equivariant elliptic cohomology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

Equivariant elliptic cohomology is, or is supposed to be, an [[equivariant cohomology|equivariant]] version of [[elliptic cohomology]], hence a higher [[chromatic homotopy theory|chromatic]] analogue of [[equivariant K-theory]].

As usual in [[equivariant cohomology]], there is a "naive" version and refinements thereof, and typically it is these refinements that one is really interested in. The traditional motivation of these from [[algebraic topology]]/[[homotopy theory]] are indicated below in

* _[Motivation from algebraic topology](#MotivationFromAlgebraicTopology)_

Despite that motivation, the precise nature of the resulting "genuine" equivariant elliptic cohomology may tend to seem a bit mysterious and also a bit baroque in its technical ingredients, some of which may appear a bit unexpected in the literature. A clear conceptual picture of what equivariant elliptic cohomology is about is obtained by regarding it as encoding aspects of low dimensional [[quantum field theory]] and [[worldsheet]] [[string theory]]; this is indicated further below in

* _[Interpretation in Quantum field theory/String theory](#InterpretationInQuantumFieldTheory)_.

### Motivation from algebraic topology
 {#MotivationFromAlgebraicTopology}

Given any [[cohomology theory]] $E$ which may be evaluated on arbitrary [[topological spaces]], then for $G$ a [[compact Lie group]] the "naive" $G$-[[equivariant cohomology|equivariant E-cohomology]] of the point is the $E$-cohomology of the [[classifying space]] $B G$ of $G$ (which is equivalently the [[delooping]]

$$
  B G \simeq \ast //G
$$

of $G$ regarded as an [[∞-group]], see at _[[∞-action]]_ for how that encodes actions on structures above it):

$$
  E_G^\bullet(\ast)_{naive} \coloneqq E^\bullet(B G)
  \,.
$$

In a discussion in the context of [[geometric homotopy theory]] it is clear what is "naive" about this definition: since $G$ has geometric structure of which $B G$ remembers only the underlying [[geometrically discrete infinity-groupoid|bare homotopy type]], one would instead want to use the something like the [[smooth stack]] $\mathbf{B}G$ (the [[moduli stack]] of $G$-[[principal bundle]]), then somehow make good sense of $\mathbf{E}^\bullet(\mathbf{B}G)$ where now $\mathbf{E}$ is some [[sheaf of spectra]] and then declare this to be the actual $G$-equivariant $E$-cohomology.

The traditional argument however proceeds as follows: if $E$ is a [[complex oriented cohomology theory]] then (essentially by definition) for $G = U(1)$ the [[circle group]] then  $E^\bullet(B U(1)) \simeq E^\bullet(\ast)[ [ c_1^E ] ]$  is the algebra of [[formal power series]] which one may think of as the [[algebra of functions]] on the [[formal geometry|formal]] [[neighbourhood]] of a point in some larger [[space]] $M_{S^1}$.

For instance in the simpler case of [[equivariant K-theory]] this has long been well understood: here the genuine $U(1)$-equivariant cohomology of the point is the [[representation ring]] $K_{U(1)}(\ast) \simeq  \mathbb{Z}[ [ t^{-1}, t] ]$ which happens to be the [[algebra of functions]] on the [[multiplicative group]]; while by [[complex oriented cohomology theory|complex orientation]] the naive equivariant cohomology $K^\bullet(B U(1)) \simeq \mathbb{Z}[ [t] ]$ is equivalently the algebra of functions on (just) the formal multiplicative group.

Based on this one may want to consider an $E$-[[(∞,1)-module bundle|∞-line bundle]] over the full space $M_{S^1}$ and take the genuine $E$-equivariant cohomology to be the [[global sections]] of that. (Specifically in [[elliptic cohomology]] that space $M_{S^1}$ is equivalent to the [[elliptic curve]] $C$ that gives the theory its name, but in some sense discussed [below](#InterpretationInQuantumFieldTheory) the spaces $M_{S^1}$ and $C$ arise conceptually differently and it is a fairly deep coincidence that they are in fact equivalent, which one may want to remember.)

