

* [[equivariant algebraic K-theory]]

* [[McKay correspondence]]

* [[equivariant ordinary cohomology]]

* [[equivariant ordinary differential cohomology]]

## References
 {#References}

### General

The idea of equivariant [[topological K-theory]] and the [[Atiyah-Segal completion theorem]] goes back to

* {#Atiyah61} [[Michael Atiyah]], _Characters and cohomology of finite groups_, Publications Mathématiques de l'IHÉS, Volume 9 (1961) , p. 23-64 ([numdam:PMIHES_1961__9__23_0]( http://www.numdam.org/item?id=PMIHES_1961__9__23_0))

* [[Michael Atiyah]], [[Friedrich Hirzebruch]], _Vector bundles and homogeneous spaces_, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, 3, 7–38 ([[AtiyahHirzebruch61.pdf:file]])

* {#Segal68} [[Graeme Segal]], _Equivariant K-theory_, Inst. Hautes Etudes Sci. Publ. Math.  No. 34 (1968) p. 129-151 ([numdam:PMIHES_1968__34__129_0](http://www.numdam.org/item/PMIHES_1968__34__129_0))

* {#SegalAtiyah69} [[Michael Atiyah]], [[Graeme Segal]], _Equivariant K-theory and completion_, J. Differential Geometry 3 (1969), 1–18.
([euclid:jdg/1214428815](https://projecteuclid.org/euclid.jdg/1214428815), MR 0259946)

and for [[algebraic K-theory]] to

* [[Robert Thomason]], _Algebraic K-theory of group scheme actions_, Algebraic topology and algebraic K-theory (Princeton, N.J., 1983), Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 539–563

with construction via [[permutative categories]] in

* [[Bertrand Guillou]], [[Peter May]], _Equivariant iterated loop space theory and permutative G-categories_, _Algebr. Geom. Topol. 17 (2017) 3259-3339_ ([arXiv:1207.3459](https://arxiv.org/abs/1207.3459))

See also at _[algebraic K-theory -- References -- On quotient stacks](algebraic+K-theory#ReferencesAlgebraicKTheoryForQuotientStacks)_.

Introductions and surveys:

* {#Greenlees05} [[John Greenlees]], _Equivariant version of real and complex connective K-theory_, Homology Homotopy Appl. Volume 7, Number 3 (2005), 63-82. ([Euclid:1139839291](http://projecteuclid.org/euclid.hha/1139839291))

* N. C. Phillips, _Equivariant K-theory for proper actions_, Pitman
Research Notes in Mathematics Series 178, Longman, Harlow, UK, 1989.

* [[Bruce Blackadar]], section 11 of _[[K-Theory for Operator Algebras]]_

* Alexander Merkujev, _Equivariant K-theory_ ([pdf](http://www.math.uiuc.edu/K-theory/0981/book/2-0925-0954.pdf))

* Zachary Maddock, _An informal discourse on equivariant K-theory_ ([pdf](http://math.columbia.edu/~ellis/ass/equiv_k-thry.pdf))

* [[Akhil Mathew]], _[Equivariant K-theory](https://amathew.wordpress.com/2011/12/03/equivariant-k-theory/)_

* {#Wilson16} [[Dylan Wilson]], _Equivariant K-theory_, 2016 ([pdf](https://www.math.uchicago.edu/~dwilson/notes/equivariant-k-theory-talk.pdf), [[WilsonKTheory16.pdf:file]])

Textbook accounts:

* {#HJJS08} [[Dale Husemöller]], [[Michael Joachim]], [[Branislav Jurčo]], [[Martin  Schottenloher]], Section 14 of: _[[Basic Bundle Theory and K-Cohomology Invariants]]_, Springer Lecture Notes in Physics __726__, 2008, ([pdf](http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726.pdf), [doi:10.1007/978-3-540-74956-1](https://link.springer.com/book/10.1007/978-3-540-74956-1))

See also

* {#BrunerGreenlees10} [[Robert Bruner]], [[John Greenlees]], *Connective Real K-Theory of Finite Groups*, Mathematical Surveys and Monographs **169** AMS 2010 ([ISBN:978-0-8218-5189-0](https://bookstore.ams.org/surv-169))

* {#Cantarero09} [[Jose Cantarero]], _Equivariant K-theory, groupoids and proper actions_, Thesis 2009 ([ubctheses:1.0068026](https://open.library.ubc.ca/cIRcle/collections/ubctheses/24/items/1.0068026), [[CantareroEquivariantKTheory.pdf:file]])

* {#Cantarero12} [[Jose Cantarero]], _Equivariant K-theory, groupoids and proper actions_, Journal of K-Theory, Volume 9, Issue 3  June 2012, pp. 475 - 501  ([arXiv:0803.3244](https://arxiv.org/abs/0803.3244), [doi:10.1017/is011011005jkt173](https://doi.org/10.1017/is011011005jkt173))

  > (short version of [Cantarero 09](#Cantarero09))

On [[Bott periodicity]] in equivariant K-theory:

* {#Atiyah68} [[Michael Atiyah]], _Bott periodicity and the index of elliptic operators_, The Quarterly Journal of Mathematics, Volume 19, Issue 1, 1968, Pages 113–140 ([doi:10.1093/qmath/19.1.113](https://doi.org/10.1093/qmath/19.1.113))

* {#Karoubi05} [[Max Karoubi]], _Bott Periodicity in Topological, Algebraic and Hermitian K-Theory_,  In: Friedlander E., Grayson D. (eds) [[Handbook of K-Theory]], Springer 2005 ([doi:10.1007/978-3-540-27855-9_4](https://doi.org/10.1007/978-3-540-27855-9_4))

Basic computations:

* {#Yang95} [[Yimin Yang]], _On the Coefficient Groups of Equivariant K-Theory_, Transactions of the American Mathematical Society
Vol. 347, No. 1 (Jan., 1995), pp. 77-98 ([jstor:2154789](https://www.jstor.org/stable/2154789))

* {#Karoubi02} [[Max Karoubi]], _Equivariant K-theory of real vector spaces and real vector bundles_, Topology and its Applications, 122, (2002) 531-456 ([arXiv:math/0509497](https://arxiv.org/abs/math/0509497))

The [[equivariant Chern character]] is discussed in

* {#Stefanich} German Stefanich, _Chern Character in Twisted and Equivariant K-Theory_ ([pdf](https://math.berkeley.edu/~germans/Chern2.pdf))

* {#SatiSchreiber20} [[Hisham Sati]], [[Urs Schreiber]], Sec. 3.4 of: _[[schreiber:The Character Map in Equivariant Twistorial Cohomotopy|The character map in equivariant twistorial Cohomotopy]]_ ([arXiv:2011.06533](https://arxiv.org/abs/2011.06533))

Discussion relating to K-theory of [[homotopy quotients]]/[[Borel constructions]] is in

* [[Jacob Lurie]], around p. 11 of _[[A Survey of Elliptic Cohomology]]_ ([pdf](http://www.math.harvard.edu/~lurie/papers/survey.pdf))

Discussion of the [[twisted ad-equivariant K-theory]] of [[compact Lie groups]]:

* [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], _[[Loop Groups and Twisted K-Theory]]_.

Discussion of K-theory of [[orbifolds]] is for instance in section 3 of

* [[Alejandro Adem]], Johanna Leida, [[Yongbin Ruan]], _Orbifolds and string topology_, Cambridge Tracts in Mathematics 171, 2007 ([pdf](http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf))

Discussion of [[differential K-theory]] of [[orbifolds]]:

* [[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))

Discussion of combined [[twisted K-theory|twisted]] and [[equivariant K-theory|equivariant]] and [[real K-theory|real]] K-theory

* [[El-kaïoum M. Moutuou]], _Twistings of KR for Real groupoids_ ([arXiv:1110.6836](http://arxiv.org/abs/1110.6836))

* [[El-kaïoum M. Moutuou]], _Graded Brauer groups of a groupoid with involution_, J. Funct. Anal. 266 (2014), no.5 ([arXiv:1202.2057](https://arxiv.org/abs/1202.2057))

* {#Freed12} [[Daniel Freed]], _Lectures on twisted K-theory and orientifolds_, lectures at ESI Vienna, 2012 ([[FreedESI2012.pdf:file]])

* {#FreedMoore13} [[Daniel Freed]], [[Gregory Moore]], Section 7 of: _Twisted equivariant matter_,  Ann. Henri Poincaré (2013) 14: 1927 ([arXiv:1208.5055](https://arxiv.org/abs/1208.5055))

* {#Gomi17} [[Kiyonori Gomi]], _Freed-Moore K-theory_ ([arXiv:1705.09134](https://arxiv.org/abs/1705.09134), [spire:1601772](http://inspirehep.net/record/1601772))

Discussion in the context of [[equivariant complex oriented cohomology theory]]:

* {#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))

For formulation and proof of the [[McKay correspondence]]:

* {#GSV83} [[Gérard Gonzalez-Sprinberg]], [[Jean-Louis Verdier]], _Construction géométrique de la correspondance de McKay_, Ann. Sci. ́École Norm. Sup. **16** 3 (1983) 409–449 ([numdam:ASENS_1983_4_16_3_409_0](http://www.numdam.org/item?id=ASENS_1983_4_16_3_409_0))

