equivariant stable cohomotopy

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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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 [[equivariant cohomology|equivariant]] [[generalized cohomology theory]] which is [[Brown representability theorem|represented]] by the [[equivariant sphere spectrum]] may also be called _equivariant stable cohomotopy_, as it is the [[equivariant stable homotopy theory]] version of [[stable cohomotopy]], hence of [[cohomotopy]]. This is to be thought of as the first order [[Goodwillie-Taylor tower|Goodwillie approximation]] of plain ("unstable") [[equivariant cohomotopy]].

Just as the plain [[sphere spectrum]] is a distinguished object of plain [[stable homotopy theory]], so the [[equivariant sphere spectrum]] is distinguished in [[equivariant stable homotopy theory]] and hence so is equivariant stable cohomotopy theory.

## Properties

### As equivariant algebraic K-theory over $\mathbb{F}_1$
 {#AsAlgebraicKTheoryOverTheFieldWithOneElement}

The following is known as the _Barratt-Priddy-Quillen theorem_:

+-- {: .num_prop #StableCohomotopyIsKTheoryOfFinSet}
###### Proposition
**([[stable cohomotopy]] is K-theory of [[FinSet]])**

Let $\mathcal{C} = $ [[FinSet]] be a [[skeleton]] of the category of [[finite sets]], regarded as a [[permutative category]]. Then the [[K-theory of a permutative category|K-theory of this permutative category]]

$$
  K(FinSet) \;\simeq\; \mathbb{S}
$$

is represented by the [[sphere spectrum]], hence is stable cohomotopy.

=--

This is due to [Barratt-Priddy 72](#BarrattPriddy72) reproved in [Segal 74, Prop. 3.5](#Segal74). See also [Priddy 73](#Priddy73), [Glasman 13](#Glasman13).

+-- {: .num_remark #StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement}
###### Remark
**([[stable cohomotopy]] as [[algebraic K-theory]] over the [[field with one element]])**

Notice that for $F$ a [[field]], the [[K-theory of a permutative category]] of its [[category of modules]] $F Mod$ is its [[algebraic K-theory]] $K F$ (see [this example](K-theory+of+a+permutative+category#OrdinaryAlgebraicKTheoryFromPermutativeCategoryOfProjectiveModules))

$$
  K F \;\simeq\; K(F Mod)
  \,.
$$

Now ([[pointed sets|pointed]]) [[finite sets]] may be regarded as the modules over the "[[field with one element]]" $\mathbb{F}_1$ (see [there](field+with+one+element#Modules)):

$$
  \mathbb{F}_1 Mod
  \;=\;
  FinSet^{\ast/}
$$

If this is understood, example \ref{StableCohomotopyIsKTheoryOfFinSet} says that [[stable cohomotopy]] is the algebraic K-theory of the [[field with one element]]:

$$
  \mathbb{S} \;\simeq\; K \mathbb{F}_1
  \,.
$$

=--

This perspective is highlighted for instance in ([Deitmar 06, p. 2](#Deitmar06), [Guillot 06](#Guillot06), [Mahanta 17](#Mahanta17), [Dundas-Goodwillie-McCarthy 13, II 1.2](#DundasGoodwillieMcCarthy13), [Morava](#MoravaSomeBackground), [Connes-Consani 16](#ConnesConsani16)).

The perspective that the [[K-theory]] $K \mathbb{F}_1$ over $\mathbb{F}_1$ should be [[stable Cohomotopy]] has been highlighted in ([Deitmar 06, p. 2](#Deitmar06), [Guillot 06](#Guillot06), [Mahanta 17](#Mahanta17), [Dundas-Goodwillie-McCarthy 13, II 1.2](#DundasGoodwillieMcCarthy13), [Morava](#MoravaSomeBackground), [Connes-Consani 16](#ConnesConsani16)). Generalized to [[equivariant stable homotopy theory]], the statement that [[equivariant K-theory]] $ K_G \mathbb{F}_1$ over $\mathbb{F}_1$ should be [[equivariant stable Cohomotopy]] is discussed in [Chu-Lorscheid-Santhanam 10, 5.3](#ChuLorscheidSanthanam10).

[[!include Segal completion -- table]]

## Examples

### Equivariant stable $\pi_3^{\mathbb{S}}$

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

### Of the point: The Burnside ring

+-- {: .num_prop #BurnsideRingIsEquivariantStableCohomotopyOfPoint}
###### Proposition
**([[Burnside ring is equivariant stable cohomotopy of the point]])**

Let $G$ be a [[finite group]], then its [[Burnside ring]] $A(G)$ is [[isomorphism|isomorphic]] to the [[equivariant stable cohomotopy]] [[cohomology ring]] $\mathbb{S}_G(\ast)$ of the [[point]] in degree 0.

$$
  A(G) \overset{\simeq}{\longrightarrow} \mathbb{S}_G(\ast)
  \,.
$$

=--

This is due to [Segal 71](#Segal71), a detailed proof is given by [tom Dieck 79, theorem 8.5.1](#tomDieck79). See also [Lück 05, theorem 1.13](#Lueck05), [tom Dieck-Petrie 78](#tomDieckPetrie78).

[[!include Segal completion -- table]]

More explicitly, this means that the Burnside ring of a group $G$ is isomorphic to the [[colimit]]

$$
  A(G) \simeq \underset{\longrightarrow}{\lim}_V [S^V,S^V]_G
$$

over $G$-[[representations]] in a complete [[G-universe]], of $G$-[[homotopy classes]] of $G$-equivariant based [[continuous functions]] from the [[representation sphere]] $S^V$ to itself ([Greenlees-May 95, p. 8](#GreenleesMay95)).

## Related concepts

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

* [[Burnside ring]]

* [[Segal conjecture]], [[Sullivan conjecture]]

## References

### Relation to Burnside ring

Relation to [[Burnside ring]]:

* {#Segal71} [[Graeme Segal]], _Equivariant stable homotopy theory_, In Actes du Congrès International des Math ématiciens (Nice, 1970), Tome 2 , pages 59–63. Gauthier-Villars, Paris, 1971 ([[SegalEquivariantStableHomotopyTheory.pdf:file]])

* {#tomDieckPetrie78} [[Tammo tom Dieck]], T. Petrie,  _Geometric modules over the Burnside ring_, Invent. Math. 47 (1978) 273-287 ([pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/tomDieck-geometric.pdf))

* {#tomDieck79} [[Tammo tom Dieck]], _[[Transformation Groups and Representation Theory]]_, Springer 1979

* {#tomDieck87} [[Tammo tom Dieck]], Section II.8 of: _[[Transformation Groups]]_, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

### Relation to Segal-Carlsson completion theorem

Relation to [[Segal-Carlsson completion theorem]]:

* Czes Kosniowski, _Equivariant cohomology and Stable Cohomotopy_, Math. Ann. 210, 83-104 (1974) ([doi:10.1007/BF01360033](https://doi.org/10.1007/BF01360033) [pdf](https://link.springer.com/content/pdf/10.1007/BF01360033.pdf))

* Erkki Laitinen, _On the Burnside ring and stable cohomotopy of a finite group_,  Mathematica Scandinavica Vol. 44, No. 1 (August 30, 1979), pp. 37-72 ([jstor:24491306](https://www.jstor.org/stable/24491306), [[Laitinen79.pdf:file]])

* [[Gunnar Carlsson]], _Equivariant Stable Homotopy and Segal's Burnside Ring Conjecture_,  Annals of Mathematics Second Series, Vol. 120, No. 2 (Sep., 1984), pp. 189-224 ([jstor:2006940](https://www.jstor.org/stable/2006940), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/carlsson.pdf))

* {#Lueck05} [[Wolfgang Lück]], _The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups_, Pure Appl. Math. Q.  1 (2005), no. 3, Special Issue: In memory of Armand Borel. Part 2, 479--541 ([arXiv:math/0504051](https://arxiv.org/abs/math/0504051))

* Noe Barcenas, _Nonlinearity, Proper Actions and Equivariant Stable Cohomotopy_ ([arXiv:1302.1712](https://arxiv.org/abs/1302.1712))

A lift of [[Seiberg-Witten invariants]] to classes in [[circle group]]-equivariant stable cohomotopy is discussed in

* Stefan Bauer, Mikio Furuta _A stable cohomotopy refinement of Seiberg-Witten invariants: I_ ([arXiv:math/0204340](http://arxiv.org/abs/math/0204340))

* Stefan Bauer, _A stable cohomotopy refinement of Seiberg-Witten invariants: II_ ([arXiv:math/0204267](http://arxiv.org/abs/math/0204267))

* Christian Okonek, Andrei Teleman, _Cohomotopy Invariants and the Universal Cohomotopy Invariant Jump Formula_, J. Math. Sci. Univ. Tokyo 15 (2008), 325-409 ([pdf](http://www.matmor.unam.mx/~barcenas/nonlinearity.pdf))

### As equivariant K-theory over the field with one element

The identification of stable cohomotopy with the [[K-theory of a permutative category|K-theory of the permutative category]] of [[finite set]] is due to

* {#BarrattPriddy72} [[Michael Barratt]], [[Stewart Priddy]], _On the homology of non-connected monoids and their associated groups_, Commentarii Mathematici Helvetici, December 1972, Volume 47, Issue 1, pp 1–14 ([doi:10.1007/BF02566785](https://doi.org/10.1007/BF02566785))

* {#Segal74} [[Graeme Segal]], _Categories and cohomology theories_, Topology vol 13, pp.  293-312,  1974  ([pdf](http://ncatlab.org/nlab/files/SegalCategoriesAndCohomologyTheories.pdf))

see also

* {#Priddy73} [[Stewart Priddy]], _Transfer, symmetric groups, and stable homotopy theory_, in _Higher K-Theories_, Springer, Berlin, Heidelberg, 1973. 244-255 ([pdf](https://link.springer.com/content/pdf/10.1007/BFb0067060.pdf))

* {#Glasman13} [[Saul Glasman]], _The multiplicative Barratt-Priddy-Quillen theorem and beyond_, talk 2013 ([pdf](http://math.mit.edu/~sglasman/bpq-beamer.pdf))

The resulting interpretation of stable cohomotopy as [[algebraic K-theory]] over the [[field with one element]] is amplified in the following texts:

* {#DundasGoodwillieMcCarthy13} [[Bjørn Dundas]], [[Thomas Goodwillie]], [[Randy McCarthy]], chapter II, section 1.2 of  _The local structure of algebraic K-theory_, Springer 2013 ([pdf](http://math.mit.edu/~nrozen/juvitop/dundas.pdf))

* {#Deitmar06} [[Anton Deitmar]], _Remarks on zeta functions and K-theory over $\mathbb{F}_1$_ ([arXiv:math/0605429](https://arxiv.org/abs/math/0605429))

* {#Guillot06} [[Pierre Guillot]], _Adams operations in cohomotopy_ ([arXiv:0612327](https://arxiv.org/abs/math/0612327))