In this way equivariant elliptic cohomology was defined in ([Grojnowski 94](#Grojnowski94), [Ginzburg-Kapranov-Vasserot 95](#GinzburgKapranovVasserot95), ), see also ([Ando 00, sections II.8, II.9](#Ando00)).

More generally then genuine $G$-equivariant elliptic cohomology should assign to every $G$-action on some [[space]] $X$ a sheaf $\mathcal{F}$ of algebras over the $G$-equivariant cohomology of the point, and then the $G$-equivariant elliptic cohomology of $X$ should be the global sections of this.

While this can be made to work, it remains maybe unclear what these spaces $M_G$ "mean" and what makes them related to equivariance and elliptic cohomology. Specifically, $M_G$ turns out to be essentially the [[moduli space of flat connections]] ($G$-[[principal connections]]) on the given elliptic curve (see remark \ref{ModuliSpaceOfFlatConnections} below), which suggests strong relations to [[Chern-Weil theory]] that are not apparent here. That is considerably clarified by regarding elliptic cohomology as the [[coefficients]] for  [[motivic quantization|cohomological quantization]] of 3d and 2d [[quantum field theory]], to which we now turn.

### Interpretation in Quantum field theory/String theory
 {#InterpretationInQuantumFieldTheory}

We now try to give a maybe more conceptual explanation of what genuine equivariant (and [[twisted cohomology|twisted]]) elliptic cohomology is about, when regarded over all [[elliptic curves]] (hence: "genuine equivariant twisted [[tmf]]").

The conceptual role of plain [[elliptic cohomology]] (not equivariant) was considerably clarified when ([Witten 87](Witten+genus#Witten87)) identified the  [[elliptic genus]] (an element in the elliptic cohomology of a point) with the ([[large volume limit]] of) the [[partition function]] of a [[2d superconformal field theory]] -- the [[worldsheet]] [[quantum field theory]] of the "[[superstring]]" -- where the [[worldsheet]] [[Riemann surface]] of the string is identified with the given [[elliptic curve]].

If the [[superstring]] here is specifically the [[heterotic string]] then its dynamics and hence its [[partition function]] depends in general not just on the [[target space|target spacetime]] $X$ (of which it yields the [[elliptic genus]]) but also on a [[background field|background]] [[gauge field]] for some [[gauge group]] $G$, underlying which is a $G$-[[principal bundle]] over that spacetime. In ([Kefeng Liu, 95](Witten+genus#KL95)) a succinct description of these "twisted" elliptic genera, twisted by a $G$-principal bundle, was given in terms of [[Kac-Weyl characters]] of [[associated bundle|associated]] [[loop group]] bundles. In ([Distler-Sharpe 07](#DistlerSharpe07)) the chiral [[WZW-model]] part of the [[heterotic string]] [[2d SCFT]] which emobodies the effect of this background gauge bundle was realized geometrically as a bundle of [[parameterized WZW models]] over $X$, and ([Ando 07](#Ando07)) highlighted (see [Distler-Sharpe 07, section 8.5](#DistlerSharpe07)) that this provides the string theoretic interpretation of ([Kefeng Liu, 95](Witten+genus#KL95)), in particular ([Ando 07](#Ando07)) indicates that the corresponding twisted Witten genus lands in $G$-equivariant elliptic cohomology.

Now in the special case that $X$ here is the point, then any [[parameterized WZW model]] over $X$ is just the plain single [[WZW model]], while the plain [[Witten genus]] of $X$ vanishes. So in this case the interpretation of ([Ando 07](#Ando07)) says that the [[partition function]] of the $G$-[[WZW model]] should be an element in the $G$-equivariant elliptic cohomology of the point. But that partition function is an element in the space of [[conformal blocks]] of the WZW-model over a [[torus]] worldsheet, hence over a complex [[elliptic curve]]. Therefore the $G$-equivariant elliptic cohomology of the point should accommodate the [[conformal blocks]] of the WZW model over the given elliptic curve. (See also below at _[Properties -- Relation to conformal blocks](#RelationToConformalBlocks)_).