### Representing equivariant spectrum
 {#ReferencesRepresentingSpectrum}

That $G$-equivariant [[topological K-theory]] is [[representable functor|represented]] by a [[topological G-space]] is due to:

* {#Matumoto71} Takao Matumoto, _Equivariant K-theory and Fredholm operators_, J. Fac. Sci. Tokyo 18 (1971/72), 109-112 ([pdf](https://repository.dl.itc.u-tokyo.ac.jp/?action=repository_action_common_download&item_id=39826&item_no=1&attribute_id=19&file_no=1), [[MatumotoEquivariantKTheory.pdf:file]])

* {#AtiyahSegal04} [[Michael Atiyah]], [[Graeme Segal]], Sec. 6 and Corollary A3.2 in:  _Twisted K-theory_, Ukrainian Math. Bull. **1**, 3 (2004) ([arXiv:math/0407054](http://arxiv.org/abs/math/0407054), [journal page](http://iamm.su/en/journals/j879/?VID=10), [published pdf](http://iamm.su/upload/iblock/45e/t1-n3-287-330.pdf))

* {#LueckOliver01} [[Wolfgang Lück]], [[Bob Oliver]], Section 1 of: _Chern characters for the equivariant K-theory of proper G-CW-complexes_, In: Aguadé J., Broto C., Casacuberta C. (eds.) _Cohomological Methods in Homotopy Theory_ Progress in Mathematics, vol 196. Birkhäuser 2001 ([doi:10.1007/978-3-0348-8312-2_15](https://doi.org/10.1007/978-3-0348-8312-2_15))

This is enhanced to a representing [[naive G-spectrum]] in:

* [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], Appendix A.5 of: _Loop groups and twisted K-theory I_, Journal of Topology, Volume 4, Issue 4, December 2011, Pages 737–798 ([arXiv:0711.1906](https://arxiv.org/abs/0711.1906))

In its incarnation (under [[Elmendorf's theorem]]) as a [[Spectra]]-valued [[presheaf]] on the $G$-[[orbit category]] this is discussed in

* [[James Davis]], [[Wolfgang Lück]], _Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory_, K-Theory 15:201–252, 1998 ([pdf](https://jfdmath.sitehost.iu.edu/teaching/m721F19/assembly.pdf))

Review:

* [[Valentin Zakharevich]], Section 2.2 of: _K-Theoretic Computation of the Verlinde Ring_, thesis 2018 ([hdl:2152/67663](http://hdl.handle.net/2152/67663), [pdf](http://www.math.jhu.edu/~vzakharevich/research/Dissertation.pdf), [[ZakharevichKTheoryAndVerlindeRing.pdf:file]])

* {#Ortiz09} Michael L. Ortiz, Theorem 2.2 in: _Differential Equivariant K-Theory_ ([arXiv:0905.0476](https://arxiv.org/abs/0905.0476))

* [Cantarero 09, Thm. 1.0.2 & Sec. 3.4](#Cantarero09)

### Rational equivariant K-theory
 {#ReferencesRationalEquivariantKTheory}

Discussion of [[rational equivariant K-theory]] (see also the references at *[[equivariant Chern character]]*):

* {#ConnesBaum89} [[Alain Connes]], [[Paul Baum]], _Chern character for discrete groups_, A Fête of Topology, Papers Dedicated to Itiro Tamura 1988, Pages 163-232 ([doi:10.1016/B978-0-12-480440-1.50015-0](https://doi.org/10.1016/B978-0-12-480440-1.50015-0))

* {#LueckOliver01} [[Wolfgang Lück]], [[Bob Oliver]], Section 1 of: _Chern characters for the equivariant K-theory of proper G-CW-complexes_, In: Aguadé J., Broto C., Casacuberta C. (eds.) _Cohomological Methods in Homotopy Theory_ Progress in Mathematics, vol 196. Birkhäuser 2001 ([doi:10.1007/978-3-0348-8312-2_15](https://doi.org/10.1007/978-3-0348-8312-2_15))

* {#MislinValette03} [[Guido Mislin]], [[Alain Valette]], Theorem 6.1 in: _Proper Group Actions and the Baum-Connes Conjecture_, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 ([doi:10.1007/978-3-0348-8089-3](https://link.springer.com/book/10.1007/978-3-0348-8089-3))

and with emphasis of [[commutative ring]]-structure:

* [[Anna Marie Bohmann]], [[Christy Hazel]], [[Jocelyne Ishak]], [[Magdalena Kędziorek]], [[Clover May]], *Naive-commutative structure on rational equivariant K-theory for abelian groups* ([arXiv:2002.01556](https://arxiv.org/abs/2002.01556))

* [[Anna Marie Bohmann]], [[Christy Hazel]], [[Jocelyne Ishak]], [[Magdalena Kędziorek]], [[Clover May]], *Genuine-commutative ring structure on rational equivariant K-theory for finite abelian groups* ([arXiv:2104.01079](https://arxiv.org/abs/2104.01079))

### For D-brane charge on orbifolds

The proposal that [[D-brane charge]] on [[orbifolds]] is given by [[equivariant K-theory]] (see at *[[D-brane charge quantization in K-theory]]*) goes back to

* {#Witten98} [[Edward Witten]], section 5.1 of _D-Branes And K-Theory_, JHEP 9812:019,1998 ([arXiv:hep-th/9810188](http://arxiv.org/abs/hep-th/9810188))

but it was pointed out that only a subgroup or quotient group of equivariant K-theory can be physically relevant, in

* {#BDHKMMS01} [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], around (137) of  _Triples, Fluxes, and Strings_, Adv.Theor.Math.Phys. 4 (2002) 995-1186 ([arXiv:hep-th/0103170](https://arxiv.org/abs/hep-th/0103170))

For further references see at _[[fractional D-brane]]_.

On [[Chern classes of linear representations]]:

* [Atiyah 61, Appendix](#Atiyah61)

* {#Evens1965} L. Evens, _On the Chern classes of representations of finite groups_, Trans. Am. Math. Soc. 115, 180-193 (1965) ([doi:10.2307/1994264](https://www.jstor.org/stable/1994264))

* F. Kamber, Ph. Tondeur, _Flat Bundles and Characteristic Classes of Group-Representations_, American Journal of Mathematics
Vol. 89, No. 4 (Oct., 1967), pp. 857-886 ([doi:10.2307/2373408](https://www.jstor.org/stable/2373408))

* {#Symonds91} [[Peter Symonds]], _A splitting principle for group representations_, Comment. Math. Helv. (1991) 66: 169 ([doi:10.1007/BF02566643](https://doi.org/10.1007/BF02566643))

[[!redirects equivariant topological K-theory]]

[[!redirects equivariant complex K-theory]]

[[!redirects equivariant KO-theory]]


equivariant K-theory of projective G-space

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#Contents#
* table of contents
{:toc}

## Idea

For $G$ an [[abelian group|abelian]] [[compact Lie group]], the [[equivariant K-theory]] [[cohomology ring|ring]] of [[projective G-spaces]] over a [[direct sum]] of complex 1-dimensinal [[linear representations]] is (eq:EquivariantKTheoryRingOfProjectiveGSpace) the [[quotient ring]] of the [[polynomial ring]] in the [[tautological equivariant line bundle]] $L$ by the [[ideals]] generated by [[virtual vector bundle|virtual differences]] $1 - 1_{{}_{V_i}} L$ between its [[external tensor product]] with each of these 1d representations and the [[trivial line bundle]]; see Prop. \ref{EquivariantKTheoryOfProjectiveGSpace} below.

This is the generalization to [[equivariant K-theory]] of the formula

\[
  \label{InOrdinaryKTheory}
  K\big(  \mathbb{C}P^n \big)
  \;\simeq\;
  \mathbb{Z}\big[ L \big]
  \big/
  \big(
    1 - L
  \big)^{n+1}
\]

(from the [[fundamental product theorem in topological K-theory]]) for the [[complex topological K-theory]] [[cohomology ring|ring]] of [[complex projective space]], where $L$ is the class of the [[tautological line bundle]] and $1- L$ the "[[Bott element]]".

In generalization of how (eq:InOrdinaryKTheory) exhibits [[complex oriented cohomology theory|complex orientation]] in [[topological complex K-theory]], so the equivariant version (eq:EquivariantKTheoryRingOfProjectiveGSpace) exhibits [[equivariant complex oriented cohomology|equivariant complex orientation]] of [[equivariant complex K-theory]].

## Statement

+-- {: .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]].

=--

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

## Applications

### Complex orientation of equivariant complex K-theory

+-- {: .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|complex orientation]] in [[equivariant complex K-theory]].

=--

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

## Related concepts

* [[fundamental product theorem in topological K-theory]]

* [[equivariant complex oriented cohomology theory]]

## References

* {#Greenlees01} [[John Greenlees]], Section 10 of: _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))


equivariant KK-theory

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#### Operator algebra
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#### Noncommutative geometry
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#Contents#
* table of contents
{:toc}

## Idea

The [[equivariant cohomology]] version of [[KK-theory]].

## Definition

Let $G$ be a [[locally compact topological space|locally compact]] [[topological group]].

+-- {: .num_defn }
###### Definition

$KK_G$ is the [[category]]...