* {#Mahanta17} [[Snigdhayan Mahanta]], _G-theory of $\mathbb{F}_1$-algebras I: the equivariant Nishida problem_, J. Homotopy Relat. Struct. 12 (4), 901-930, 2017 ([arXiv:1110.6001](https://arxiv.org/abs/1110.6001))

* {#ChuLorscheidSanthanam10} Chenghao Chu, [[Oliver Lorscheid]], Rekha Santhanam, _Sheaves and K-theory for $\mathbb{F}_1$-schemes_, Advances in Mathematics, Volume 229, Issue 4, 1 March 2012, Pages 2239-2286 ([arxiv:1010.2896](https://arxiv.org/abs/1010.2896))

see also

* {#MoravaSomeBackground} [[Jack Morava]], _Some background on Manin's theorem $K(\mathbb{F}_1) \sim \mathbb{S}$_ ([pdf](http://www.alainconnes.org/docs/Morava.pdf), [[MoravaSomeBackground.pdf:file]])

* {#ConnesConsani16} [[Alain Connes]], [[Caterina Consani]], _Absolute algebra and Segal's Gamma sets_, Journal of Number Theory 162 (2016): 518-551 ([arXiv:1502.05585](https://arxiv.org/abs/1502.05585))

* {#Berman18} [[John D. Berman]], p. 92 of: _Categorified algebra and equivariant homotopy theory_, PhD thesis 2018  ([arXiv:1805.08745](https://arxiv.org/abs/1805.08745))

### Relation to equivariant cobordism theory

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))

### In M-brane charge quantization

Discussion for [[M-brane]] physics:

* [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_ ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987))

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

* {#SatiSchreiber19} [[Hisham Sati]], [[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 stable Cohomotopy]]

[[!redirects equivariant stable cohomotopy theory]]
[[!redirects equivariant stable cohomotopy theories]]

[[!redirects equivariant stable Cohomotopy theory]]
[[!redirects equivariant stable Cohomotopy theories]]

[[!redirects stable equivariant cohomotopy]]
[[!redirects stable equivariant Cohomotopy]]


equivariant stable homotopy category

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###Context###
#### 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 generalization of the [[stable homotopy category]] from [[stable homotopy theory]] to [[equivariant stable homotopy theory]].

## Definition

A [[homomorphism]] $f \colon E_1 \longrightarrow E_2$ between two [[G-spectra]], indexed on a [[G-universe]] $\mathcal{U}$, is called an equivariant [[weak homotopy equivalence]] if the following equivalent conditions hold

1. For each $V\in \mathcal{U}$ the component map $f_V \colon f$ induces ordinary [[weak homotopy equivalences]] $(E_1)_V^H \to (E_2)_V^H$ on all [[fixed point]] spaces for all closed [[subgroups]] $H \hookrightarrow G$.

1. For each $n \in \mathbb{Z}$ and each closed subgroup $H \hookrightarrow G$ the morphism $f$ induces an [[isomorphism]] of [[Mackey functors]] of [[equivariant homotopy groups]] $\pi_n^H(E_1) \stackrel{\simeq}{\longrightarrow} \pi_n^H(E_2)$.

(The equivalence of these conditions is part of the [[equivariant Whitehead theorem]].)

The $G$-equivariant stable homotopy category is the [[homotopy category]] of [[G-spectra]] with respect to these weak equivalences.

## Properties

### Relation to Mackey functors

The  [[full subcategory]] $\mathcal{B}_G$ of the equivariant stable homotopy category on the objects of the form

$$
  G/H_* \wedge \Sigma^\infty S^n
$$

is, as an [[additive category]], the [[domain]] of [[Mackey functors]], such as the [[equivariant homotopy group]]-functors.

## References

* {#GreenleesMay95} [[John Greenlees]], [[Peter May]], section 2 of _Equivariant stable homotopy theory_, in I.M. James (ed.), _Handbook of Algebraic Topology_ , pp. 279-325. 1995. ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))

* {#Bohmann} [[Anna Marie Bohmann]], _Basic notions of equivariant stable homotopy theory_, ([pdf](https://math.vanderbilt.edu/bohmanar/basicequivnotions.pdf))


Equivariant stable homotopy theory

This page collects links related to

* [[John Greenlees]], [[Peter May]],

  _Equivariant stable homotopy theory_,

  in [[Ioan James]] (ed.),

  _[[Handbook of Algebraic Topology]]_ (1995)

  pp. 279-325.

  [doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7)

  [pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf)

on basics of [[equivariant stable homotopy theory]]

category: reference


equivariant stable homotopy theory

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###Context###
#### Stable Homotopy theory
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[[!include stable homotopy theory - contents]]
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#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Equivariant stable homotopy theory over some [[topological group]] $G$ is the [[stable homotopy theory]] of [[G-spectra]]. This includes the [[naive G-spectra]] which constitute the actual [[stabilization]] of [[equivariant homotopy theory]], but is more general, one speaks of _genuine $G$-spectra_. Notably a genuine $G$-spectrum has [[homotopy groups]] graded not by the group of [[integers]], but by the [[representation ring]] of $G$ (usually called [[RO(G)-grading]]).

The concept of [[cohomology]] in equivariant stable homotopy theory is _[[equivariant cohomology]]_:

[[!include equivariant cohomology -- table]]

## Basic definitions

### In terms of looping by representation spheres

The definition of _[[G-spectrum]]_ is typically given in generalization of the definition of [[coordinate-free spectrum]].

A **[[G-universe]]** in this context is (e.g. [Greenlees-May, p. 10](#GreenleesMay)) an infinite dimensional real [[inner product space]] equipped with a linear $G$-[[action]] that is the [[direct sum]] of [[countable|countably]] many copies of a given [[set]] of (finite dimensional? -DMR) [[representations]] of $G$, at least containing the trivial representation on $\mathbb{R}$ (so that $U$ contains at least a copy of $\mathbb{R}^\infty$).

Each such subspace of $U$ (representation contained in $U$? -DMR) is called an _indexing space_ ([[RO(G)-grading]]). For $V \subset W$ indexing spaces, write $W-V$ for the [[orthogonal complement]] of $V$ in $W$. Write $S^V$ for the [[one-point compactification]] of $V$; and for $X$ any (pointed) [[topological space]] write $\Omega^V := [S^V,X]$ for the corresponding (based) sphere space.

A **[[G-space]]** in the following means a [[pointed object|pointed]] [[topological space]] equipped with a continuous [[action]] of the [[topological group]] $G$ that fixes the base point. A  morphism of $G$-spaces is a continuous map that fixes the basepoints and is $G$-equivariant.

A weak equivalence of $G$-spaces is a morphism that induces isomorphism on all $H$-fixed homotopy groups (...)

A **$G$-spectrum** $E$ (indexed on the chosen universe $U$) is

* for each indexing space $V \subset U$ a $G$-space $E V$;

* for each pair $V \subset W$ of indexing spaces a $G$-equivariant [[homeomorphism]]

  $$
    E V \stackrel{\simeq}{\to} \Omega^{W-V} E W
    \,.
  $$

A [[morphism]] $f : E \to E'$ of $G$-spectra is for each indexing space $V$ a morphism of $G$-spaces $f_V : E V \to E' V$, such that this makes the obvious diagrams commute.

This becomes a [[category with weak equivalences]] by setting:

a morphism $f$ of $G$-spectra is a **weak equivalence of $G$-spectra** if for every indexing space $V$ the component $f_V$ is a weak equivalence of $G$-spaces (as defined above).

This may be expressed directly in terms of the notion of **homotopy group of a $G$-spectrum**: this is ...

### In terms of orthogonal spectra with $G$-action

... ([Schwede 15](#Schwede15))...

### In terms of Mackey-functors

A _[[Mackey functor]]_ with values in [[spectra]] ("spectral Mackey functor") is an [[(∞,1)-functor]] on a suitable  [[(∞,1)-category of correspondences]]
$Corr_1^{eff}(\mathcal{C}) \hookrightarrow Corr_1(\mathcal{C})$ which sends [[coproducts]] to [[smash product]]. (This is similar to the concept of [[sheaf with transfer]].)

$$
  S \;\colon\; Corr_1^{eff}(\mathcal{C}) \longrightarrow Spectra
$$

For $G$ a [[finite group]] and $\mathcal{C}= G Set$ its category of [[permutation representations]], we have that $S$ is a genuine $G$-[[equivariant spectrum]] ([Guillou-May 11](#GuillouMay11)).
 So in this case the [[homotopy theory]] of spectral Mackey functors is a presentation for [[equivariant stable homotopy theory]] ([Guillou-May 11](#GuillouMay11), [Barwick 14](#Barwick14)).

For $\mathcal{C}$ an [[abelian category]] this definition reduces ([Barwick 14](#Barwick14)) Mackey functors as originally defined in ([Dress 71](#Dress71)).

## Examples

* [[equivariant sphere spectrum]], [[equivariant stable cohomotopy]]

* [[equivariant bordism homology theory]], [[equivariant cobordism cohomology theory]]

* [[equivariant connective topological K-theory]]

[[!include Segal completion -- table]]

* [[KR cohomology theory]]

* [[equivariant complex cobordism cohomology theory]]

* [[equivariant Morava K-theory]]

## Equivariant cohomology

The notion of [[cohomology]] relevant in equivariant stable homotopy theory is the flavor of [[equivariant cohomology]] (see there for details) called [[Bredon cohomology]]. (See also at _[[orbifold cohomology]]_.)