Next, by the [[holographic principle]] of the [[AdS3-CFT2 and CS-WZW correspondence|3dCS/2dWZW-correspondence]], the space of [[conformal blocks]] of the [[WZW model]] on a surface is identified with the [[space of quantum states]] of [[Chern-Simons theory]] over that surface. This in turn, by the general rules of [[geometric quantization]] and specifically by the discussion at _[[quantization of 3d Chern-Simons theory]]_, is the space of [[holomorphic sections]] of a [[prequantum line bundle]] over the [[moduli space of flat connections]] ($G$-[[principal connections]]) $M_G$ over the given elliptic curve. And _that_ is indeed what $G$-equivariant elliptic cohomology assigns to the point.

In other words, universal $G$-equivariant elliptic cohomology (meaning: we vary over the [[moduli space of elliptic curves]]), hence _$G$-equivariant [[tmf]]_ of the point, is essentially the [[modular functor]] of [[3d Chern-Simons theory]]. This last statement appears as ([Lurie 09, remark 5.2](#Lurie)).

But observe that actually it is a bit more: a [[modular functor]] assigns just an abstract vector space to a surface, which however is meant to be obtained by the process of [[quantization]] of [[3d Chern-Simons theory]], explicitly as the space of holomorphic sections of the [[prequantum line bundle]] (over [[phase space]], which here is the [[moduli space of flat connections]] $M_G$ on the given elliptic curve). (Beware that, while this is true over the complex numbers, as discussed [here](moduli+space+of+connections#FlatConnectionsOverATorus), it is at least subtle in the algebro geometric context of elliptic cohomology, see Jacob Lurie's MO comment [here](http://mathoverflow.net/a/226716/381)). Equivariant elliptic cohomology/tmf actually remembers this [[quantization]] process and not just the resulting [[space of quantum states]] in that it actually assigns to an [[elliptic curve]] $C$ and suitable [[Lie group]] $G$ that [[prequantum line bundle]] over the [[moduli space of elliptic curves]] (or equivalently its [[sheaf]] of sections). Notice that this [[local prequantum field theory|pre-quantum]] information is crucial for deep aspects in the context of [[3d Chern-Simons theory]] and the 2d [[Wess-Zumino-Witten model]]: the [[holographic principle|holographic relation]] that identifies the latter as the [[boundary field theory]] of the former (explicitly so by the [[FRS-theorem on rational 2d CFT]]) needs as input not just the quantized Chern-Simons [[3d TQFT]], which will assign an "abstract" vector space to a surface, but needs to know how this space arose via [[quantization]] by choosing [[polarizations]] in the form of [[conformal structures]] on the elliptic curves, such as to be actually identified with a space of [[conformal blocks]]. (In the context of the [[Reshetikhin-Turaev construction]] of the Chern-Simons [[3d TQFT]] this information is in a choice of [[equivalence of categories|equivalence]] of the given [[modular tensor category]] with the [[category of representations]] of a rational [[vertex operator algebra]]).

In summary we have as a **slogan** that:

* _$G$-Equivariant $tmf$ over the point is essentially an incarnation of the [[local prequantum field theory|pre-quantum]] [[modular functor]]_ of [[3d Chern-Simons theory|3d G-Chern-Simons theory]] over [[genus of a surface|genus]]-1 surfaces/[[elliptic curves]]_, together with the [[quantization]]-process of that to the actual [[modular functor]]_.

Moreover, by the above reasoning via ([Ando 07](#Ando07)) and using the [[AdS3-CFT2 and CS-WZW correspondence|3dCS/2dWZW holographic correspondence]] we also have the interpretation of $G$-equivariant tmf (universal $G$-equivariant elliptic cohomology) over a more general space $X$: the space of [[conformal blocks]] of a bundle of [[parameterized WZW models]] over $X$, regarded pointwise as the gauge coupling part of the twisted [[Witten genus]].

Here all the statements on the QFT/string theory side involve a parameter called the "level", which is the [[characteristic class]] of the [[universal Chern-Simons circle 3-bundle]] that is the [[prequantum n-bundle|prequantum 3-bundle]] governing the [[3d Chern-Simons theory]] (whose [[transgression]] to the [[moduli space of flat connections]] is the "theta"-[[prequantum line bundle]] there). On the cohomological side this corresponds to a [[twisted cohomology|twist]] of the cohomology theory.