=--

([Blackadar, section 20.2](#Blackadar)) The [[composition]] is in ([Blackadar, theorem 20.3.1](#Blackadar)) .

+-- {: .num_defn #KKTheoreticRepresentationRing}
###### Definition

The KK-theoretic [[representation ring]] of $G$ is the [[ring]]

$$
  R(G) \simeq KK_G(\mathbb{C}, \mathbb{C})
  \,.
$$

=--

([Blackadar def. 20.4.2](#Blackadar))

## Properties

The [[Green-Julg theorem]]:

+-- {: .num_theorem #GreenJulgTheorem}
###### Theorem
**(Green-Julg theorem)**

Let $G$ be a [[topological group]] acting on a [[C*-algebra]] $A$.

1. If $G$ is a [[compact topological group]] then the descent map

   $$
     KK^G(\mathbb{C}, A) \to KK(\mathbb{C},G\ltimes A)
   $$

   is an [[isomorphism]], identifying the [[equivariant operator K-theory]] of $A$ with the ordinary [[operator K-theory]] of the [[crossed product C*-algebra]] $G \ltimes A$.

1. if $G$ is a [[discrete group]] then the descent map

   $$
     KK^G(A, \mathbb{C}) \to KK(G \ltimes A, \mathbb{C})
   $$

   is an [[isomorphism]], identifying the equivariant [[K-homology]] of $A$ with the ordinary [[K-homology]] of the [[crossed product C*-algebra]] $G \ltimes A$.

=--

([Blackadar, 20.2.7](#Blackadar)).

+-- {: .num_prop }
###### Proposition

If $G$ is a [[compact topological group]], then the KK-theoretic representation ring, def. \ref{KKTheoreticRepresentationRing} coincides with the ordinary [[representation ring]] of $G$.

=--

([Blackadar, prop. 20.4.4](#Blackadar))

## Related concepts

* [[Baum-Connes conjecture]]

## References

Section 20 of

* [[Bruce Blackadar]], _[[K-Theory for Operator Algebras]]_
 {#Blackadar}


equivariant localization

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## Idea

Localization in [[equivariant de Rham cohomology]].

## Related concepts

* [[equivariant cohomology]]

* [[equivariant localization and elimination of nodes]],

* [[moment map]], [[heat kernel]], [[semiclassical approximation]]

## References

Original articles:

* J. J. Duistermaat, G. J. Heckman, _On the variation in the cohomology in the symplectic form of the reduced phase space_, Invent. Math. __69__ (1982) 259-268.

* [[Nicole Berline]], [[Michèle Vergne]], _Classes caractéristiques équivariantes. Formule de localisation en cohomologie équivariante_, C. R. Acad. Sci. Paris Sér. I Math.  295  (1982), no. 9, 539--541, [MR83m:58002](http://www.ams.org/mathscinet-getitem?mr=83m:58002)

* [[Michael Atiyah]], [[Raoul Bott]], _The moment map and equivariant cohomology_, Topology __23__, 1, 1-28 (1984) (doi:10.1016/0040-9383(84)90021-1)

* [[Victor Guillemin]], [[Shlomo Sternberg]], _Supersymmetry and equivariant de Rham theory_, Springer, 1999.

* [[Victor Guillemin]], [[Shlomo Sternberg]], _Geometric asymptotics_, AMS 1977, [online](http://www.ams.org/online_bks/surv14)

Review:

* [[Richard Szabo]], _Equivariant cohomology and localization of path integrals_, Lecture Notes in Physics, N.S. Monographs __63__. Springer  2000. xii+315 pp. (preprint version: _Equivariant localization of path integrals_, [hep-th/9608068](http://arxiv.org/abs/hep-th/9608068))

* [[Loring Tu]], _Localization in equivariant cohomology_, from biography web on [[Raoul Bott]], [node21](http://www.math.harvard.edu/history/bott/bottbio/node21.html)

* [[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))

Textbook account:

* [[Loring Tu]], Part V in: *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:

* Wikipedia, _[Localization formula for equivariant cohomology](https://en.m.wikipedia.org/wiki/Localization_formula_for_equivariant_cohomology)

[[!redirects localization in equivariant cohomology]]

[[!redirects Atiyah-Bott localization theorem]]


equivariant localization and elimination of nodes

This is one rambling paragraph previously at [[symplectic geometry]]. We should also have [[equivariant localization]] per se. Hopefully this entry will be cleaned up later. It obviously has its origin from the informal discussion [here](http://golem.ph.utexas.edu/category/2009/10/structural_foundations_of_quan.html#c028550)

**Jacobi's elimination of nodes and Witten's "Two dimensional gauge theories revisited."**

Conservation laws arising from symmetries have been formalized as moment maps by Kirillov, Kostant and Souriau in late 1960s. Elimination of nodes procedure has been made rigorous by Marsden and Weinstein and, independently, by Meyer, as [[symplectic reduction]] (symplectic quotient construction).

In the early 1980s Mumford observed that symplectic quotients are closely related to [[geometric invariant theory]] quotients and that many [[moduli space]]s
important in [[algebraic geometry]] and in mathematical physics can be realized as [[symplectic quotient]]s.  Atiyah and Bott used this point of view in "The moment map and equivariant cohomology" to construct
[[cohomology]] classes of [[moduli space]]s of flat [[connection on a bundle|connection]]s on [[Riemann
surface]]s.  In "Two dimensional gauge theories revisited" [[Edward Witten|Witten]] conjectured a method for computing the [[intersection pairing]]s of cohomology classes of symplectic quotients.  The work of Atiyah and Bott and Witten's conjecture stimulated a large research effort to understand the topology of symplectic quotients in terms of the [[equivariant cohomology]] of the original spaces. Witten's conjecture was
proved by Jeffrey and Kirwan several years later.


equivariant MFr

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## Idea

The [[equivariant bordism homology theory|equivariant]] version of [[framed bordism theory]], [[Brown representability theorem|represented]] by the [[equivariant sphere spectrum]], dually given by [[equivariant stable Cohomotopy]].

## Related concepts

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

## References

Proof that [[equivariant framed bordism homology theory]] is [[Brown representability theorem|co-represented]] by the [[equivariant sphere spectrum]]:

* Czes Kosniowski, _Equivariant Stable Homotopy and Framed Bordism_, Transactions of the American Mathematical Society
Vol. 219 (1976), pp. 225-234 ([jstor:1997591](https://www.jstor.org/stable/1997591))

[[!redirects equivariant framed bordism homology theory]]
[[!redirects equivariant framed bordism cohomology theory]]


equivariant Morava K-theory

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=--
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#Contents#
* table of contents
{:toc}

## Idea

The lift of [[Morava K-theory]] to [[equivariant stable homotopy theory]].

## Related concepts

* [[equivariant complex cobordism cohomology theory]]

## 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))


equivariant motivic homotopy theory

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#Contents#
* table of contents
{:toc}

## Idea

The combination of [[motivic homotopy theory]] with [[equivariant homotopy theory]], particularly with [[equivariant stable homotopy theory]].

## Related concepts

* [[equivariant homotopy theory]]

* [[stable equivariant homotopy theory]]

## References

* {#CJ2014} [[Gunnar Carlsson]], [[Roy Joshua]], _Equivariant motivic homotopy theory_,([arXiv:1404.1597](http://arxiv.org/abs/1404.1597))

* [[Marc Hoyois]], *The six operations in equivariant motivic homotopy theory*, Adv. Math. **305** (2017) 197-279 [[arXiv:1509.02145](https://arxiv.org/abs/1509.02145), [doi:10.1016/j.aim.2016.09.031](https://doi.org/10.1016/j.aim.2016.09.031)]

  > (cf. [[Grothendieck's six operations]])


equivariant noncommutative algebraic geometry

This entry is intended to be about symmetry objects acting on noncommutative spaces and objects/constructions which are invariant or equivariant under their action.

##Motivation##

One would like to have symmetry objects like [[algebraic group]]s and [[Lie algebra]]s in [[noncommutative geometry]] (including algebraic flavour). The group-like objects should be noncommutative spaces themselves, they should have representation theory, they should act on other noncommutative spaces, define quotient spaces and so on. One also expects to have a equivalence between the category of $G$-equivariant "sheaves" on a total noncommutative principal $G$-bundle (where $G$ is an appropriate symmetry object) and usual "sheaves" on the quotient. A first massive appearance were quantum groups, and one should be warned that quantum groups are not cogroup objects in the category of noncommutative rings, because they are Hopf algebras with respect to the tensor product rather than categorical coproduct of algebras.

##Main characters##

Here one should write about [[quantum groups]] (Drinfeld, Manin, Woronowicz, Jimbo, Lusztig, Faddeev-Reshetikin-Tahtajan, Majid), [[Hopf algebras]], Hopf algebroids (quantum groupoids), quantum Lie algebras, entwinings/[[distributive law]]s, quantum flag varieties, (co)module (co)algebras, quantum principal bundles, associated bundles, Drinfel'd center, equivariant cyclic homology etc.

##Literature##

* V. G. Drinfel'd, _Quantum groups_, Proc. Int. Cong. Math. 1986, Vol. 1, 2 798–820, AMS 1987.

* S. Majid, _Foundations of quantum group theory_, Cambridge University Press 1995, 2000.