## Related concepts

* [[equivariant differential topology]]

* [[global equivariant homotopy theory]]

* [[rational equivariant stable homotopy theory]]

* [[equivariant motivic homotopy theory]]

* [[equivariant spectrum]],

  [[equivariant sphere spectrum]], [[equivariant suspension spectrum]], [[equivariant homotopy group]], [[equivariant stable homotopy category]], [[tom Dieck splitting]], [[slice spectral sequence]]

* [[G-spectra]], [[naive G-spectra]], [[spectra with G-action]]

  [[G-spaces]]

* [[Burnside category]], [[Burnside ring]]

  [[Mackey functor]]

## References

Original articles include

* {#Segal71} [[Graeme Segal]], _Equivariant stable homotopy theory_, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 59--63. Gauthier-Villars, Paris, 1971. ([[SegalEquivariantStableHomotopyTheory.pdf:file]])

Textbook accounts:

* {#LewisMaySteinberger86} [[L. Gaunce Lewis]], [[Peter May]], [[Mark Steinberger]] (with contributions by [[James McClure|J.E. McClure]]), _Equivariant stable homotopy theory_, Springer Lecture Notes in Mathematics  **1213** (1986) [[pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf), [doi:10.1007/BFb0075778](https://link.springer.com/book/10.1007/BFb0075778)]

* {#tomDieck87} [[Tammo tom Dieck]], Section II.6 of: _[[Transformation Groups]]_, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

and a more modern version taking into account the theory of [[symmetric monoidal categories of spectra]] is in

* [[Michael  Mandell]], [[Peter May]], _Equivariant orthogonal spectra and S-modules_, Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108. MR 2003i:55012 ([pdf](http://www.math.uiuc.edu/K-theory/0408/MMM.pdf), [K-theory archive](http://www.math.uiuc.edu/K-theory/0408/))

Solving the [[Arf-Kervaire invariant problem]] with methods of [[equivariant stable homotopy theory]] (and reviewing these):

* {#HillHopkinsRavenel09} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], *On the non-existence of elements of Kervaire invariant one*, Annals of Mathematics **184** 1 (2016)[[doi:10.4007/annals.2016.184.1.1](https://doi.org/10.4007/annals.2016.184.1.1), [arXiv:0908.3724](http://arxiv.org/abs/0908.3724), [talk slides](https://www.math.rochester.edu/people/faculty/doug/otherpapers/Skye_handout.pdf)]

* {#HillHopkinsRavenel21} [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], *Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem*, New Mathematical Monographs, Cambridge University Press (2021) [[doi:10.1017/9781108917278](https://doi.org/10.1017/9781108917278)]

Lecture notes are in

* {#Blumberg17} [[Andrew Blumberg]], _The Burnside category_, 2017 ([pdf](https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf), [GitHub](https://github.com/adebray/equivariant_homotopy_theory))

Further introductions and surveys include the following

* {#Carlsson92} [[Gunnar Carlsson]], _A survey of equivariant stable homotopy theory_,Topology, Vol 31, No. 1, pp. 1-27, 1992 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/carlsson1.pdf))

* {#Bohmann} [[Anna Marie Bohmann]], _Basic notions of equivariant stable homotopy theory_ ([pdf](https://math.vanderbilt.edu/bohmanar/basicequivnotions.pdf))

* {#GreenleesMay} [[John Greenlees]], [[Peter May]], _[[Equivariant stable homotopy theory]]_, in [[Ioan James]] (ed.), _[[Handbook of Algebraic Topology]]_ , pp. 279-325. 1995. ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))

* [[Brooke Shipley]], _An introduction to equivariant homotopy theory_ ([pdf](http://homepages.math.uic.edu/~bshipley/bonn.2.pdf))

* [Talbot workshop 2016](http://math.mit.edu/conferences/talbot/index.php?year=2016)

Lecture notes on [[G-spectra]] modeled as [[orthogonal spectra]] with $G$-actions are

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

An alternative perspective on this is in

* [[Mark Hovey]], [[David White]], _An alternative approach to equivariant stable homotopy theory_ ([arXiv:1312.3846](http://arxiv.org/abs/1312.3846))

Generalization from equivariance under [[compact Lie groups]] to [[compact topological groups]] (Hausdorff) and in particular to [[profinite groups]] and [[pro-homotopy theory]] is in

* {#Fausk06} [[Halvard Fausk]], _Equivariant homotopy theory for pro-spectra_ ([arXiv:math/0609635](http://arxiv.org/abs/math/0609635))

The [[May recognition theorem]] for [[G-spaces]] and [[genuine G-spectra]] is discussed in

* Costenoble and Warner, _Fixed set systems of equivariant infinite loop spaces_ Transactions of the American mathematical society, volume 326, Number 2 (1991) ([JSTOR](http://www.jstor.org/pss/2001770))

Characterization of [[G-spectra]] [via excisive functors](spectrum+object#ViaExcisiveFunctors) on [[G-spaces]] is in

* [[Andrew Blumberg]], _Continuous functors as a model for the equivariant stable homotopy category_ ([arXiv:math.AT/0505512](http://arxiv.org/abs/math.AT/0505512))

The characterization of $G$-equivariant functors in terms of topological [[Mackey functors]] is discussed in example 3.4 (i) of

* [[Stefan Schwede]], [[Brooke Shipley]], _Classification of stable model categories_ ([pdf](http://hopf.math.purdue.edu/Schwede-Shipley/class.final.pdf))

A construction of equivariant stable homotopy theory in terms of [[spectral Mackey functors]] is due to

* [[Bert Guillou]], [[Peter May]],  _Models of $G$-spectra as presheaves of spectra, ([arXiv:1110.3571](http://arxiv.org/abs/1110.3571))

  _Permutative $G$-categories in equivariant infinite loop space theory ([arXiv:1207.3459](http://arxiv.org/abs/1207.3459))

see at _[[spectral Mackey functor]]_ for more references.

A fully [[(∞,1)-category theory|(∞,1)-category theoretic]] formulation:

* {#Barwick14} [[Clark Barwick]], _Spectral Mackey functors and equivariant algebraic K-theory (I)_, Adv. Math., 304:646–727 ([arXiv:1404.0108](http://arxiv.org/abs/1404.0108))

A [[universal property]] characterizing equivariant stable homotopy theory:

* [[Tim Campion]], _Free duals and a new universal property for stable equivariant homotopy theory_ [[arXiv:2302.04207](https://arxiv.org/abs/2302.04207)]

[[!redirects stable equivariant homotopy theory]]


equivariant structure

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## Idea

Given a [[structure]] on some [[space]] $X$ which is equipped with the [[action]] of a [[group]] $G$, then a lift to a _equivariant structure_ is an extension/[[descent]] to the [[action groupoid]] $X//G$.

## Examples

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

## Related concepts

* [[equivariance]]

* [[general covariance]], [[gauge invariance]], [[principle of equivalence]]

* [[equivariant cohomology]]

* [[equivariant homotopy theory]]

* [[equivariant stable homotopy theory]]

* [[global equivariant stable homotopy theory]]

* [[rational equivariant stable homotopy theory]]

[[!redirects equivariant structures]]


equivariant Sullivan model

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## Idea

Equivariant Sullivan models are [[equivariant dgc-algebras]] that generalize the [[Sullivan model]] [[dgc-algebras]] in [[rational homotopy theory]] to [[equivariant rational homotopy theory]].

## Definition

[Triantafillou 82, Construction 5.10](#Triantafillou82), [Scull 01, Section 11](#Scull01), [Scull 08, Section 4](#Scull08)

## 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))

* {#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))

* {#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))

For [[cyclic groups]]:

* Thierry Lambre, _Modèle Minimal Équivariant et Formalité_, Transactions of the American Mathematical Society Vol. 327, No. 2 (Oct., 1991), pp. 621-639 ([doi:2001817](https://www.jstor.org/stable/2001817))

[[!redirects equivariant Sullivan models]]


equivariant suspension spectrum

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## Idea

The generalization of the concept of [[suspension spectrum]] from [[stable homotopy theory]] to $G$-[[equivariant stable homotopy theory]].

## Definition

### As a spectrum indexed on a $G$-universe

Let $X$ be a [[pointed topological space|pointed]] [[topological G-space]]. For a [[representation]] $V$ in the [[G-universe]] $U$, write $S^V$ for its [[representation sphere]].

As a [[G-spectrum]] indexed on a [[G-universe]]:

1. the _suspension $G$-pre-spectrum_ is $\Pi^\infty X \colon V \mapsto S^V \wedge X$;

1. the suspension $G$-spectrum is $\Sigma^\infty X \colon V \mapsto Q(S^V \wedge X)$

where $Q (-) = \underset{V \in U}{\cup} \Omega^V \Sigma^V X$.

([Greenlees-May 95](#GreenleesMay95))

### As an orthogonal spectrum

The _equivariant suspension spectrum_ $\Sigma^\infty_G X$ of a [[pointed topological space|pointed]] [[topological G-space]] $X$ is the [[G-spectrum]] which, modeled as an [[orthogonal spectrum]] with $G$-action, is in degree $n$ given by the [[smash product]]

$$
  (\Sigma^\infty_G X)_n \coloneqq X \wedge S^n
$$

of $X$ with the [[n-sphere]], equipped with the canonical [[action]] of the [[orthogonal group]] $O(n)$ just on the $S^n$-factor and equipped with the given action  of $G$ on just $X$.

(e.g. [Schwede 15, example 2.11](#Schwede15))

## Properties
 {#Properties}

### Monoidalness
 {#Monoidalness}

In generalization to the [[strong monoidal functor|strong monoidal]]-[[structure]] on the ordinary [[suspension spectrum]] functor with respect to the [[symmetric monoidal smash product of spectra]] (see [there](suspension+spectrum#StrongMonoidalness))
also the equivariant suspension spectrum functor ought to constitute a [[monoidal (infinity,1)-functor]] from $G$-[[equivariant homotopy theory]] to $G$-[[equivariant stable homotopy theory]].

This follows from general properties of [[stabilization]] when regarding [[equivariant stable homotopy theory]] as the result of inverting [[smash product]] with all [[representation spheres]], via [Robalo 12, last clause of Prop. 4.1 with last clause of Prop. 4.10 (1) ](stabilization#Robalo12), generalized to sets of objects as in [Hoyois 15, section 6.1](stabilization#Hoyois15), see also [Hoyois 15, Def. 6.1](stabilization#Hoyois15).

Alternatively, under the [[equivalence of (infinity,1)-categories|equivalence]] of [[genuine G-spectra]] with [[spectral Mackey functors]] on the [[Burnside category]], it follows as in [Nardin 12, Remark A.12](#Nardin12).

### $RO(G)$-degrees
 {#ROGDegrees}

For $V$ an orthogonal linear $G$-[[representation]] then the value of the equivariant suspension spectrum in that [[RO(G)-grading|RO(G)-degree]] is the [[smash product]] of $X$ with the corresponding [[representation sphere]].

$$
  (\Sigma^\infty_G X)(V) \simeq X \wedge S^V
$$

### Relation between genuine and Bredon-equivariant suspension

For $U$ a complete $G$-[[universe]] and $U^G$ its [[fixed point]] universe, then the inclusion $i \colon U^G \longrightarrow U$ induces an [[adjunction]]

$$
  Spectra(G Top) \stackrel{\overset{i^\ast}{\longleftarrow}}{\underset{i_\ast}{\longrightarrow}}G Spectra
$$

between [[naive G-spectra]] and [[genuine G-spectra]]. The genuine $G$-suspension spectrum is the naive $G$-suspension spectrum followed by $i$:

$$
  i_\ast \circ \Sigma^\infty_{U^G }  \simeq \Sigma^\infty_U
  \,.
$$

([Greenlees-May 95, p. 16](#GreenleesMay95))

## Examples

The $G$-[[equivariant sphere spectrum]] is

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

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]] $\mathbb{S}(V) \simeq S^V$.

### Equivariant homotopy groups and fixed point spectra

The [[equivariant homotopy groups]] and the [[fixed point spectra]] of equivariant suspension spectra $\Sigma^\infty_G X$ decompose into the naive [[fixed points]] of the $G$-action on $X$. This is the _[[tom Dieck splitting]]_, see there for details.