Now with equivariant $tmf$ identified with the [[quantization of Chern-Simons theory]] in [[dimension]] 2 this way (the [[modular functor]] together with its pre-quantum origin via [[geometric quantization]]), the physical desireability of [[local quantum field theory]] ("[[extended TQFT]]") suggests to ask for a refinement of this also to dimensions 1 and 0, such that the higher dimensional data arises by "tracing"/[[transgression]]. There is such a [[local prequantum field theory]] refinement of 3d Chern-Simons theory, governed in dimension 0 by the [[universal Chern-Simons circle 3-bundle]] regarded as a [[prequantum n-bundle|prequantum 3-bundle]]. Indeed, the [[transgression]] of that to the [[moduli space of flat connections]] is precisely the [[prequantum bundle]] over $M_G$ that appears in the above discussion (e.g. [[schreiber:Extended higher cup-product Chern-Simons theories|FSS 12]], [[schreiber:A higher stacky perspective on Chern-Simons theory|FSS 13]]).

Now that [[universal Chern-Simons circle 3-bundle]] in turn is modulated by the geometric refinement of the universal [[second Chern class]]/[[first fractional Pontryagin class]] given by a map of [[smooth infinity-stacks]] of the form $\mathbf{B}G \to \mathbf{B}^3 U(1)$. This exhibits a homomorphism of [[smooth infinity-group]] $G \to \mathbf{B}^2 U(1)$ (to the [[circle n-group|circle 3-group]]) and so one might wonder if there is a way to "globalize" the equivariance of equivariant elliptic cohomology (in the sense of "[[global equivariant homotopy theory]]") such that it may be evaluated also on [[n-group|3-groups]] such as $\mathbf{B}^2 U(1)$ and such that the homomorphism above then _induces_ the previous 1-equivariant data by transgression.

Such a "localization" of equivariant elliptic cohomology seems to be just what is being vaguely hinted at in ([Lurie, section 5.1](#Lurie)) under the name "[2-equivariant elliptic cohomology](#2EquivariantEllipticCohomology)", we discuss this in more detail [below](#2EquivariantEllipticCohomology).

Hence we arrive at a refinement of the above **slogan**:

* _2-Equivariant $tmf$ over the point should be [[3d Chern-Simons theory]] in dimension 2 (hence the [[modular functor]]) [[extended TQFT|localized]] to dimensions 1 and 0 as a [[local prequantum field theory]] and including its [[higher geometric quantization]] in these dimensions 0,1, and 2_.

> A formal systematic discussion of this story in [[schreiber:Quantization via Linear homotopy types|cohomological quantization]] is going to be in ([Nuiten-S.](#NS)). It essentially amounts to the discussion of diagram ([0.0.4 b)](#ESI14).

[[!include moduli of higher lines -- table]]

## Definition

### 1-Equivariant elliptic cohomology

Let $G$ be a [[compact Lie group]]. Write $T \hookrightarrow G$ for its [[maximal torus]] and $W$ for its [[Weyl group]].

Let $E \in CRing_\infty$ be an [[elliptic spectrum|elliptic]] [[E-∞ ring]] [[spectrum]] with [[elliptic curve]] $A \to Spec E$.

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

Write

$$
  A_G \coloneqq \left[\left[T,\mathbb{T}\right],A\right]/W
  \;\;
  \in Sch_{/Spec E}
$$

for the [[derived scheme]] formed from the [[character group]] of the [[maximal torus]] mapped into the given [[elliptic curve]].

=--

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

This $A_G$ is the [[moduli space|moduli scheme]] of [[stable bundle|semistable]] $G$-[[principal bundles]] over the dual elliptic curve $A^\vee$ ([Ginzburg-Kapranov-Vasserot 95, (1.4.5)](#GinzburgKapranovVasserot95)).

=--

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

For geometry over the [[complex numbers]] and  $A = \mathbb{C}/\tau$ a 2-[[torus]], the scheme $A_G$ is the [[moduli space of flat connections]] on $A$, by the discussion at _[moduli space of connections -- flat connections over a torus](moduli+space+of+connections#FlatConnectionsOverATorus)_.

=--

Wtite $L Top_G$ for the collection of [[G-CW complexes]].
Write $Orb(G)$ for the [[orbit category]] of $G$.