* B. Parshall, J.Wang, _Quantum linear groups_, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.

* [[Zoran Škoda]], _Some equivariant constructions in noncommutative geometry_, Georgian Math. J. 16 (2009) 1; 183--202 [arXiv:0811.4770](https://arxiv.org/abs/0811.4770)

* [[Yuri Manin]], _Quantum groups and noncommutative geometry_, CRM, Montreal (1988)

* Ludwig Faddeev, N. Reshetikin, L. Tahtajan, _Quantization of Lie groups and Lie algebras_, Algebra i Analiz 1 (1989) 178 (transl.  Leningrad Math. J. 1 (1990), 193-225)

* [[Susan Montgomery]], _Hopf algebras and their action on rings_, AMS 1994, 240p.

* [[Yan Soibelman]], _On the quantum flag manifold_, Funk. analiz i ego pril. __26__, 3, 90--92, 1992

* Z. Škoda, _Localizations for construction of quantum coset spaces_, [math.QA/0301090](http://front.math.ucdavis.edu/math.QA/0301090),  Banach Center Publ. vol.61, pp. 265--298, Warszawa 2003.

*  [[Nikolai Reshetikhin]], A. A. Voronov, [[Alan Weinstein]],  _Semiquantum geometry_, [math.q-alg/9606007](https://arxiv.org/abs/q-alg/9606007)

* [[Tomasz Brzeziński]], [[Shahn Majid]], _Coalgebra bundles_, Comm. Math. Phys.  191  (1998),  no. 2, 467--492 ([arXiv version](https://arxiv.org/abs/q-alg/9602022)).

* [[Brian Day]], [[Ross Street]], _Monoidal bicategories and Hopf algebroids_, Adv. Math. __129__, (1997) 99-157 [doi](https://doi.org/10.1006/aima.1997.1649)

category: algebraic geometry, noncommutative geometry


equivariant object

Given a [[fibered category]] $\pi=\pi_F: F\to C$, a [[presheaf]] of [[group]]s $\hat{G}:C^{\mathrm{op}}\to\mathrm{Grp}$ (possibly [[representable functor|representable]]), an object $X$ in $C$, and an [[action]] $\hat\nu: \hat{G}\times h_X\to h_X$ (generalizing the action $G\times X\to X$ in the representable case), one says that the object $\rho$ in the fiber $F_X = \pi^{-1}(\mathrm{id}_X)$ is __$\hat{G}$-equivariant__ if $\hat{G}\circ\pi$ is equipped with an action on $h_\rho$, i.e., it is equipped with a natural transformation $\sigma:(\hat{G}\circ\pi)\times h_\rho : F^{\mathrm{op}}\to\mathrm{Set}$ such that for any $\xi\in F$, $\sigma_\xi:\hat{G}(\pi(\xi))\times\mathrm{Hom}_F(\xi,\rho)\to\mathrm{Hom}_F(\xi,\rho)$ satisfies the action axiom of a group $\hat{G}(\pi(\xi))$ on a set $\mathrm{Hom}_F(\xi,\rho)$, and such that $\pi(\sigma)=\hat\nu$. The $\hat{G}$-equivariant objects naturally form a fibered category $\pi^{\hat{G}}:F^{\hat{G}}\to C$ of equivariant objects.

Alternatively, if $F$ and $C$ are [[complete category|complete]] and $\hat{G}$ is represented by $G$ in $C$, the equivariant fiber over a $G$-object $X$ is simply the fiber over the [[simplicial object]] $[ X/G ]$ (the simplicial [[Borel construction]]), that is the category of Cartesian functors $(\sigma^F,\sigma^C):\mathrm{Id}_{\Delta^{\mathrm{op}}}\to\pi_F$ from the opposite of the category of [[simplex|simplices]] $\Delta^{\mathrm{op}}$ considered as a fibered category $\mathrm{Id}_{\Delta^{\mathrm{op}}}:\Delta^{\mathrm{op}}\to\Delta^{\mathrm{op}}$ to the fibered category $\pi_F:F\to C$, such that the bottom component is $\sigma_C = [X/G]:\Delta^{\mathrm{op}}\to C$.

Equivariant objects in fibered categories generalize [[equivariant sheaf|equivariant sheaves]] which in turn generalize (sheaves of sections of) $G$-[[equivariant bundle]]s. The description of $G$-equivariant sheaves via cartesian sections over the simplicial Borel construction translates into the statement that they form the category equivalent to the subcategory of the category of [[Deligne sheaf|Deligne sheaves]] over the simplicial Borel construction all of whose structure morphisms are isomorphisms. Mumford has expressed $G$-equivariant sheaves over a $G$-space $(X,\nu)$ as sheaves $\rho$ equipped with an isomorphism $\theta:\nu^*(\rho)\to p^*(\rho)$ where $p:G\times X\to X$ is the projection and $\nu:G\times X\to X$ is the action, and such that $\theta$ satisfies a certain cocycle condition, which is an identity of sheaves over $G\times G\times X$.

For $G$-equivariant sheaves over a *topological* $G$-space $(X,\nu)$ (as opposed to a $G$-object $(X,\nu)$ in a Grothendieck [[site]]), one can extend the canonical equivalence between étale spaces over $X$ and sheaves over $X$ to a canonical equivalence between étale $G$-spaces over $(X,\nu)$ and $G$-equivariant sheaves over $(X,\nu)$.

##Literature##
* P. Deligne, Théorie de Hodge : III. Publications Mathématiques de l'IHÉS, 44 (1974), p. 5--77
([numdam](http://www.numdam.org/item?id=PMIHES_1974__44__5_0))
* S. MacLane, I. Moerdijk, Sheaves in geometry and logic, Springer 1992.
* D. Mumford, Geometric invariant theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34. Springer 1965. 2nd edition with J. Fogarty, Springer 1982.
* Z. Škoda, Some equivariant constructions in noncommutative algebraic geometry, accepted to Georgian Math. J., [arXiv:0811.4770](http://arxiv.org/abs/0811.4770).
* A. Vistoli, Grothendieck topologies, fibered categories and descent theory. Fundamental algebraic geometry, 1--104, Math. Surveys Monogr., 123, AMS 2005, [arXiv:math.AG/0412512](http://front.math.ucdavis.edu/0412.5512).


equivariant open cover

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#### Representation theory
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notions of *equivariant open cover* and of *equivariant good open cover* are the generalization of the notions of *[[open cover]]* and *[[good open cover]]* from [[differential topology]] to [[equivariant differential topology]], hence from [[topological spaces]] to [[topological G-spaces]] and [[G-manifolds]].

Accordingly, for some [[equivariance group]] $G$, an equivariant (good) open cover of a [[topological G-space]] is a ([[good open cover|good]]) [[open cover]] of its [[underlying]] [[topological space]] which is compatible with the [[group action]] in a suitable way. At a minimum this means that the cover is itself a [[topological G-space]] and that the covering map is an [[equivariant function]], but for purposes of [[proper equivariant homotopy theory|proper]] [[equivariant homotopy theory]] one will typically need that passage to $H$-[[fixed loci]], for suitable [[subgroups]] $H \subset G$, preserves the (good) covering properties.

## Definition

In the following, $kTopSp$ denotes the [[convenient category of topological spaces|convenient category]] of [[compactly generated weak Hausdorff spaces]].

For $G \,\in\, Grp(kTopSp)$ a (cgwh-)[[topological group]], and $H \,\subset\, G$ a [[subgroup]], we write

\[
  \label{HFixedLocusFunctor}
  G Act(kTopSp)
  \xrightarrow{\; (-)^H \;}
  kTopSp
\]

for the [[functor]] from the [[category]] of (cgwh-)[[topological G-spaces]] which assigns $H$-[[fixed loci]].

\begin{definition}
\label{ProperlyEquivariantOpenCover}
**(properly equivariant open covers)**
\linebreak
  For $G \,\in\, Grp(SmthMfd) \xhookrightarrow{\;} Grp(kTopSp)$ a [[topological group]] [[underlying]] a [[Lie group]], and $G \curvearrowright X \,\in\, G Act(kTopSp)$ a [[topological G-space]], say that an [[open cover]]

$$
  \widehat{X}
  \;\coloneqq\;
    \underset{i \in I}{\sqcup}
    U_i
  \;\;
  \xrightarrow{\;\; p \;\;}
  \;\;
  X
$$

is

1. *equivariant* (often: "invariant") if the $G$-action on $X$ pulls back to a $G$-action on $\widehat{X}$

   $$
     G \curvearrowright \widehat{X}
     \;\coloneqq\;
     G \curvearrowright
     \big(
       \underset{i \in I}{\sqcup}
       U_i
     \big)
     \;\;
     \xrightarrow{\;\; p \;\;}
     \;\;
     G \curvearrowright X
     \,;
   $$