## References

* {#Blumberg17} [[Andrew Blumberg]], Example 2.5.9 in _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))

* {#GreenleesMay95} [[John Greenlees]], [[Peter May]], _Equivariant stable homotopy theory_, in I.M. James (ed.), _Handbook of Algebraic Topology_ , pp. 279-325. 1995. ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))

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

Discussion in terms of [[spectral Mackey functors]] on the [[Burnside category]]:

* {#Nardin12} [[Denis Nardin]], section A.2 of _Stability and distributivity over orbital ∞-categories_, 2012 ([hdl.handle.net/1721.1/112895]( http://hdl.handle.net/1721.1/112895), [pdf](https://www.math.univ-paris13.fr/~nardin/thesis.pdf))

[[!redirects equivariant suspension spectra]]


equivariant symmetric monoidal category

#Content#
* table of contents
{:toc}

## Idea

An _equivariant symmetric monoidal category_ ([Hill-Hopkins 16](#HillHopkins16)) is like a [[symmetric monoidal category]] but with the symmetric monoidal [[tensor product]] generalized to symmetric monoidal powers indexed by [[finite set|finite]] [[G-sets]], for some [[group]] $G$.

Motivating applications come from [[equivariant homotopy theory]].

## References

* {#HillHopkins16} [[Michael Hill]], [[Michael Hopkins]], _Equivariant symmetric monoidal structures_ ([arXiv:1610.03114](https://arxiv.org/abs/1610.03114))

[[!redirects equivariant symmetric monoidal categories]]


equivariant Tietze extension theorem

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* table of contents
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## Idea

A generalization of the [[Tietze extension theorem]] to [[equivariant functions]] provides conditions under which a [[continuous function|continuous]] and [[equivariant function]] from a [[subspace]] of a [[topological G-space]] to another [[topological G-space]] has an [[extension]] to a [[continuous function|continuous]] and [[equivariant function]] to the full $G$-space.

## Statement

+-- {: .num_theorem #TietzeGleasonExtensionTheorem}
###### Theorem
**(Tietze-Gleason extension theorem)

Let

* $G$ be a [[compact Lie group]],

* $X$ a [[topological G-space]],

* $A \subset X$ a [[topological subspace|sub]] [[G-space]] of $X$,

* $E$ be a [[finite-dimensional vector space|finite-dimensional]] [[orthogonal group|orthogonal]] $G$-[[linear representation]], regarded as a [[Euclidean space|Euclidean]] [[topological G-space]],

* $A \overset{f}{\longrightarrow} E$ a $G$-[[equivariant function|equivariant]] [[continuous function]].

If

* $A \subset X$ is a [[compact subspace]]

[[or]]

* $X$ is a [[normal topological space]] [[and]] $A \subset X$ is a [[closed subspace]]

then $f$ has an [[extension]] to an [[equivariant function|equivariant]] [[continuous function]] $\widehat f$ on all of $X$.

$$
  \array{
    A &\overset{f}{\longrightarrow}& E
    \\
    \cap & \nearrow_{\mathrlap{ \widehat{f} }}
    \\
    X
  }
$$

=--

([Gleason 50](#Gleason50), see [Palais 60, Theorem 1.4.3](#Palais60))

Other/more general conditions for the equivariant extension to exist:

\begin{prop}\label{JaworowskiExtensionTheorem}
**(Jaworowski-extension theorem)** \linebreak

If

1. the ambient [[domain]] [[G-space]] $X$ is a

   1. [[locally compact topological space|locally comact]]

   1. [[separable metric space]]

   1. of [[finite number|finite]] [[dimension of a separable metric space|dimension]]

   1. with a [[finite number]] of [[orbit type|orbit types]];

1. the [[domain]] $A \subset X$ is a

   * [[closed subspace|closed]]

   * [[topological subspace|sub]]-[[G-space]];

1. the [[codomain]] [[G-space]] is a

   1. [[locally compact topological space|locally comact]]

   1. [[separable metric space]]

   1. such that for every $G$-[[orbit type]] $(H)$ in the [[complement]] $X \setminus A$

      the [[fixed locus]] $E^H$ is an [[absolute neighbourhood retract]].

Then every [[continuous function]] $f \colon A \to E$ has an [[extension]] to a $G$-[[equivariant function|equivariant]] [[continuous function]] $\widehat f$ on an [[open neighbourhood]] $A \subset O_A \subset X$

$$
  \left(
    \underset{ G/H \subset X \setminus A  }{\forall}
    \;
    E^H
    \;\;
    \text{is absolute neighbourhood retract}
  \right)
  \;\;\;\;
    \Rightarrow
  \;\;\;\;
  \left(
    \underset{\widehat f}{\exists}
    \;\;\;\;
    \array{
      A &\overset{f}{\longrightarrow}& E
      \\
      \cap
      &
      \nearrow_{\mathrlap{ \widehat{f} }}
      \\
      O_A
      \\
      \cap
      \\
      X
     }
   \right)
   \,.
$$

Moreover, if the above [[fixed loci]] $E^H$ are even [[absolute retracts]], then an extension $\widehat f$ exists on all of $X$:

$$
  \left(
    \underset{ G/H \subset X \setminus A  }{\forall}
    \;
    E^H
    \;\;
    \text{is absolute retract}
  \right)
  \;\;\;\;
    \Rightarrow
  \;\;\;\;
  \left(
    \underset{\widehat f}{\exists}
    \;\;\;\;
    \array{
      A &\overset{f}{\longrightarrow}& E
      \\
      \cap
      &
      \nearrow_{\mathrlap{ \widehat{f} }}
      \\
      X
     }
   \right)
   \,.
$$

\end{prop}

([Jaworowski 76](#Jaworowski76), [Lashof 81](#Lashof81))

## Related concepts

[[!include extension theorems -- table]]

* [[equivariant principal bundle]]

* [[slice theorem]]

## References
 {#References}

* {#Gleason50} [[Andrew Gleason]], _Spaces with a compact Lie group of transformations_,  Proc. Amer. Math. Soc. 1 (1950), 35-43 ([doi:10.1090/S0002-9939-1950-0033830-7](https://doi.org/10.1090/S0002-9939-1950-0033830-7))

* {#Palais60} [[Richard Palais]], Theorem 1.4.3 in: _The classification of $G$-spaces_, Memoirs of the American Mathematical Society, Number 36, 1960 ([ISBN:978-0-8218-9979-3](https://bookstore.ams.org/memo-1-36) [pdf](http://vmm.math.uci.edu/PalaisPapers/ClassificationOfG-Spaces.pdf), [[Palais_ClassificationOfGSpaces.pdf:file]])

* {#Jaworowski73} [[Jan Jaworowski]], _Equivariant extensions of maps_, Pacific J. Math. Volume 45, Number 1 (1973), 229-244 ([euclid:1102947720](https://projecteuclid.org/euclid.pjm/1102947720))

* {#Jaworowski73b} [[Jan Jaworowski]], _Extending equivariant maps for compact Lie group actions_, Bull. Amer. Math. Soc. Volume 79, Number 4 (1973), 698-701 ([euclid:1183534741](https://projecteuclid.org/euclid.bams/1183534741))

* {#Jaworowski76} [[Jan Jaworowski]], _Extensions of $G$-maps and Euclidean $G$-retracts_, Math Z (1976) 146: 143 ([doi:10.1007/BF01187702](https://doi.org/10.1007/BF01187702))

* {#Jaworowski81} [[Jan Jaworowski]], _An equivariant extension theorem and $G$-retracts with a finite structure_, Manuscripta Math (1981) 35: 323 ([doi:10.1007/BF01263266](https://doi.org/10.1007/BF01263266))

* {#Lashof81} [[Richard Lashof]], _The Equivariant Extension Theorem_, Proceedings of the American Mathematical Society Vol. 83, No. 1 (Sep., 1981), pp. 138-140 ([jstor:2043909](https://www.jstor.org/stable/2043909))

[[!redirects equivariant Tietze extension theorems]]

[[!redirects Tietze-Gleason extension theorem]]
[[!redirects Tietze-Gleason extension theorems]]

[[!redirects equivariant extension theorem]]
[[!redirects equivariant extension theorems]]


equivariant triangulation

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## Idea

For $X$ a [[manifold]] equipped with an [[action]] of a [[group]] $G$, a _$G$-equivariant triangulation_ of $X$ is a [[triangulation]] of $X$ which is compatible with this $G$-[[action]], in that this action restricts to [[bijections]] on sets of $k$-[[cells]] of the triangulation, for each $k \in \mathbb{N}$.

## Properties

The _[[equivariant triangulation theorem]]_ ([Illman 78](#Illman78), [Illman 83](#Illman83)) asserts that for $G$ a [[compact Lie group]] (for instance a [[finite group]]) equivariant triangulations of [[smooth manifolds]] always exist.

## Related concepts

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

* [[equivariant triangulation theorem]]

* [[equivariant differential topology]]

## References

* {#Illman78} [[Sören Illman]], _Smooth equivariant triangulations of $G$-manifolds for $G$ a finite group_, Math. Ann. (1978) 233: 199 ([doi:10.1007/BF01405351](https://doi.org/10.1007/BF01405351))

* {#Illman83} [[Sören Illman]], _The Equivariant Triangulation Theorem for Actions of Compact Lie Groups_, Mathematische Annalen (1983) Volume: 262, page 487-502 ([dml:163720](https://eudml.org/doc/163720))

[[!redirects equivariant triangulations]]

[[!redirects G-equivariant triangulation]]
[[!redirects G-equivariant triangulations]]


equivariant triangulation theorem

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#Contents#
* table of contents
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## Idea

The _equivariant triangulation theorem_ ([Illman 78](#Illman78), [Illman 83](#Illman83)) says that for $G$ a [[compact Lie group]] (for instance a [[finite group]]) and $X$ a [[compact topological space|compact]] [[smooth manifold]] equipped with a [[smooth function|smooth]] $G$-[[action]], there exists a $G$-[[equivariant triangulation]] of $X$.

([Illman 72, Thm. 3.1](#Illman72), [Illman 83, theorem 7.1, corollary 7.2](#Illman83)) Recalled as ([ALR 07, theorem 3.2](#ALR07)). See also [Waner 80, p. 6](#Waner80) who attributes this to [Matumoto 71](#Matumoto71)

Moreover, if the manifold does have a [[manifold with boundary|boundary]], then its G-CW complex may be chosen such that the boundary is a G-subcomplex. ([Illman 83, last sentence above theorem 7.1](#Illman83))

These results continue to hold when $G$ is not [[compact]], see [Illman00](#Illman00).