By [[Elmendorf's theorem]] we have a [[equivalence of (∞,1)-categories]]

$$
  L Top_G
  \stackrel{\simeq}{\longrightarrow}
  PSh_\infty(Orb(G))
  \,.
$$

Let the [[global points]] of the [[elliptic curve]] $A$ over $Spec E$ be equipped with an _orientation_ in the sense of a non-degenerate [[∞-group]] homomorphism of the form

$$
  B U(1) \longrightarrow A(Spec E)
$$

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

Induced form this (...) is an [[essential geometric morphism]]

$$
  PSh_\infty(Orb(G))
  \stackrel{\overset{(-)\otimes_G A}{\longrightarrow}}{\stackrel{\overset{}{\leftarrow}}{\underset{}{\longrightarrow}}}
  Sh_\infty(Aff_E)_{/A_G}
$$

to the [[slice (∞,1)-topos]] over $A_G$.

=--

([Gepner 05, theorem 3](#Gepner05))

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

Let

$$

  \mathcal{O}
  \;\colon\;
  Sh_\infty(Aff_E)
  \longrightarrow
  Aff_E
  \simeq
  E Alg^{op}
$$

be the [[left adjoint]] to the [[(∞,1)-Yoneda embedding]] as discussed at _[[function algebras on ∞-stacks]]_.

=--

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

The composite

$$
  L Top_G
  \hookrightarrow
  PSh_\infty(Orb(G))
   \stackrel{(-)\otimes_G A}{\longrightarrow}
  Sh_\infty(Aff_E)_{/A_G}
  \stackrel{\mathcal{O}}{\longrightarrow}
  E Alg^{op}
$$

takes a space with $G$-[[action]] to its $G$-equivariant elliptic cohomology spectrum.

=--

([Gepner 05, theorem 4](#Gepner05))

### 2-Equivariant elliptic cohomology
 {#2EquivariantEllipticCohomology}

> under construction, tentative

In ([Lurie, section 5.1](#Lurie)) is a vague mentioning of a more general perspective, where one evaluates elliptic cohomology not just on [[action groupoids]] of a [[group]], such as $B G$ but also on [[homotopy quotients]] of [[2-groups]], such as notably the [[string 2-group]], and how that gives a more conceptual picture.

The following are some remarks on how to possibly realize this and at the same time refine it to geometric cohomology ([[differential cohomology]]). Tentative. Handle with care.

So let $G$ be a simple, simply connected [[compact Lie group]].

Regard $\mathbf{B}G$ in [[Smooth∞Grpd]] = $Sh_\infty(SmthMfd)$. Then by the discussion at [[Lie group cohomology]] we have

$$
  \pi_0\mathbf{H}(\mathbf{B}G, \mathbf{B}\mathbb{C}^\times)
  \simeq
  H(B G, K(\mathbb{Z},4))
  \simeq
  \mathbb{Z}
  \,.
$$

The [[∞-group extension]] classified by $k \in \mathbb{Z} \in \pi_0\mathbf{H}(\mathbf{B}G, \mathbf{B}\mathbb{C}^\times)$ is the [[string 2-group]] at level $k$

$$
  \array{
    \mathbf{B}\mathbb{C}^\times &\longrightarrow& \mathbf{B}String_k(G)
    \\
    && \downarrow
    \\
    && \mathbf{B}G &\stackrel{k\mathbf{c}}{\longrightarrow}& \mathbf{B}^3 \mathbb{C}^\times
  }
$$

This cocycle has a [[differential cohomology]]-refinement to the [[universal Chern-Simons 3-connection]]

$$
  k \mathbf{L} \;\colon\; \mathbf{B}G_{conn} \longrightarrow \mathbf{B}^3 \mathbb{C}^\times_{conn}
$$

Now given a [[torus]] $E = T^2$, regarded, for the moment, as a [[smooth manifold]], we have the [[transgression]] of the definition cocycle

$$
  \exp\left(
    \tfrac{i}{\hbar}
    \int_{E} k \mathbf{L}
  \right)
  \;\colon\;
  [E, \mathbf{B}G_{conn}]
   \stackrel{[E, k \mathbf{L}]}{\longrightarrow}
  [E, \mathbf{B}^3 \mathbb{C}^\times_{conn}]
   \stackrel{\exp\left(\tfrac{i}{\hbar} \int_{E}(-)\right)}{\longrightarrow}
  \mathbf{B}\mathbb{C}^\times_{conn}
$$

which now defines a $\mathbb{C}^\times$ [[bundle with connection]] on the [[moduli stack of connections]] on $E$.  We can restrict to the [[moduli stack of flat connections]], the [[phase space]] of $G$-[[Chern-Simons theory]]. This is the [[Hitchin connection]].