1. *regular* if it satisfies ([Yang 2014, Def. 2.7](#Yang14))

   \[
     \label{RegularityCondition}
     \begin{array}{ll}
       (a)
       &
       \underset{i,j \in I}{\forall}
       \left(
         U_i \,=\, U_j
         \;\;\;\;
         \Rightarrow
         \;\;\;\;
         i \,=\, j
       \right)
       \\
       (b)
       &
       \underset{i \in I}{\forall}
       \,\,
       \underset{g \in G}{\forall}
       \;
       \Big(
         U_i
          \,\cap\,
         g \cdot U_j
         \;\neq\;
         \varnothing
         \;\;\Rightarrow\;\;
         U_i \,=\, g \cdot U_i
       \Big)
       \\
       (c)
       &
       \underset{n \in \mathbb{N}}{\forall}
       \;
       \underset{
          { i_0, \cdots, i_n \in I }
          \atop
          { g_0, \cdots, g_n \,\in\, G }
       }{\forall}
       \left(
         \begin{array}{rcl}
           U_{i_0} \cap \cdots \cap U_{i_n}
           & \neq
           & \varnothing
           \mathrlap{,}
           \\
           g_0 \cdot U_{i_0} \cap \cdots g_n \cdot U_{i_n}
           & \neq
           & \varnothing
         \end{array}
         \;\;\Rightarrow\;\;
         \underset{g \in G}{\exists}
         \;
         \underset{0 \leq k \leq n}{\forall}
         \;
         g \cdot U_{i_k} \,=\, g_k \cdot U_{i_k}
       \right)
     \end{array}
   \]

1. *properly equivariant* if for all [[compact subgroups]] $H \,\subset\, G$ its restriction (eq:HFixedLocusFunctor) to $H$-[[fixed loci]]  is a plain [[open cover]] of [[topological spaces]]:

   \[
     \label{OpenCoverOnAllFixedLoci}
     \underset{
       H \underset{cpt}{\subset} G
     }{\forall}
     \;\;\;
     \widehat{X}^H
     \underoverset
       { open\,cover }
       {p^H}
       {\longrightarrow}
     X^H
   \]

1. *properly equivariant good* if it is regular and proper equivariant and the restrictions (eq:OpenCoverOnAllFixedLoci) are all [[good open covers]] ([Yang 2014, Def. 2.10](#Yang14)).

\end{definition}

\begin{remark}
  If an equivariant open cover
  is regular (eq:RegularityCondition)
  then the index set $I$ inherits a unique $G$-action $i \mapsto g \cdot i$ such that
$$
  g \cdot U_i \,\subset\, U_{g_i}
$$
and such that the [[stabilizer subgroup]] $G_x \subset G$ of any $x \in U_i \subset X$ also stabilizes $i$:
$$
  G_x \cdot U_i \,=\, U_i
  \,.
$$
\end{remark}

## Properties

### Existence

\begin{proposition}
  For $G \,\in\, Grp(FinSet) \xhookrightarrow{Grp(Dsc)} Grp(kTopSp)$ a [[finite group]], the [[topological G-space]] [[underlying]] a [[smooth manifold|smooth]] [[G-manifold]] admits a regular properly equivariant good open cover (Def. \ref{ProperlyEquivariantOpenCover}).
\end{proposition}

([Yang 2014, Thm. 2.11](#Yang14), using the [[equivariant triangulation theorem]])

## Related entries

* [[equivariant triangulation theorem]]

## References

* {#Yang14} Haibo Yang, *Equivariant cohomology and sheaves*, Journal of Algebra, **412** 2014, 230-254 ([doi:10.1016/j.jalgebra.2014.05.009](https://doi.org/10.1016/j.jalgebra.2014.05.009))

[[!redirects equivariant good open covers]]

[[!redirects equivariant open cover]]
[[!redirects equivariant open covers]]

[[!redirects equivariant good open cover]]
[[!redirects equivariant good open covers]]


equivariant operator K-theory

#Contents#
* table of contents
{:toc}

## Idea

The [[equivariant cohomology]] version of [[operator K-theory]]

## Related concepts

* [[equivariant K-theory]]

* [[equivariant KK-theory]]

## References

Section 11 of

* [[Bruce Blackadar]], _[[K-Theory for Operator Algebras]]_


equivariant ordinary cohomology

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Equivariant ordinary cohomology is the [[equivariant cohomology]]-version of [[ordinary cohomology]].

By default this is often understood to be the [[Borel equivariant cohomology]]-version of [[ordinary cohomology]], but more generally this must be understood as [[Bredon cohomology]], i.e. as the [[proper equivariant homotopy theory|proper-equivariant]] version of [[ordinary cohomology]].

## Properties

### Equivariant Chern character

There is a [[Chern character]] map from [[equivariant K-theory]] to [[equivariant ordinary cohomology]].

(e.g. [Stefanich](#Stefanich))

## Related concepts

* [[equivariant de Rham cohomology]]

* [[equivariant ordinary differential cohomology]]

[[!include Segal completion -- table]]

## References

### As Borel-equivariant cohomology theory

* [[Steven Costenoble]], [[Stefan Waner]], _Equivariant ordinary homology and cohomology_, Springer, 2016 ([arXiv:math/0310237](https://arxiv.org/abs/math/0310237))

* {#Stefanich} German Stefanich, _Chern Character in Twisted and Equivariant K-Theory_ ([pdf](https://math.berkeley.edu/~germans/Chern2.pdf))

### As Bredon-equivariant cohomology theory

Discussion over the point but in arbitrary [[RO(G)-degree]]:

for [[equivariance group]] a [[dihedral group]] of [[order of a group|order]] $2p$:

* [[Igor Kriz]], [[Yunze Lu]], *On the $RO(G)$-graded coefficients of dihedral equivariant cohomology*, Mathematical Research Letters **27** 4 (2020) ([arXiv:2005.01225](https://arxiv.org/abs/2005.01225), [doi:10.4310/MRL.2020.v27.n4.a7](https://dx.doi.org/10.4310/MRL.2020.v27.n4.a7))

for [[equivariance group]] the [[quaternion group]]:

* [[Yunze Lu]], *On the $RO(G)$-graded coefficients of $Q_8$ equivariant cohomology*, Topology and its Applications, 2021 ([doi:10.1016/j.topol.2021.107921](https://doi.org/10.1016/j.topol.2021.107921))

[[!redirects equivariant ordinary cohomology theory]]

[[!redirects ordinary equivariant cohomology]]


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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Equivariant ordinary differential cohomology_ or _equivariant Deligne cohomology_ is the [[equivariant cohomology]]-enhancement of [[ordinary differential cohomology]] ([[Deligne cohomology]]), equivalently the [[differential cohomology]]-enhancement of [[ordinary equivariant cohomology]].

There is one evident general abstract definition of what this means concretely: Given a $G$-[[action]] on some $X$, the (Borel-)$G$-equivariant ordinary differential cohomology of $X$ in degree $n$ is the [[(∞,1)-categorical hom-space]] in the [[cohesive (∞,1)-topos]] of [[Smooth∞Groupoids]] from the [[quotient stack]] $X \sslash G$ to the [[Deligne complex]] $\mathbf{B}^n U(1)_{conn}$.

That various definitions in the literature coincide with this one (in particular [Kübel-Thom 15](#KubelThom15), and for [[finite groups]] also [Lupercio-Uribe 01](#LupercioUribe01), [Gomi 05](#Gomi05)) is discussed in ([Park-Redden 19](#ParkRedden19)).

## Related concepts

* [[discrete torsion]]

* [[equivariant de Rham cohomology]]

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

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

* [[twisted ordinary differential cohomology]]

## References

The main definitions can be found in

* {#KubelThom15} [[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))

* [[Corbett Redden]], _Differential Borel equivariant cohomology via connections_, New York Journal of Mathematics, Volume 23 (2017) 441-487 ([nyjm:23-20](http://nyjm.albany.edu/j/2017/23-20.html), [arXiv:1602.06921](https://arxiv.org/abs/1602.06921))

An earlier definition thst works (only) for [[finite groups]] can be found in:

* {#Gomi05} [[Kiyonori Gomi]], _Equivariant smooth Deligne cohomology_, Osaka J. Math. Volume 42, Number 2 (2005), 309-337 ([euclid:ojm/1153494380](https://projecteuclid.org/euclid.ojm/1153494380))

Other references:

* {#LupercioUribe01} [[Ernesto Lupercio]], [[Bernardo Uribe]], _Deligne Cohomology for Orbifolds, discrete torsion and B-fields_, in: _Geometric and topological methods for quantum field theory_ Proceedings, Summer School, Villa de Leyva, Colombia, July 9-27, 2001 ([arXiv:hep-th/0201184](https://arxiv.org/abs/hep-th/0201184), [spire:582101](https://inspirehep.net/literature/582101))

* Cheng-Yong Du Xiaojuan Zhao, _Spark and Deligne-Beilinson cohomology on orbifolds_, Journal of Geometry and Physics Volume 104, June 2016, Pages 277-290 ([doi:10.1016/j.geomphys.2016.02.011](https://doi.org/10.1016/j.geomphys.2016.02.011))

* [[Corbett Redden]], _An alternate description of equivariant connections_, Differential Geometry and its Applications Volume 56, February 2018, Pages 81-94 ([doi:10.1016/j.difgeo.2017.11.003](https://doi.org/10.1016/j.difgeo.2017.11.003) [arXiv:1608.01297](https://arxiv.org/abs/1608.01297))

* {#ParkRedden19} [[Byungdo Park]], [[Corbett Redden]], _A classification of equivariant gerbe connections_, Communications in Contemporary MathematicsVol. 21, No. 02, 1850001 (2019) ([doi:10.1142/S0219199718500013](https://doi.org/10.1142/S0219199718500013), [arXiv:1709.06003](https://arxiv.org/abs/1709.06003))

* Cheng-Yong Du, Lili Shen, Xiaojuan Zhao, _Spark complexes on good effective orbifold atlases categorically_, Theory and Applications of Categories, 33(26):784-812, 2018 ([tac:33-26](http://www.tac.mta.ca/tac/volumes/33/26/33-26abs.html), [arXiv:1708.07551](https://arxiv.org/abs/1708.07551))

Relation to [[action functionals]] for [[topological field theories]] (such as [[Chern-Simons theory]]):

* Joe Davighi, [[Ben Gripaios]], [[Oscar Randal-Williams]], _Differential cohomology and topological actions in physics_ ([arXiv:2011.05768](https://arxiv.org/abs/2011.05768))

[[!redirects ordinary equivariant differential cohomology]]


equivariant PL de Rham complex

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#### Algebra
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[[!include higher algebra - contents]]
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#### Rational homotopy theory
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[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

+-- {: .num_defn #EquivariantPLDeRhamComplex}
###### Definition
**(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]] $X^H \subset X$.