## Applications

* [[G-representation spheres are G-CW-complexes]]

## Related concepts

* [[triangulation theorem]]

* [[equivariant good open cover]]

* [[equivariant differential topology]]

## References

* {#Illman72} [[Sören Illman]], Theorem 3.1 in: *Equivariant algebraic topology*, Princeton University 1972 ([[Illman_EquivariantAlgebraicTopology1972.pdf:file]])

* {#Illman78} [[Sören Illman]], _Smooth equivariant triangulations of $G$-manifolds for $G$ a finite group_, Math. Ann. **233** (1978) 199-220 [[doi:10.1007/BF01405351](https://doi.org/10.1007/BF01405351)]

* {#Illman83} [[Sören Illman]], _The Equivariant Triangulation Theorem for Actions of Compact Lie Groups_, Mathematische Annalen **262** (1983) 487-502 [[dml:163720](https://eudml.org/doc/163720)]

* {#Illman00} [[Sören Illman]], _Existence and uniqueness of equivariant triangulations of smooth proper $G$-manifolds with some applications to equivariant Whitehead torsion_,  J. Reine Angew. Math. **524** (2000) 129–183.  [[doi:10.1515/crll.2000.054](https://doi.org/10.1515/crll.2000.054)]

See also

*  {#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]])

* {#Waner80} [[Stefan Waner]], _Equivariant Homotopy Theory and Milnor's Theorem_, Transactions of the American Mathematical Society Vol. 258, No. 2 (Apr., 1980), pp. 351-368 ([JSTOR](http://www.jstor.org/stable/1998061))

* {#ALR07} A. Adem, J. Leida and Y. Ruan, _Orbifolds and Stringy Topology_, Cambridge Tracts in Mathematics **171** (2007) ([pdf](http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf))

[[!redirects equivariant triangulation theorems]]

[[!redirects smooth triangulation]]
[[!redirects smooth triangulations]]


equivariant tubular neighbourhood

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## Idea

The notion of _equivariant tubular neighbourhood_ (often called "invariant"!) is the generalization of the notion of _[[tubular neighbourhood]]_ from [[differential topology]] to [[equivariant differential topology]].

## Definition

+-- {: .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 $\Sigma$.

=--

## Properties

### Existence

+-- {: .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).

## References

* {#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))

* {#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))

* {#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))

[[!redirects equivariant tubular neighbourhoods]]

[[!redirects invariant tubular neighbourhood]]
[[!redirects invariant tubular neighbourhoods]]


equivariant tubular neighbourhoods

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
=--
=--

#Contents#
* table of contents
{: toc}

### Introduction ###

Let $M$ be a finite dimensional smooth manifold. Its [[smooth
loop space]], $L M \coloneqq C^{\infty}(S^{1},M)$, is an example
of a [[mapping space]] and is one of the simplest and most
commonly encountered mapping spaces. It has many interesting
submanifolds: some are defined by *coincidences*, such as the
[[based loop space]], and others are defined by considering fixed
point sets of the natural circle action. An extremely useful tool
when studying a submanifold is a tubular neighbourhood. Not all
submanifolds have tubular neighbourhoods, so it is good to
establish some results that establish that they exist. The
situation for coindicence submanifolds is considered in [[tubular
neighbourhood of a mapping space]]. Here we shall consider
submanifolds defined by the circle action.

Although in the above we concentrated on the loop space, these
results generalise to more general mapping spaces.

+-- {: .num_theorem}
###### Theorem ######
Let $E \to B$ be a fibre bundle with typical fibre $S$. Suppose
that $E$ is compact, and that $S$ has a measure such that the
transition functions are measure-preserving. Consider the
submanifold of $C^{\infty}(E,M)$ of maps $E \to M$ which are
constant on the fibres of $E \to B$. Then this submanifold has a
tubular neighbourhood which is invariant under fibrewise
measure-preserving diffeomorphisms of $E \to B$.
=--

As a corollary of this, $G \subseteq S^{1}$ is a compact subgroup
then the fixed point set $L M^{G}$ is a submanifold of $L M$ with
an equivariant tubular neighbourhood.

This theorem is itself a corollary of a more specific situation
where the base consists of a single point. In this case, we
consider maps $S \to M$ and look at the submanifold of constant
maps. We construct a tubular neighbourhood for this that is
invariant as described, and the invariance allows us to extend
the result to the more general situation.

The construction of the tubular neighbourhood depends on the
existence of a *local averaging* function. This should be
compared with the [[local addition]] which was needed to
construct the [[manifold structure of mapping spaces|manifold
structure]] on the mapping space in the first place. A *local
addition* is a way of saying: if two points are close, we can
move one of them to the other. A *local averaging* function
generalises this to a family of points indexed by some group and
says that if such a family is close then we can move all of them
towards each other in an equivariant fashion.

The reason that this is needed is because we want to define what
it means for a map to be close to one in the fixed point set. The
simplest case is where the source is a two point set, say
$\{1,-1\}$. We want to say what it means for a map $\{1,-1\} \to
M$ to be "close" to a constant map, but we want to say this in
such a way that if we swap the labels, nothing changes.

### A Local Averaging Function ###

The crucial ingredient in this is what we call a *local averaging
function*. For a local averaging function, we need to be able to
take a smooth map $\alpha \colon S \to M$ which is "small" in
some sense, and define a single point in $M$ which is the
"average" of $\alpha$. This should invariant under the action of
measure-preserving diffeomorphisms of $S$ on $\alpha$.

Let us illustrate this with $S = \{1,-1\}$. So we have $\alpha
\colon \{1,-1\} \to M$, which means two points $a_{1}, a_{-1} \in
M$. We need to give a general rule as to how to move them
together to a point $a_{0} \in M$ with the property that if we
swap $a_{1}$ and $a_{-1}$ then we produce the same point. Once we
have $a_{0}$ we can use a local addition (based at $a_{0}$) to
move both $a_{1}$ and $a_{-1}$ towards it. This will be invariant
under the group action because it only depends on the points
$a_{1}$ and $a_{-1}$ and not on the labels.
Thus the key is to find $a_{0}$, the "average" of $a_{1}$ and
$a_{-1}$. If $M = \mathbb{R}^{n}$ then there is an obvious such
point: $\frac{1}{2}a_{1} + \frac{1}{2}a_{-1}$. However, this will
not work on $M$ because the sum does not make sense. A local
addition could solve this, except that a local addition requires
a choice of point and that is precisely what we are trying to
avoid.

Our solution is to put $M$ inside $\mathbb{R}^{n}$. In
$\mathbb{R}^{n}$, we can form $a_{0} \coloneqq \frac{1}{2}a_{1} +
\frac{1}{2}a_{-1}$. However, the point $a_{0}$ may not lie on
$M$, so we need a way to return it to $M$. We can do that if we
know that $a_{0}$ is in a tubular neighbourhood of $M$. Then
having found $a_{0}$, we can use a local addition at $a_{0}$ to
move $a_{1}$ and $a_{-1}$ together.

There is one more subtlety. As we move $a_{1}$ and $a_{-1}$
together, we want to ensure that their "average" point stays at
$a_{0}$. That is, we need to ensure that the local addition and
the local average are independent. To see that this is a
reasonable thing, note that the local addition is defined using
the tangent space of $M$ whilst the tubular neighbourhood uses
the normal bundle of the embedding. Thus their effects should be
separable. To make this concrete, we use the orthogonal structure
on the ambient Euclidean space.

Thus we start with an embedding of $M$ in some Euclidean space,
say $\iota \colon M \to \mathbb{R}^{n}$. Then $d_{p} \iota \colon
T_{p} M \to T_{\iota(p)} \mathbb{R}^{n}$ embeds $T_{p} M$ in
$T_{\iota(p)} \mathbb{R}^{n}$. Using the orthogonal structure on
$\mathbb{R}^{n}$, we can form the [[orthogonal complement]] of $T_{p}
M$ in $T_{\iota(p)} \mathbb{R}^{n}$. Let us write this as
$N_{p}$. Then $N_{p}$ is the fibre of the normal bundle of the
embedding at $p$.

Let us identify $M$ with its image. Let us also identify $T_{x}
\mathbb{R}^{n}$ with $\mathbb{R}^{n}$ except that we translate it
so that the origin lies at $x$. The identification of $M$ with
its image also allows us to identify $T_{p} M$ and $N_{p}$ with
their images in $T_{p} \mathbb{R}^{n}$, and thus with their
images in $\mathbb{R}^{n}$. They are then affine spaces anchored
at $p$ and are orthogonal (once the translation is taken into
account).

Our first task is to construct a tubular neighbourhood. From the
above, we have a map $\nu \colon N \to \mathbb{R}^{n}$ which
restricts on the zero section to the inclusion of $M$. At the
zero section, the derivative of this map is the identification of
$T_{p} M \oplus N_{p}$ with $T_{p} \mathbb{R}^{n}$. As this is an
isomorphism, the [[inverse function theorem]] comes in to play
and says that there is a neighbourhood $W$ of the zero section in
$N$ and a neighbourhood $X$ of $M$ in $\mathbb{R}^{n}$ such that
$\nu$ restricts to a diffeomorphism $W \to X$, which we shall
also denote by $\nu$. Reversing $\nu$, we get a map $X \to W
\subseteq N$ which we can combine with the projection to $M$. Let
us write $\mu \colon X \to M$ for this composition.

This map $\mu$ is the **local averaging function** that we need.
Well, technically the local averaging function is a bit more
complicated but to define it in full one needs more data.
However, that data will depend on the source of our maps so $\mu$
is the data that is solely dependent on $M$ that goes in to the
mix that defines the local averaging function and so deserves
that name.

### A Compatible Local Addition ###

The local averaging function is only half of what we need to
construct a suitable tubular neighbourhood. The other half is
provided by a [[local addition]]. It needs to be compatible with
the local averaging function and so we assume that the
construction of the previous section has been carried out.

Turning to the tangent spaces, for $p \in M$, the space $T_{p} M$
sits as an affine subspace in $\mathbb{R}^{n}$ with complement
$N_{p}$. There is an orthogonal projection map $\mathbb{R}^{n}
\to T_{p} M$, projecting along $N_{p}$. Restricting this to $M$,
we obtain a map $\lambda_{p} \colon M \to T_{p} M$.

Note that $\lambda_{p}(p) = p$, which is the zero of $T_{p} M$,
and that $d \lambda_{p} \colon T_{p} M \to T_{p}(T_{p} M) = T_{p}
M$ is the identity. Allowing $p$ to vary, we obtain a smooth map
$\lambda \colon M \times M \to T M$. If we think of $M \times M$
as a bundle over $M$ (via the projection on to the first factor)
then $\lambda \colon M \times M \to T M$ is a bundle map.

Now we don't actually need this to be defined on the whole of
$M$, just on a small piece of it near $p$, but what "small" means
here depends on $X$. Using the metric on $\mathbb{R}^{n}$ coming
from the orthogonal structure, we choose a continuous function
$\epsilon \colon M \to (0,\infty)$ with the property that the
closed ball of radius $\epsilon(p)$ at $p$ is contained in $X$.
Let us write $X_{p}$ for the corresponding *open* ball. Then we
restrict $\lambda_{p}$ to $X_{p} \cap M$ and thus restrict
$\lambda$ to the set
$$
X_{M} \coloneqq \{(p,q) \in M \times M : q \in X_{p}\} = \{(p,q) \in M \times M : d(p,q) \lt \epsilon(p)\}
$$
which is an open neighbourhood of the diagonal.