Consider then a collection of tori $E$ parameterized trivially over some parameter space $B$.

$$
  E \times B \to B
  \,.
$$

Then the above yields

$$
  [(\Pi E) \times B, \mathbf{B}G]
   \stackrel{}{\longrightarrow}
  \mathbf{B}[B,\mathbb{C}^\times]
$$

hence yields a $[B,\mathbb{C}^\times]$-bundle over the [[moduli space]] of $B$-collections of flat connections on $E$.

Now we want to consider this for the case that $B$ is a space in [[spectral geometry]].

To that end, pass to the larger [[(∞,1)-topos]] of [[smooth E-∞ groupoids]] over the [[complex numbers]].

Let $\mathbb{G}_m$ there denote the object which to a pair consisting of a [[smooth manifold]] $U$ and an [[E-∞ ring]] $R$ assigns

$$
  \mathbb{G}_m \;\colon\; (U, R) \mapsto GL_1(R) \otimes C^\infty(U,\mathbb{C}^\times)
$$

hence the [[tensor product]] of the [[∞-group of units]] of $R$ with the underlying [[abelian group]] of [[smooth functions]] on $X$ with values in $\mathbb{C}^\times$.

Let then $A \in CAlg_\infty$ be an [[E-∞ ring]], and take now $B = Spec(A)$. Write

$$
  E \to Spec(A)
$$

for a $B$-collection of [[tori]], now taken to be an [[elliptic curve]] over $Spec(A)$.

Since for a [[torus]] its [[fundamental group]] is [[isomorphic]] to its [[character group]] (via the canonical non-degenrarate [[bilinear form]] on both), we take the [[fundamental groupoid]] $\Pi(E)$ now to be

$$

  \mathbf{B}[E, \mathbb{G}_m]
  \,.
$$

Then since $\mathbb{C}^\times = \mathbb{G}_m$ is the [[multiplicative group]] in this context, we have now (and there is a subtlety here...) that maps

$$
  Spec(A) \longrightarrow \mathbb{G}_m
$$

are equivalently elements in the [[∞-group of units]] of $A$. So we should get an $A$-[[(∞,1)-module bundle]] modulated by

$$
  \chi
   \colon
  [ \mathbf{B}[E,\mathbb{G}_m], \mathbf{B}G ]
  \longrightarrow
  \mathbf{B}GL_1(A)
  \,.
$$

Forming its space of co-sections yields, by the discussion at [[Thom spectrum]], the $\chi$-[[twisted cohomology|twisted]] A-[[cohomology]] [[spectrum]]

$$
  A^\chi([ \mathbf{B}[E,\mathbb{G}_m], \mathbf{B}G ])
  \,.
$$

And that should be the $G$-"equivariant" elliptic cohomology of the point. Actually the [[motivic quantization]] of $G$-[[Chern-Simons theory]].

(...)

## Properties
 {#Properties}

### Relation to conformal blocks of the WZW model
 {#RelationToConformalBlocks}

For $G$ a compact, simple and simply connected Lie group,
consider the [[string 2-group]] [[∞-group extension]]

$$
  \array{
    \mathbf{B}^2 U(1) &\to& \mathbf{B}String
    \\
    && \downarrow
    \\
    && \mathbf{B}G
  }
  \,.
$$

The corresponding [[higher moduli stacks]] of [[flat ∞-connections]] on an [[elliptic curve]] $T$ form the [[∞-group extension]]

$$
  \array{
    [\Pi(T),\mathbf{B}^2 U(1)] &\to& [\Pi(T),\mathbf{B}String]
    \\
    && \downarrow
    \\
    && [\Pi(T), \mathbf{B}G]
  }
  \,.
$$

Now passing to the [[0-truncation]] turns the bottom piece into the [[moduli space of flat connections]] on the torus, which is $A_G$, def. \ref{TheModuliSpace}, remark \ref{ModuliSpaceOfFlatConnections}.