=--

## Properties

+-- {: .num_prop #EquivariantPLDeRhamComplexIsInjectiveInDualVectorGSpaces}
###### Proposition
**([[equivariant PL de Rham complex]] is degreewise [[injective object|injective]] in [[dual vector G-spaces]])**

The [[dual vector G-space]] underlying the equivariant PL de Rham complex (Def. \ref{EquivariantPLDeRhamComplex})

$$
  \array{
   G Orbits
    &
    \overset{
      \Omega^\bullet_{PLdR}
      \big(
        Maps(-,X)^G
      \big)
    }{\longrightarrow}
    &
    dgcAlgebras
    &\overset{}{\longrightarrow}&
    VectorSpaces_{\mathbb{Q}}
  }
$$

is degreewise an [[injective object]].

=--

([Triantafillou 82, Prop. 4.3](#Triantafillou82))

+-- {: .num_cor }
###### Corollary

Any equivariant PL de Rham complex (Def. \ref{EquivariantPLDeRhamComplex}) is a [[fibrant object]] in the [[model structure on equivariant connective dgc-algebras]].

=--

(also [Scull 08, Lemma 5.2](#Scull08))

In fact:

+-- {: .num_prop}
###### Proposition
**([[Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras]])**

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

The $G$-[[equivariant PL de Rham complex]]-construction is the [[left adjoint]] in a [[Quillen adjunction]] between

* the [[opposite model structure|opposite]] of the [[projective model structure on equivariant connective dgc-algebras]]

* the [[model structure on equivariant simplicial sets]]

  (i.e.: the global projective [[model structure on functors]] from the [[opposite category|opposite]] of the [[orbit category]] to the [[classical model structure on simplicial sets]])

$$
  \big(
    G dgcAlgebras^{\geq 0}_{k}
  \big)^{op}_{proj}
  \underoverset
    {
      \underset
        {\;\;\; exp \;\;\;}
        {\longrightarrow}
    }
    {
      \overset
        {\;\;\;\Omega^\bullet_{PLdR}\;\;\;}
        {\longleftarrow}
    }
    {\bot_{\mathrlap{Qu}}}
  G SimplicialSets_{Qu}
$$

=--

([Scull 08, Prop. 5.1](#Scull08))

\linebreak

* [[equivariant PL de Rham theorem]]

## Related concepts

* [[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))

* {#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))


equivariant PL de Rham theorem

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The equivariant PL de Rham theorem ([Triantafillou 82, Theorem 4.9](#Triantafillou82))
is the generalization of the [[PL de Rham theorem]] to the [[equivariant PL de Rham complex]] in [[equivariant rational homotopy theory]].

## Related concepts

* [[equivariant de Rham theorem]]

* [[de Rham theorem]]

## References

* {#Triantafillou82} [[Georgia Triantafillou]], Theorem 4.9 in: _Equivariant minimal models_, Trans. Amer. Math. Soc. vol 274 pp 509-532 (1982) ([jstor:1999119](http://www.jstor.org/stable/1999119))


equivariant Pontrjagin theorem

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#### Algebraic topology
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _equivariant Pontrjagin theorem_ should generalize the [[Pontrjagin theorem]] from plain [[homotopy theory]]/[[cobordism theory]] to [[equivariant homotopy theory]]/[[equivariant cobordism theory]]:

Where the plain [[Pontrjagin theorem]] identifies the [[Cohomotopy]] of a [[differentiable manifold]] with its [[cobordism classes]] of [[normally framed submanifolds]], an equivariant Pontrjagin theorem should identify [[equivariant Cohomotopy]] with [[fixed locus|fixed]] [[submanifolds]] whose [[normally framed submanifold|normal framing]] inherits the [[linear representation]] of the given [[RO(G)-degree]].

### General

A fully general version of an equivariant Pontrjagin theorem is not known (?) and not expected to exist (?) because [[Thom's transversality theorem]] (which, in one formulation or another, underlies the proof of the ordinary [[Pontrjagin theorem]]) is known to fail equivariantly: Since [[equivariant functions]] need to send [[fixed points]] to [[fixed points]], an equivariant map from an [[open neighbourhood]] of fixed points to an isolated fixed point has no equivariant deformation whatsoever, let along a [[regular point|regular]] one.

### For free $G$-manifolds

This problem with transversaliuty goes away when the [[action]] on the [[domain]] [[G-manifold]] is [[free action]]. In this case the equivariant Pontrjagin theorem in [Cruickshank 99, Thm. 5.06](#Cruickshank99).

Of course, this case is only "mildly equivariant": When the $G$-action on the domain [[G-manifold]] is [[free action|free]] then its [[equivariant homotopy theory]] should essentially reduce to plain [[homotopy theory]] of the [[quotient space]]. Indeed, [Cruickshank 99, Cor. 6.0.13](#Cruickshank99) identifies, in this case, [[equivariant Cohomotopy]] with [[cobordism classes]] of [[normally twisted-framed submanifolds]], i.e. identifies it with the [[twisted Pontrjagin theorem]] ([Cruickshank 03, Lemma 5.2](#Cruickshank03)) on the quotient.

## Related concepts

* [[Pontrjagin theorem]], [[twisted Pontrjagin theorem]]

* [[Pontrjagin-Thom theorem]]

* [[Hopf degree theorem]], [[equivariant Hopf degree theorem]]

* [[equivariant differential topology]]

## References

[[!include Pontryagin-Thom construction -- references]]


equivariant rational homotopy theory

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[equivariance|equivariant]] version of [[rational homotopy theory]], dealing with [[rationalization]] in ([[proper equivariant homotopy theory|proper]]) [[equivariant homotopy theory]], detected by [[Bredon cohomology|Bredon equivariant]] [[rational cohomology]].

## Related concepts

* [[equivariant rationalization]]

* [[vector G-space]]

* [[equivariant chain complex]], [[model structure on equivariant chain complexes]]

* [[equivariant dgc-algebra]]

* [[equivariant PL de Rham complex]]

* [[equivariant Sullivan model]]

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

* [[Quillen adjunction between equivariant simplicial sets and equivariant connective dgc-algebras]]

* [[fundamental theorem of dg-algebraic equivariant rational homotopy theory]]

\linebreak

* [[rational homotopy theory]]

* [[Borel-equivariant rational homotopy theory]]

* [[rational stable homotopy theory]]

* [[rational equivariant stable homotopy theory]]

* [[rational parameterized stable homotopy theory]]

* [[equivariant differential topology]]

## References
  {#References}

The original reference for [[finite groups]] is

* [[Georgia Triantafillou]], _Equivariant rational homotopy theory_, chapter III of [[Peter May]], _Equivariant homotopy and cohomology theory_, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comeza˜na, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. ([pdf](http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf), [ISBN: 978-0-8218-0319-6](https://bookstore.ams.org/cbms-91))

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

but beware that [Scull 01](#Scull01) claims that the statement about [[minimal models]] there is not correct. Corrected statements for finite groups as well as generalization to [[compact Lie groups]], at least to the [[circle group]], is due to:

* {#Scull01} [[Laura Scull]], _Rational $S^1$-equivariant homotopy theory_, Transactions of the AMS, Volume 354, Number 1, Pages 1-45 2001 ([pdf](http://www.ams.org/journals/tran/2002-354-01/S0002-9947-01-02790-8/S0002-9947-01-02790-8.pdf), [doi:10.1090/S0002-9947-01-02790-8](https://doi.org/10.1090/S0002-9947-01-02790-8))

Further discussion in:

* {#Golasinski97a} [[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]])

* {#Golasinski97b} [[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))

The [[model structure on equivariant dgc-algebras]], generalizing the [[projective model structure on dgc-algebras]], in which equivariant [[minimal Sullivan models]] are [[cofibrant objects]]:

* {#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))

See also

* Peter J. Kahn, _Rational Moore G-Spaces_, Transactions of the American Mathematical Society
Vol. 298, No. 1 (1986), pp. 245-271 ([jstor:2000619](https://www.jstor.org/stable/2000619))

* C. Allday, V. Puppe, sections 3.3 and 3.4 of _Cohomological methods in transformation groups_, Cambridge 1993 (doi:10.1017/CBO9780511526275)

[[!redirects rational equivariant homotopy theory]]


equivariant rationalization

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#### Rational homotopy theory
+--{: .hide}
[[!include differential graded objects - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $G$ be a [[finite group]] (or more generally a [[compact Lie group]]).