From the above, we see that (again by the inverse function
theorem), there is a neighbourhood $V$ of the diagonal and a
neighbourhood $U$ of the zero section such that $\lambda$
restricts to a diffeomorphism $\lambda \colon V \to U$. As we
have already restricted $\lambda$ to $X_{M}$, $V$ is a subset of
it.

Let $\eta \colon U \to M$ be the composition:
$$
T M \supseteq
U \xrightarrow{\lambda^{-1}}V \subseteq M \times M \to M
$$
where
the projection is on to the second factor. As $\lambda$ is a
bundle map, $\lambda^{-1} = \pi \times \eta$ and thus $\eta$ is a
[[local addition]].

Let $U_{p} = U \cap T_{p} M$. Then $\eta(U_{p})$ is a subset of
$\{p\} \times M$. Let $\eta_{p} \colon U_{p} \to V_{p} \subseteq
M$ be the obvious restriction. Since $\lambda^{-1} = \pi \times
\eta$ and the domain of $\lambda$ is a subset of $X_{M}$, the set
$\eta(U_{p})$ is a subset of $X_{M} \cap \{p\} \times M = \{p\}
\times X_{p}$. Thus $\eta_{p}(U_{p}) \subseteq X_{p}$. What this
means is that the set $\eta_{p}(U_{p})$, which is a subset of $M$
and therefore may be all twisted and turned, nevertheless has the
property that the closure of its convex hull is contained inside
$X$, the tubular neighbourhood of $M$.

Thus we have the maps
$$
\begin{aligned}
\eta &\colon T M \supseteq U \xrightarrow{\cong}V \subseteq M
\times M, \\   \eta_{p} & \colon T_{p} M \supseteq U_{p}
\xrightarrow{\cong}V_{p} \subseteq M, \\   \nu &\colon N
\supseteq W \xrightarrow{\cong}X \subseteq \mathbb{R}^{n}, \\ \mu
&\colon \mathbb{R}^{n} \supseteq X \xrightarrow{\cong}W \subseteq
N \xrightarrow{\pi}M.
\end{aligned}
$$
The map $\eta_{p}$ works by projecting $U_{p} \subseteq T_{p}
M$ on to $M$ along $N_{p}$. The map $\mu$ projects $X$ on to $M$
also by projecting along $N_{p}$.

What we have so far can be portrayed in the following diagram.

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=--

### The Tubular Neighbourhood ###

At this point it is time to we reintroduce our group, although at
this stage it isn't important that our source be a group. We need
a compact [[smooth space]]; let us write this as $S$. We also
need a measure on $S$ with the property that $S$ has measure $1$.
The idea is to consider smooth maps $S \to M$ with the property
that the closed convex hull of the image of $S$ is contained in
$X$. Then given such a smooth map $\alpha \colon S \to M$ we can
consider it as a map in to $\mathbb{R}^{n}$ and take its
integral. As the closed convex hull of the image of $S$ is in
$X$, the value of this integral will also be in $X$ and thus can
be projected to $M$ via $\mu$. Let us write $\Cvx(S,M)$ for the
set of smooth maps $S \to M$ with the property that the closed
convex hull of $S$ lies in $X$ and let us write $\tau \colon
\Cvx(S,M) \to M$ for this *local averaging* map.

Now that we know what is the "average" of a "small" map $\alpha
\colon S \to M$, we can consider the question as to whether or
not we can use the local addition to contract $\alpha$ to its
average. To be able to do this, we need to further restrict our
attention to those maps $\alpha \colon S \to M$ with the property
that $(\tau \alpha, \alpha(s)) \in V$ for all $s \in S$. Knowing
this, we can compose $\alpha$ with $\eta_{\tau \alpha}^{-1}$ to
produce a map $S \to U$.

The construction is set up so that the resulting map $S \to U$
lies in a single fibre and has the property that it averages to
$0$ (viewed in $T_{p} M$). This will define a diffeomorphism
between the set of smooth maps $S \to U$ with those properties
and an open subset of $C^{\infty}(S,M)$. Let us define:
$$
C^{\infty}_{0}(S,U) \coloneqq \{ \alpha \colon S \to U : \pi
\alpha\; \text{is constant}, \int_{S} \alpha = 0_{\pi \alpha}\}.
$$

The reverse map is straightforward to describe: given a map
$\alpha \in C^{\infty}_{0}(S,U)$, we compose it with $\eta_{\pi
\alpha}$ to produce a map $S \to M$.

+-- {: .num_lemma}
###### Lemma ######
The map $C^{\infty}_{0}(S,U) \to C^{\infty}(S,M)$ defined by
composition with $\eta$ is a diffeomorphism on to its image which
is open in $C^{\infty}(S,M)$.
=--

+-- {: .proof}
To see that its image is open, let us first note that $\Cvx(S,M)$
is open since $X$ is open in $\mathbb{R}^{n}$ and $S$ is compact.
Then within that, the condition on a map $\alpha \colon S \to M$
that $(\tau \alpha, \alpha(s)) \in V$ for all $s \in S$ is also
an open condition, again because $S$ is compact. Hence the set of
smooth maps $\alpha \colon S \to M$ with the property that the
closed convex hull of $\alpha(S)$ lies in $X$ (whence $\tau
\alpha$ is defined) and that $(\tau \alpha, \alpha(s)) \in V$ for
all $s \in S$ is an open subset of $C^{\infty}(S,M)$.

Let us write this as $Y$. We have a map $C^{\infty}_{0}(S,U) \to
Y$ given by composition: $\alpha \mapsto \eta_{\pi \alpha} \circ
\alpha$; and a map $Y \to C^{\infty}_{0}(S,U)$ given by
$\eta_{\tau \beta}^{-1} \circ \beta$. That the second is
well-defined is not completely obvious: although $\eta_{\tau
\beta}^{-1} \circ \beta$ lies in $U_{\tau \beta}$ by
construction, we also have to show that its average is the zero
in $T_{\tau \beta} M$.

This follows from the way that $\eta_{\tau \beta}$ and $\tau$
were constructed. Since averaging is a linear map, the average of
$\beta$ splits as the sum of a point in $T_{\tau \beta} M$ and a
point in $N_{\tau \beta} M$. However, by definition, $\tau \beta$
lies in $N_{\tau \beta}$ and thus the contribution from $T_{\tau
\beta} M$ must be the zero vector.

To show that these are inverse, it is enough to show that $\tau
(\eta_{\pi \alpha} \circ \alpha) = \pi \alpha$ and
$\pi(\eta_{\tau \beta}^{-1} \circ \beta) = \tau \beta$.

The second is by construction: $\eta_{\tau \beta}$ has image in
$U_{\tau \beta}$. The first follows by the argument on splitting
the average: the average of $\eta_{\pi \alpha} \circ \alpha$
splits as the average of its projection to $T_{\pi \alpha} M$
plus the average of its projection to $N_{\pi \alpha}$. Since the
former is $0_{\pi \alpha}$, the average of $\eta_{\pi \alpha}
\circ \alpha$ lies in $N_{\pi \alpha}$ and hence $\tau \eta_{\pi
\alpha} \circ \alpha = \pi \alpha$.
=--

The last step in constructing the tubular neighbourhood is to
construct a diffeomorphism between $C^{\infty}_{0}(S,U)$ and
$C^{\infty}_{0}(S,T M)$. This is straightforward: we restrict $U$
(if necessary) so that it is diffeomorphic as a fibre bundle to
$T M$, say via $\rho \colon U \to T M$, and then define the map
$C^{\infty}_{0}(S,U) \to C^{\infty}_{0}(S, T M)$ via:
$$
\alpha \mapsto \rho \circ \alpha - \int_{S} \rho \circ \alpha.
$$

### Invariance ###
The above construction produces a tubular neighbourhood of the
space of constant maps $S \to M$ in the space of smooth maps $S
\to M$. The construction is invariant under measure-preserving
diffeomorphisms of $S$. In particular, if $S$ is a group and the
measure is invariant under the group action, the tubular
neighbourhood is equivariant.

### Fixed Point Submanifolds ###

This construction extends to the case where the source is a fibre
bundle $E \to B$ with fibre $S$ and the submanifold under
consideration are those maps $E \to M$ which are constant on the
fibres of $E \to B$. We need to assume that the transition
functions preserve the measure on the fibres. The required
tubular neighbourhood consists of those maps $E \to M$ with the
property that when restricted to a fibre (identified with $S$ via
a measure-preserving diffeomorphism), the map lies in the image
of $C^{\infty}_{0}(S,M)$.


equivariant weak homotopy equivalence

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

#Contents#
* table of contents
{:toc}

## Idea

In generalization of the notion of [[weak homotopy equivalences]] from plain [[homotopy theory]] to [[equivariant homotopy theory]], a [[map]] of [[G-spaces]] is called an *equivariant weak homotopy equivalence* if it induces [[isomorphisms]] on all [[equivariant homotopy groups]].

## Related concepts

* [[equivariant Whitehead theorem]]

* [[Elmendorf's theorem]]

* [[fine model structure on topological G-spaces]]

## References

See the references at *[[equivariant homotopy group]]*.

Detection in terms of free homotopy sets:

* [[Takao Matumoto]], [[Norihiko Minami]], [[Masahiro Sugawara]], *On the set of free homotopy classes and Brown's construction*, Hiroshima Math. J. 14(2): 359-369 (1984) ([doi:10.32917/hmj/1206133043](https://projecteuclid.org/journals/hiroshima-mathematical-journal/volume-14/issue-2/On-the-set-of-free-homotopy-classes-and-Browns-construction/10.32917/hmj/1206133043.full))

[[!redirects equivariant weak homotopy equivalences]]


equivariant Whitehead theorem

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## Idea

The _equivariant Whitehead theorem_ is the generalization of the [[Whitehead theorem]] from ([[stable homotopy theory|stable]]) [[homotopy]] to ([[equivariant stable homotopy theory|stable]]) [[equivariant homotopy theory]]:

Assume that the [[equivariance group]] be a [[compact Lie group]] ([Matumoto 71](#Matumoto71), [Illman 72](#Illman72), [James-Segal 78](#JamesSegal78), [Waner 80](#Waner80)). (This assumption is used, e.g. in [Waner 80, Rem. 7.4](#Waner80), to guarantee that [[Cartesian products]] of [[coset spaces]] $G/H$ are themselves [[G-CW-complexes]], which follows for [[compact Lie groups]], e.g. by the [[equivariant triangulation theorem]].)

Then: *$G$-[[homotopy equivalences]] $f \colon X \longrightarrow Y$ between [[G-CW complexes]] are equivalent to [[equivariant weak homotopy equivalences]], hence to [[maps]] that induce [[weak homotopy equivalences]] $f^H \colon X^H \longrightarrow Y^H$ on all [[fixed point spaces]] for all [[closed topological space|closed]] [[subgroups|subgroups]] $H \hookrightarrow G$*.