Say that an equivariantly [[connected topological space|connected]] and [[nilpotent topological space|nilpotent]] [[topological G-space]] $Y$ (i.e. all [[fixed loci]] $X^H$ are connected and nilpotent, for $H \subset_{clsd} G$, with a common bound of nilpotency as $H$ ranges) is _rational_ if all its [[equivariant homotopy groups]] $\pi_n\big( X^H \big)$ (for $n \in \mathbb{N}$ and $H \subset_{cld} G$) admit the [[mathematical structure|structure]] of [[rational vector spaces]].

Given any [[topological G-space]] $X$, a _rationalization_ of $X$ is a morphism (a $G$-[[equivariant function|equivariant]] [[continuous function]])

$$
  X
    \overset{
      \;\;\; \eta^{\mathbb{Q}}_X \;\;\;
    }{\longrightarrow}
  L_{\mathbb{Q}}X
$$

to a rational $G$-space $L_{\mathbb{Q}}X$ which induces [[isomorphisms]] on all rationalized equivariant homotopy groups:

$$
  \big(
    \eta^{\mathbb{Q}}_X(G/H)
  \big)_\ast
  \;\colon\;
  \pi_\bullet( X^H )
  \otimes \mathbb{Q}
  \overset{\simeq}{\longrightarrow}
  \pi_\bullet
  \Big(
     \big(
       L_{\mathbb{Q}}X
     \big)^H
  \Big)
  \,.
$$

## Properties

### Via Elmendorf's theorem

In other words, after regarding them, via [[Elmendorf's theorem]], as [[(∞,1)-presheaves]] on the [[orbit category]] $G Orbits$ of $G$, the [[equivariant homotopy types]] of rational $G$-spaces and their rationalizations are equivalently stage-wise over $G/H \in G Orbits$ plain [[rational spaces]] and [[rationalizations]], respectively.

It follows from the [[fundamental theorem of dg-algebraic equivariant rational homotopy theory]] that, at least on equivariantly [[simply connected topological space|simply connected]] [[topological G-spaces]], equivariant rationalization is given by the [[derived adjunction|derived]] [[adjunction unit]] of the [[equivariant PL de Rham complex]]-[[Quillen adjunction]].

## Related concepts

* [[equivariant Chern character]]

* [[equivariant Chern-Dold character]]

* [[rational equivariant homotopy theory]]

* [[rational stable equivariant homotopy theory]]

## References

* {#May96} [[Peter May]], Section II.3 in: _Equivariant homotopy and cohomology theory_, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. ([ISBN: 978-0-8218-0319-6](https://bookstore.ams.org/cbms-91/?startBookmarkIdx=200)  [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/alaska1.pdf), [[MayEtAlEquivariant96.pdf:file]])

* [[Georgia Triantafillou]], Section 2.6 in _Equivariant minimal models_, Trans. Amer. Math. Soc. vol 274 pp 509-532 (1982) ([jstor:1999119](http://www.jstor.org/stable/1999119))

* [[Laura Scull]], p. 11 of: _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 rationalizations]]

[[!redirects rational G-space]]
[[!redirects rational G-spaces]]

[[!redirects rational topological G-space]]
[[!redirects rational topological G-spaces]]


equivariant sheaf

Let $G$ be a [[group object|group]] in some category $C$ of [[space]]s. We assume that for every space $X$ in $C$ some category of [[sheaf|sheaves]] $F_X$ is given, in a way making a [[fibered category]] $F$ over $C$. For example, we can consider sheaves of abelian [[group]]s over [[topological space]]s.

Consider a $G$-space $X$ with [[action]] $\rho: G\times X\to X$ and projection $p: G\times X\to X$.  These data give rise to an [[action groupoid]] in the category of spaces, denoted by $[X/G]$ or sometimes $X//G$.  This groupoid can also be identified with its [[nerve]], which is a [[simplicial object]] in spaces.

An __equivariant sheaf__ over a $G$-space $X$ is a $G$-[[equivariant object]] in $F_X$. In other words, it is a sheaf $x$ over $X$ together with an isomorphism $\theta : p^* x \to \rho^* x$ of sheaves over $G\times X$ satisfying the usual cocycle condition on $G\times G\times X$. All equivariant sheaves form the equivariant fiber $F_X^G$ of $F$ over $X$. The equivariant fiber over $X$ is thought of as a fiber over the simplicial object $[X/G]$.

If the fibered category $F$ is a [[stack]] for some [[subcanonical topology|subcanonical Grothendieck topology]] on $C$ and $X$ a $G$-[[torsor]] over some true base space $X/G$ in $C$ then there is a descent along torsor: the equivariant fiber $F_X^G$ is canonically equivalent to the usual fiber over $X/G$. In other words the fibers over $[X/G]$ and $X/G$ are equivalent.

If $F$ is not a [[stack]] one can instead of a [[Grothendieck topology]] in $C$ use the effective descent topology and include into the condition for a [[torsor]] instead of local triviality in a Grothendieck topology that $P$ over $X/G$ is an effective descent epimorphism relative to the fibered category $F$ over $C$.

Notice that being an equivariant sheaf is an additional structure on a sheaf, rather than a property. Mumford introduced this notion in the geometric invariant theory under the name $G$-linearization of a sheaf. $G$-equivariant sheaves generalize (sheaves of sections of) $G$-equivariant bundles, as studied for example in representation theory earlier by Borel, Weil and Bott.

Note also that if "space" means [[topological space]] and $G$ is a [[topological group]] acting on the [[point]] $1$, then equivariant sheaves are just the same as continuous $G$-sets.  In particular, if $G$ is a discrete group, then equivariant sheaves are just the same as ordinary $G$-sets.

Observe that the definition of equivariant sheaf only depends on the [[action groupoid]], and thus can be generalized to equivariant sheaves on any internal groupoid in the category of spaces.  If "space" means [[locale]], then every [[Grothendieck topos]] can be presented as the category of equivariant sheaves on some [[localic groupoid]].  This is a theorem of [[André Joyal]] and [[Miles Tierney]], and can be found in chapter C5 of the [[Elephant]].

The appropriate notion of the "[[equivariant derived category]]" is in general not equal to the derived category of the [[abelian category]] of equivariant sheaves; one needs to resolve appropriately.

category: sheaf theory
[[!redirects equivariant sheaves]]


equivariant sphere spectrum

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#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#### Representation theory
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[[!include representation theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The [[sphere spectrum]] in ([[global equivariant stable homotopy theory|global]]) [[equivariant stable homotopy theory]].

Its [[RO(G)-grading|RO(G)-graded]] [[homotopy groups]] are the equivariant version of the [[stable homotopy groups of spheres]].

## Definition

The $G$-equivariant sphere spectrum is the [[equivariant suspension spectrum]] of the [[0-sphere]] $S^0 = \ast_+$

$$
  \mathbb{S} = \Sigma^\infty_G S^0
$$

for $S^0$ regarded as equipped with the (necessarily) trivial $G$-action. It follows that for $V$ an orthogonal linear $G$-representation then in [[RO(G)-degree]] $V$ the equivariant sphere spectrum is the corresponding [[representation sphere]] $\mathbb{S}(V) \simeq S^V$.

(e.g. [Schwede 15, example 2.10](#Schwede15))

## Properties

### Equivariant homotopy groups
 {#EquivariantHomotopyGroups}

Just as for the plain [[sphere spectrum]], the [[equivariant homotopy groups]] of the equivariant sphere spectrum in ordinary [[integer]] degrees $n$ are all [[torsion subgroup|torsion]], except at $n = 0$:

$$
  \pi_n^H(\mathbb{S})\otimes \mathbb{Q} =
  \left\{
    \array{
       \cdots & for \; n = 0
       \\
       0 & otherwise
    }
  \right.
$$

([Greenlees-May 95, prop. A.3](#GreenleesMay95))

But in some [[RO(G)-degrees]] there may appear further non-torsion groups, see the [examples](#Examples) below.