This is due to [Matumoto 71, Thm. 5.3](#Matumoto71), [Illman 72, Prop. 2.5](#Illman72), [Waner 80, Theorem 3.4](#Waner80), see also [James-Segal 78, Thm. 1.1](#JamesSegal78) with [Kwasik 81](#Kwasik81),
review in [Shah 10](#Shah10), [Blumberg 17, Cor. 1.2.14](#Blumberg17).

An analogous statement holds in [[stable equivariant homotopy theory]]:

For maps $F \colon E \longrightarrow F$ between genuine [[G-spectra]], they are weak equivalences (isomorphisms in the [[equivariant stable homotopy category]]) if they induce isomorphisms on all [[equivariant homotopy group]] [[Mackey functors]] $\pi_n(f)\colon \pi_n(E) \longrightarrow \pi_n(F) $ (e. g. [Greenlees-May 95, theorem 2.4](#GreenleesMay95), [Bohmann, theorem 3.2](#Bohmann)).

## Related concepts

* [[G-CW approximation theorem]], [[equivariant cellular approximation theorem]]

* [[Elmendorf's theorem]]

* [[tom Dieck splitting]]

## References

Proofs for general [[G-CW-complexes]] (for $G$ a [[compact Lie group]]) are due to

* {#Matumoto71} [[Takao Matumoto]], Theorem 5.3 in: _On $G$-CW complexes and a theorem of JHC Whitehead_, J. Fac. Sci. Univ. Tokyo Sect. IA 18, 363-374, 1971 ([[matumoto.pdf:file]])

following a partial result in

*  [[Takao Matumoto]], Lemma 4.3 in: _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]])

and, independently, due to

* {#Illman72} [[Sören Illman]], Prop. 3.3 in: *Equivariant Algebraic Topology*, Princeton University 1972 ([[Illman_EquivariantAlgebraicTopology1972.pdf:file]])

* {#Illman72} [[Sören Illman]], Prop. 2.5 in: *Equivariant singular homology and cohomology for actions of compact lie groups* ([doi:10.1007/BFb0070055](https://doi.org/10.1007/BFb0070055)) In: H. T. Ku, L. N. Mann, J. L. Sicks, J. C. Su (eds.), *Proceedings of the Second Conference on Compact Transformation Groups* Lecture Notes in Mathematics, vol 298. Springer 1972 ([doi:10.1007/BFb0070029](https://link.springer.com/book/10.1007/BFb0070029))

See also:

* {#Waner80} [[Stefan Waner]], Theorem 3.4 in: _Equivariant Homotopy Theory and Milnor's Theorem_, Transactions of the American Mathematical Society Vol. 258, No. 2 (Apr., 1980), pp. 351-368 ([jstor:1998061](http://www.jstor.org/stable/1998061))

Discussion in [[equivariant stable homotopy theory]]:

* {#GreenleesMay95} [[John Greenlees]], [[Peter May]], _Equivariant stable homotopy theory_, in I.M. James (ed.), _Handbook of Algebraic Topology_ , pp. 279-325. 1995. ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))

Review and Lecture notes:

* {#Blumberg17} [[Andrew Blumberg]], Section 1.2 in: _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))

A proof for $G$-[[absolute neighbourhood retract|ANRs]] is due to:

* {#JamesSegal78}  [[Ioan James]], [[Graeme Segal]], Theorem 1.1 in: _On equivariant homotopy type_, Topology Volume 17, Issue 3, 1978, Pages 267-272 (doi:10.1016/0040-9383(78)90030-7)

Proof that these $G$-[[absolute neighbourhood retract|ANRs]] have the [[equivariant homotopy type]] of [[G-CW-complexes]] (for $G$ a [[compact Lie group]]):

* {#Kwasik81} [[Slawomir Kwasik]], _On the Equivariant Homotopy Type of $G$-ANR's_, Proceedings of the American Mathematical Society Vol. 83, No. 1 (Sep., 1981), pp. 193-194 (2 pages) ([jstor:2043921](https://www.jstor.org/stable/2043921))

Textbook account for $G$-ANRs:

* {#tomDieck79} [[Tammo tom Dieck]], Prop. 8.2.4, 8.2.6 in: _[[Transformation Groups and Representation Theory]]_, Lecture Notes in Mathematics 766, Springer 1979 ([doi:10.1007/BFb0085965](https://link.springer.com/book/10.1007/BFb0085965))

For the case of [[stable equivariant homotopy theory]]:

* {#Bohmann} [[Anna Marie Bohmann]], _Basic notions of equivariant stable homotopy theory_ ([pdf](http://math.northwestern.edu/~bohmann/basicequivnotions.pdf))


equivariantization

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On this whole page, assume that $G$ is a finite group. Denote by $\underline{G}$ the corresponding discrete monoidal category, and by $\operatorname{Rep}_G$ its finite dimensional $k$-representations. Further, for a monoidal category $\mathcal{C}$, denote by $\mathcal{C}-\mathcal{MOD}$ the bicategory of $k$-linear [[module category|$\mathcal{C}$-module categories]], module functors and natural transformations.

# Main idea

There are 2-functors $D : \operatorname{Rep}_G-\mathcal{MOD} \rightleftarrows \underline{G}-\mathcal{MOD} : E$, called equivariantisation and deequivariantisation, respectively.
They become weak inverses once we restrict to semisimple module categories.
In a sense, (de)equivariantisation generalises the Morita equivalence between $\operatorname{Rep}_G$ and $\operatorname{Vec}_G$ for finite groups.

## Detailed definitions

A $\underline{G}$-action (a $\underline{G}$-module category structure) on a $k$-linear semisimple category $\mathcal{C}$ amounts to the following data:

* for each group element $g \in G$ a (linear) equivalence $\rho(g)\colon \mathcal{C} \to \mathcal{C}$
* natural isomorphisms $\rho^{2}(g,h)_X\colon \rho(g)(\rho(h) X) \to \rho(gh)X$
* a natural isomorphism $\rho^0_X\colon X \to \rho(1)X$
* satisfying certain obvious coherence axioms

### Equivariantisation

__Definition__ A _$G$-equivariant object_ in $\mathcal{C}$ is then an object $X$ in $\mathcal{C}$ and a family of isomorphisms $u_g\colon \rho(g) X \to X$ compatible with the action and $\gamma$.

The $G$-equivariant objects form a category (where morphisms need to commute with the $u_g$), which is denoted by $\mathcal{C}^G$.

The primordial example for this construction is the category of finite dimensional vector spaces $k-\operatorname{Vect}$, which has a trivial $G$-action. Its $G$-equivariant objects $k-\operatorname{Vect}^G$ are simply $\operatorname{Rep}_G$.

Since $k-\operatorname{Vect}$ has a natural action on any $k$-linear category like $\mathcal{C}$, $k-\operatorname{Vect}^G \simeq \operatorname{Rep}_G$ acquires an action on $\mathcal{C}^G$. Explicitly, let $(V, \phi\colon G \to \operatorname{End}(V))$ a representation, and $(X, u_g)$ an equivariant object, then the module structure $-\triangleright-\colon \operatorname{Rep}(G) \boxtimes \mathcal{C}^G \to \mathcal{C}^G$ is defined as $(V, \phi) \triangleright (X, u_g) = (V \triangleright X, \phi(g) \otimes u_g)$. Thus we define the 2-functor $E$ as $E\mathcal{C} = \mathcal{C}^G$.

### Deequivariantisation

First note that the function algebra $k(G)$ is an object in $\operatorname{Rep}(G)$ by means of left multiplication with $G$. It has an additional $G$-representation by right inverse multiplication. Furthermore, it is an internal algebra.

__Definition__ Let $\mathcal{C}$ be a $\mathcal{M}$-module category, and $A$ an algebra internal to $\mathcal{M}$. An $A$-module in $\mathcal{C}$ is an object $X$ in $\mathcal{C}$ and a morphism $A \triangleright X \to X$ satisfying the obvious action axioms.

We can therefore form the category of $k(G)$-modules in $\mathcal{C}$, since we have a  $\operatorname{Rep}(G)$ module structure, and denote it by $\mathcal{C}_G$. Since $k(G)$ has the additional right $G$-representation, $\mathcal{C}_G$ inherits the $G$-action. Thus the 2-functor $D\mathcal{C} = \mathcal{C}_G$ is defined.

### Mutual inverses

__Theorem__ The previously defined 2-functors $D$ and $E$ are mutually (weakly) inverse 2-equivalences between the bicategory of semisimple $k$-linear categories with $G$-action, and the bicategory of semisimple $k$-linear categories with $\operatorname{Rep}_g$-action. In particular, $\mathcal{C} \simeq (\mathcal{C}^G)_G$.

### Forgetful and induction functors

* There is an obvious faithful forgetful functor $U \colon \mathcal{C}^G \to \mathcal{C}$.
* There is a left and right adjoint to $U$, the induction functor $IX = \oplus_{g \in G} \rho(g) X$.
* There is an obvious forgetful functor $U'\colon \mathcal{C}_G \to \mathcal{C}$.
* There is a left adjoint to $U$, the canonical functor $FX = k(G) \otimes X$, with the trivial action on $X$.

# Monoidal and braided categories, and central functors

The previous constructions generalise easily when our categories acquire monoidal or braided structures.

Actions of groups on monoidal categories are simply a monoidal equivalence for every group element with the same extra structure as mentioned before. Actions of groups on braided categories are additionally required to preserve the braiding. (Following the idea of [[stuff, structure, and properties]], an action on a symmetric category are simply actions on the underlying braided category.)

The action of $k-\operatorname{Vect}$ on a _monoidal_ linear category $\mathcal{C}$ can also be understood as the canonical monoidal functor $k-\operatorname{Vect} \to \mathcal{C}$ sending $k$ to the monoidal unit, followed by the tensor product.
In this situation, the $\operatorname{Rep}_G$ module structure on $\mathcal{C}$ can then also be understood as the canonical functor $\operatorname{Rep}_G \to \mathcal{C}^G$.

By abstract nonsense, an action on an algebraic object should be a morphism into its [[center]]. Consequently, the action of a braided category $\mathcal{B}$ on a monoidal category $\mathcal{C}$ should be given by a functor $\mathcal{B} \to \mathcal{Z}(\mathcal{C})$, where $\mathcal{Z}$ is the [[Drinfeld center]]. This is called a _central functor_. Similarly, the action of a _symmetric category_ $\mathcal{A}$ on a braided category $\mathcal{B}$ should be a functor $\mathcal{A} \to \mathcal{B}'$, where $\mathcal{B}'$ is the "symmetric center", or "Müger center".

The previously mentioned functor $\operatorname{Rep}_G \to \mathcal{C}^G$ factors through the forgetful functor $\mathcal{Z}(\mathcal{C}) \to \mathcal{C}$ (or, in the case of a braided category, through the inclusion $\mathcal{C}' \hookrightarrow \mathcal{C}$), so we have an action in this sense.

__Theorem__ By equivariantisation, the bicategory of fusion categories with $G$-actions is 2-equivalent to the bicategory of fusion categories with central functors from $\operatorname{Rep}_G$. Similarly, the bicategory of braided fusion categories with braided $G$-actions is 2-equivalent to the bicategory of fusion categories (symmetric central functors from $\operatorname{Rep}_G$.

## Modularisation

Each  braided fusion category $\mathcal{B}$ has a canonical symmetric subcategory, its symmetric centre $\mathcal{B}'$. By choosing a trivial twist, $\mathcal{B}'$ has a canonical spherical structure. If all the quantum dimensions are positive, that is, if $\mathcal{B}'$ is _tannakian_, it is equivalent to $\operatorname{Rep}_G$ for a unique group $G$ (since there is a canonical fibre functor). The obvious construction is then to deequivariantise over $\mathcal{B}' \simeq \operatorname{Rep}_G$, which yields $\mathcal{B}_G$, the __modularisation__ of $\mathcal{B}$.

There is an "exact sequence" of braided fusion categories:
$$ k-\operatorname{Vect} \hookrightarrow \mathcal{B}' \simeq \operatorname{Rep}_G \hookrightarrow \mathcal{B} \twoheadrightarrow \mathcal{B}_G$$
It is exact in the sense that $\mathcal{B}'$ is the maximal symmetric subcategory of $\mathcal{B}$, and that $\mathcal{B}_G$ is the maximal modular category such that $\mathcal{B}'$ is sent to sums of the monoidal unit.

# References

* Vladimir Drinfeld, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, _On braided fusion categories I_, section 4 [arXiv](https://arxiv.org/abs/0906.0620)
* Alain Bruguières, _Catégories prémodulaires, modularisations et invariants des variétés de dimension 3_ (french) [pdf](http://www.math.univ-montp2.fr/~bruguieres/docs/modular.pdf)
* Michael Müger, _Galois Theory for Braided Tensor Categories and the Modular Closure_
[arXiv](https://arxiv.org/abs/math/9812040)

[[!redirects equivariantisation]]
[[!redirects deequivariantisation]]
[[!redirects de-equivariantisation]]
[[!redirects deequivariantization]]
[[!redirects de-equivariantization]]


ER = EPR

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## Idea
 {#Idea}

The symbols *ER = EPR* (pronounced: "ee arr is ee pee arr") serve as a slogan for a hypothesis about [[quantum gravity]], claiming, somewhat vaguely, that [[entangled particles]] (the [[EPR paradox|Einstein-Podolsky-Rosen phenomenon]]: "EPR") "are" connected by a [[wormhole]] ([[Einstein-Rosen bridge]]: "ER").

This slogan is due to [Maldacena & Susskind (2013)](#MaldacenaSusskind13) (there motivated from a search for a solution for the perceived [[black hole firewall problem]]) but the idea may in this form have circulated earlier [cf. [Verlinde & Verlinde (2022)](#VerlindeVerlinde22)]. In fact, what is arguably a precise and provable version of the statement (in lattice models) has appeared years earlier: the [[Ryu-Takayanagi formula]], see [below](#AsASloganForTheRyuTakayanagiFormula) (which, however, applies in a [[holographic principle|holographic]] context, a qualification not necessarily brought out by [Maldacena & Susskind](#MaldacenaSusskind13)).

At face value the statement "ER = EPR", even if parsed benevolently, suffers from a [[type checking|typing error]] between [[quantum physics]] and [[classical field theory|classical]] [[gravity]]. However, [like with a Zen koan](https://twitter.com/schreiberurs/status/1134503200550203392?lang=en), one may imagine the [[paradox|paradoxical]] formulation to propel the intuitive grasp of indeed rather deep principles of [[holographic principle|holography]] and particularly of [[holographic entanglement entropy]], such as expressed with precision by the [Ryu-Takayanagi formula](#AsASloganForTheRyuTakayanagiFormula).

Another source of fun with the slogan "ER = EPR" is to thereby see the two classical and seminal papers by [[Albert Einstein]] et al. from 1935 [[Einstein & Rosen (1935)](wormhole#EinsteinRosen35), [Einstein, Podoldsky & Rosen (1935)](EPR+paradox#EinsteinPodoldskyRosen35)] --- which to [[Einstein]] must have seemed unrelated --- to secretly be about two sides of the same medal.

## As a slogan for the Ryu-Takayanagi formula
 {#AsASloganForTheRyuTakayanagiFormula}

The idea meant to be expressed by the slogan "ER = EPR" may usefully be compared to -- and may be a way of turning into prose -- the *[[Ryu-Takayanagi formula]]* [[Ryu & Takayanagi (2006)](#RyuTakayanagi06a)] (which has a more well-defined content as a *[[theorem]]* in a class of discrete [[tensor network]]-models). The RT-formula equates:

* **(quantum entanglement:)** the [[entanglement entropy]] of a [[boundary field theory|boundary]] [[quantum field theory]] in a subspace of an [[asymptotic boundary]]

with

* **(global spacetime structure:)** the [[minimal surface|minimal area]] of a [[coboundary|cobounding]] [[hypersurface]] in an ambient [[curvature|curved]] [[bulk]] [[spacetime]].

Concretely, in [[tensor network]]-models of [[AdS2/CFT1 duality]], the [[entanglement entropy]] $S_A$ of any [[Majorana dimer code]]-[[tensor network]] [[quantum state|state]] turns out to count the number of [[dimers]] that cross between a ([[connected topological space|connected]]) subregion $A$ and its [[complement]]:

\begin{imagefromfile}
    "file_name": "ChordDiagramEntanglementEntropy.jpg",
    "width": 460,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [JGPE 19](holographic+entanglement+entropy#JGPE19)"
\end{imagefromfile}

\begin{imagefromfile}
    "file_name": "HanUniversalHolographicCode.jpg",
    "float": "right",
    "width": 380,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "From [Yan 20](holographic+entanglement+entropy#Yan20)"
\end{imagefromfile}

In the case that the [[dimers]] correspond to the [[hyperbolic tesselation]] $\{5,4\}$ from the [[HaPPY code]], this entropy formula

$$
  S_A
  \;=\;
  \tfrac{1}{2}
  \ln
  \big(
    2^{ \left\vert Chords(A, \bar A) \right\vert }
  \big)
$$

recovers the [[Ryu-Takayanagi formula]] ([JGPE 19 (78)](holographic+entanglement+entropy#JGPE19)), as here the number of chords crossing any hyperbolic [[geodesic]] grows linearly with the [[length]] of this geodesic.

A precursor to this picture is the "bit-thread"-interpretation of entanglement entropy due to [Freedman & Headrick 16](holographic+entanglement+entropy#FreedmanHeadrick16) (notice the use of "EPR pair" in their [Figure 4](https://arxiv.org/pdf/1604.00354.pdf#page=4).)

\begin{imagefromfile}
    "file_name": "ChordDiagram.jpg",
    "float": "right",
    "width": 230,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "from [Sati & Schreiber 19c](#SatiSchreiber19c)"
\end{imagefromfile}

Following [Sati-Schreiber 19c](#SatiSchreiber19c) we recognize the above [[Majorana dimer]]/bit-thread networks from [JGPE 19](#JGPE19), [Yan 19](#Yan19) as *[[chord diagram]]*-encodings of holographic bulks ([p. 38](https://arxiv.org/pdf/1912.10425v3.pdf#page=38)).

Notice that the formula for the [[entanglement entropy]] in the [[Majorana dimer code]]  holds generally, for *any* configuration and hence any underlying [[chord diagram]].

In conclusion, note how each [[chord]] here reflects at the same time:

1. (ER) one [[entanglement|entangled pair]] of [[qbits]] in the [[boundary field theory|boundary]] [[quantum system]];

1. (EPR) a [[geodesic]] through the [[hyperbolic plane]] [[bulk]] spacetime,

which is a (rigorous) state of affairs clearly reminiscent of the "ER = EPR" slogan (except that wormholes are replaced by minimal area hypersurfaces, here: geodesics).

For more on this see at *[[holographic entanglement entropy]]*.

## Related entries

* [[entanglement]], [[EPR paradox]]

* [[wormhole]]

* [[holographic entanglement entropy]]

* [[quantum teleportation]]

* [[black hole firewall problem]]

## References

The original result of the [[Ryu-Takayanagi formula]]:

* {#RyuTakayanagi06a} [[Shinsei Ryu]], [[Tadashi Takayanagi]], *Holographic Derivation of Entanglement Entropy from AdS/CFT*, Phys. Rev. Lett. **96** 181602 (2006) [[arXiv:hep-th/0603001](https://arxiv.org/abs/hep-th/0603001), [doi:10.1103/PhysRevLett.96.181602](https://doi.org/10.1103/PhysRevLett.96.181602)]

The slogan "ER = EPR" is due to:

 * {#MaldacenaSusskind13} [[Juan Maldacena]], [[Leonard Susskind]], _Cool horizons for entangled black holes_, Fortsch. Phys. 61 (2013) 781-811 [[arXiv:1306.0533](http://arxiv.org/abs/1306.0533), [doi:10.1002/prop.201300020](https://doi.org/10.1002/prop.201300020)]

* {#Susskind14} [[Leonard Susskind]], *ER=EPR, GHZ, and the consistency of quantum measurements*, Fortsch. Phys. **64** 1 (2016) 72-83 [[arXiv:1412.8483](https://arxiv.org/abs/1412.8483), [doi:10.1002/prop.201500094](https://doi.org/10.1002/prop.201500094)]

An email exchange of similar content that happened half a year earlier is recalled in:

* {#VerlindeVerlinde22} [[Erik Verlinde]], [[Herman Verlinde]], *A Conversation on ER = EPR* [[arXiv:2212.09389](https://arxiv.org/abs/2212.09389)]

See also

* Wikipedia, *[ER=EPR](https://en.wikipedia.org/wiki/ER%3DEPR)*

[[!redirects ER=EPR]]
[[!redirects er=epr]]
[[!redirects ER= EPR]]
[[!redirects ER =EPR]]


ER-theory

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## Idea

The [[real-oriented cohomology theory]]-version of [[Morava E-theory]].

## Related concepts

* [[w4-orientation of EO(2)-theory]]

## References

ER-theory was constructed in

* {#Kriz01} [[Igor Kriz]], _Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence_, Topology 40 (2001) 317-399 ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/hukriz.pdf))

based on Araki's discussion of [[BPR-theory]].

The [[orientation in generalized cohomology]] for $ER$-theory by $w_4$-structure is discussed in

* {#KrizSati04} [[Igor Kriz]], [[Hisham Sati]], _M Theory, Type IIA Superstrings, and Elliptic Cohomology_, Adv.Theor.Math.Phys. 8 (2004) 345-395 ([arXiv:hep-th/0404013](http://arxiv.org/abs/hep-th/0404013))

Some relations to versions of [[tmf]] are discussed in

* [[Mark Mahowald]], [[Charles Rezk]], _Topological Modular Forms of Level 3_, Pure and Applied Math. Quarterly, 5 (2009) 853-872 ([arXiv:0812.2009](http://arxiv.org/abs/0812.2009))