In degree 0, the [[tom Dieck splitting]] applied to the [[equivariant suspension spectrum]] $\mathbb{S} = \Sigma^\infty_G S^0$ gives that $\pi_0^G(\mathbb{S})$ is the [[free abelian group]] on the set of [[conjugacy classes]] of [[subgroups]] of $G$:

\[
  \label{pi_0ofEquivariantSphere}
  \pi_0^G(\mathbb{S})
  \simeq
  \underset{[H \subset G]}{\oplus} \pi_0^{W_G H}(\Sigma_+^\infty E (W_G H))
  \simeq
  \mathbb{Z}[conjugacy\;classes\;of\;subgroups]
\]

(e.g. [Schwede 15, p. 64](#Schwede15))

## Examples
 {#Examples}

### $\mathbb{Z}_2$-equivariance
 {#Z2equivariance}

Consider $G= \mathbb{Z}_2$ the [[cyclic group of order 2]] and write $\pi^S_{p,q}$ for the homotopy group in [[RO(G)-grading|RO(G)-degree]] given by the representation on $\mathbb{R}^{p+q}$ where $\mathbb{Z}_2$ acts by [[reflection]] on the first $p$ coordinates, and trivially on the remaining $q$ coordinates:

The following groups contain $\mathbb{Z}$-summands:

* $\pi^S_{0,0} = \mathbb{Z} \times \mathbb{Z}$ (by the [[equivariant Hopf degree theorem]], [this Example](Hopf+degree+theorem#EquivariantCohomotopyOfRepresentationSphereOfSignRepresentationInThatDegree))

* $\pi^S_{1,0}$ ([Araki-Iriye 82, theorem 8.7, p. 25](#ArakiIriye82)) generated by the [[complex Hopf fibration]] $\hat \eta$ ([Araki-Iriye 82, p. 24](#ArakiIriye82))

* $\pi^S_{2,0}$ generated by $\hat \eta^2$ ([Araki-Iriye 82, theorem 9.6, p. 27](#ArakiIriye82))

* $\pi^S_{3,0}$ generated by $\hat \eta^3$ ([Araki-Iriye 82, theorem 10.11, p. 31](#ArakiIriye82))

* $\pi^S_{4,0}$ generated by $\hat \eta^4$ ([Araki-Iriye 82, theorem 11.3, p. 32](#ArakiIriye82))

* $\pi^S_{5,0}$ ([Araki-Iriye 82, theorem 12.4, p. 34](#ArakiIriye82))

* $\pi^S_{6,0}$ ([Araki-Iriye 82, theorem 13.12, p. 38](#ArakiIriye82))

* $\pi^S_{7,0}$ ([Araki-Iriye 82, theorem 14.18, p. 45](#ArakiIriye82))

* $\pi^S_{8,0}$ ([Araki-Iriye 82, theorem 15.26, p. 54](#ArakiIriye82))

In addition we have

* $\pi^S_{2,1} \simeq \mathbb{Z}/24$ generated by the [[quaternionic Hopf fibration]] $\hat \nu$  ([Araki-Iriye 82, prop. 10.1, p. 28 with theorem 10.11, p. 31](#ArakiIriye82))

{#ArakiIriyeResultsTable} In summary and more generally we have the following $\mathbb{Z}/2$-equivariant stable homotopy groups of spheres in low bidegree:

\begin{center}
<img src="https://ncatlab.org/nlab/files/ArakiIriyeResults.jpg" width="600">
\end{center}

> The table shows the $\mathbb{Z}/2$-equivariant stable homotopy groups of spheres $\pi^S_{p,q}$ with $p+q$ increasing horizontally to the right, and $p$ increasing vertically upwards. The origin is the double-circled $\pi^S_{0,0} = \mathbb{Z}^2$.  The [[complex Hopf fibration]] $\hat\eta$ generates $\pi^S_{1,0} = \mathbb{Z}$, and the [[quaternionic Hopf fibration]] generates $\pi^S_{2,1} = \mathbb{Z}/24$

> graphics grabbed from [Dugger 08](#Dugger08), based on [Araki-Iriye 82](#ArakiIriye82)

> (beware that [Dugger 08](#Dugger08) uses a different bi-degree labeling convention: the $(p,q)$ here is $(p+q,p)$ in [Dugger 08](#Dugger08), matching the coordinates of the above table)

|  |  $n_{sgn} + n $ | $0$ | $1$ | $2$ | $3$ |
|--|----------------|--|----|---|--|
| $n_{sgn}$ | $\pi^{st}_{n_{sgn} + n}$ |  |  |  |  |
| $2$ |   |  $0$ |  $0$ | $\mathbb{Z} = \langle (h_{\mathbb{C}})^2\rangle$ | $\mathbb{Z}_{24} = \langle h_{\mathbb{H}}\rangle$  |
| $1$ |   |  $0$ |  $\mathbb{Z} = \langle h_{\mathbb{C}}\rangle$ | $\pi_1^{st} \oplus \mathbb{Z}_2$ | $\pi^{st}_2 \oplus \mathbb{Z}_2 $ |
| $0$ |   |  $\pi_0^{st} \oplus \mathbb{Z}$ |  $\pi_1^{st} \oplus (\mathbb{Z}_2)^2 $ | $\pi_2^{st} \oplus (\mathbb{Z}_2)^2$ | $\pi_3^{st} \oplus \mathbb{Z}_8 \oplus \mathbb{Z}_{24}$ |

### Cyclic ($G_D \hookrightarrow SO(2)$-)equivariance

The global equivariant sphere spectrum for all the [[cyclic groups]] over the [[circle group]] is canonically a [[cyclotomic spectrum]] and as such is the [[tensor unit]] in the [[monoidal (infinity,1)-category]] of [[cyclotomic spectra]] (see there).

### $G_{ADE} \hookrightarrow SO(3)$-equivariance

See at _[quaternionic Hopf fibration --  Class in equivariant stable homotopy theory](quaternionic+Hopf+fibration#ClassInEquivariantStableHomotopyTheory)_

## Related concepts

* The [[cohomology theory]] [[Brown representability theorem|represented]] by the equivariant sphere spectrum is _[[equivariant stable cohomotopy]]_.

* [[sphere spectrum]]

* [[2-periodic sphere spectrum]]

## References

### General

General lecture notes include

* {#Schwede15} [[Stefan Schwede]], example 2.10 in _[[Lectures on Equivariant Stable Homotopy Theory]]_

Discussion in [[rational equivariant stable homotopy theory]] includes

* {#GreenleesMay95} [[John Greenlees]], [[Peter May]], appendix A of _Generalized Tate cohomology_, Mem. Amer. Math. Soc. 113 (1995) no 543 ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/GM-Tate-543.pdf))

The sphere spectrum in [[global equivariant homotopy theory]] is discussed in

* [[John Rognes]], _Galois extensions of structured ring spectra_ ([arXiv:math/0502183](http://arxiv.org/abs/math/0502183))

* Markus Hausmann, Dominik Ostermayr, _Filtrations of global equivariant K-theory_ ([arXiv:1510.04011](http://arxiv.org/abs/1510.04011))

### Relation to equivariant framed bordism

Proof that [[equivariant framed bordism homology theory]] is [[Brown representability theorem|co-represented]] by the [[equivariant sphere spectrum]]:

* Czes Kosniowski, _Equivariant Stable Homotopy and Framed Bordism_, Transactions of the American Mathematical Society
Vol. 219 (1976), pp. 225-234 ([jstor:1997591](https://www.jstor.org/stable/1997591))

### $\mathbb{Z}/2$-equivariance

Discussion of $G$-equivariant homotopy groups for $G = \mathbb{Z}/2$ is in

* [[Peter Landweber]], _On Equivariant Maps Between Spheres with Involutions_, Annals of Mathematics Second Series, Vol. 89, No. 1 (Jan., 1969), pp. 125-137 ([jstor]( http://www.jstor.org/stable/1970812))

* {#ArakiIriye82} [[Shôrô Araki]], [[Kouyemon Iriye]], _Equivariant stable homotopy groups of spheres with involutions. I_, Osaka J. Math. Volume 19, Number 1 (1982), 1-55. ([Euclid:1200774828](http://projecteuclid.org/euclid.ojm/1200774828))

* {#Iriye82} [[Kouyemon Iriye]], _Equivariant stable homotopy groups of spheres with involutions. II_, Osaka J. Math. Volume 19, Number 4 (1982), 733-743 ([euclid:1200775536](https://projecteuclid.org/euclid.ojm/1200775536))

* {#DuggerIsaksen16} [[Daniel Dugger]], [[Daniel Isaksen]], _$\mathbb{Z}/2$-equivariant and R-motivic stable stems_, Proceedings of the American Mathematical Society 145.8 (2017): 3617-3627 ([arXiv:1603.09305](https://arxiv.org/abs/1603.09305))

with exposition in

* {#Dugger08} [[Daniel Dugger]], _Motivic stable homotopy groups of spheres_, talk at [Third Arolla Conference on Algebraic Topology](http://sma.epfl.ch/~hessbell/arolla/) 2008 ([pdf](http://sma.epfl.ch/~hessbell/arolla/slides12/Dugger.pdf), [[DuggerMotivicStableHomotopy08.pdf:file]])

### $\mathbb{Z}/4$-equivariance

Discussion for $G = \mathbb{Z}/4$ is in

* M. C. Crabb, _Periodicty in $\mathbb{Z}/4$ equivariant stable homotopy theory_, in [[Mark Mahowald]] and Stewart Priddy (eds.) _Algebraic Topology_, Proceedings of the
International Conference held March 21-24 (1988) Contemporary Mathematics, volume 96, 1989 ([pdf](http://www.ams.org/books/conm/096/conm096.pdf), [publisher page](http://www.ams.org/books/conm/096/))

General background includes

* {#Schwede15} [[Stefan Schwede]], _[[Lectures on Equivariant Stable Homotopy Theory]]_, 2015 ([pdf](http://www.math.uni-bonn.de/people/schwede/equivariant.pdf))

[[!redirects equivariant sphere spectra]]

[[!redirects global equivariant sphere spectrum]]
[[!redirects global sphere spectrum]]