By the discussion at [smooth higher holonomy](smooth+infinity-groupoid#StrucChernSimons) the 0-truncation of the top left piece is $U(1)$, so under 0-truncation we should get a $U(1)$-[[principal bundle]]

$$
  \array{
    U(1) &\longrightarrow& \tau_0 [\Pi(T),\mathbf{B}String]
    \\
    && \downarrow
    \\
    && A_G
  }
  \,.
$$

This state of affairs is hinted at in ([Lurie, section 5.1](#Lurie)).

More in detail, notice that the [[string 2-group]] extension is modulated by a map

$$
  \mathbf{c} \;\colon\; \mathbf{B}G_{conn} \longrightarrow \mathbf{B}^3 U(1)_{conn}
$$

and the above circle-bundle is modulated by the [[transgression]] of that

$$
  \exp\left(
    \tfrac{i}{\hbar}
    \int_{T} \mathbf{c}
  \right)
  \;\colon\;
  [T, \mathbf{B}G_{conn}]
   \stackrel{[T,\mathbf{c}]}{\longrightarrow}
  [T, \mathbf{B}^3 U(1)_{conn}]
   \stackrel{\exp(\tfrac{i}{\hbar} (-) )}{\longrightarrow}
  \mathbf{B}U(1)_{conn}
  \,.
$$

(By the discussion at [[schreiber:differential cohomology in a cohesive topos|dcct]].)

By the general discussion at [[quantization of Chern-Simons theory]] and the [[holography|holograpic]] [[AdS3-CFT2 and CS-WZW correspondence|CS-WZW correspondence]], the space of [[sections]] of this line bundle is the space of [[conformal blocks]] of the [[Wess-Zumino-Witten model]] on $T$.

(This statement also appears as ([Lurie, remark 5.2](#Lurie))).

### Relation to loop group representations
 {#RelationToLoopGroupRepresentations}

When restricting the above general construction
to the [[Tate curve]], then the conformal blocks become
[[loop group representations]] (when over the complex numbers, at least, [Ando00, theorem 10.10](#Ando00)).

In terms of differential geometry ([[schreiber:differential cohomology in a cohesive topos|dcct]]) consider the map

$$
  G \longrightarrow [S^1, \mathbf{B}G_{conn}]
$$

which locally sends a group elemnent $g$ to the constant [[principal connection]] on the circle with $g$ as its [[holonomy]].

This induces an inclusion

$$
  [S^1, G]
    \hookrightarrow
  [S^1, [S^1, \mathbf{B}G_{conn}]]
    \simeq
  [T, \mathbf{B}G_{conn}]
$$

and pulling the above WZW circle bundle back along this inclusion
yields the bundle on the [[loop group]] which is the [[prequantum bundle]]
whose [[geometric quantization]] yields the [[loop group representations]]
of [[positive energy representations|positive energy]].

Algebraically, this corresponds to evaluating equivariant elliptic cohomology on the [[Tate curve]], this is ([Lurie, theorem 5.1](#Lurie)).

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

In the full [[derived algebraic geometry]] the space of sections of the line
bundle on the moduli space has the structure of a $K((q))$-[[∞-module]], hence of an actual [[spectrum]] ([Lurie, below remark 5.4](#Lurie)).

=--

(...)

### Relation to the Chern-Simons $\infty$-line bundle on $\mathbf{B}G$
 {#RelationToLineBundleOnBSpin}

Given an [[E-∞ ring]] $A$ with an oriented [[derived elliptic curve]] $\Sigma \to Spec(A)$ there are a priori two different $A$-[[∞-line bundles]] on $B Spin$.

On the one hand there is the bundle classified by

$$
  J_A
  \;\colon\;
  B Spin
  \stackrel{}{\longrightarrow}
  B O
  \stackrel{J}{\longrightarrow}
  B GL_1(\mathbb{S})
  \longrightarrow
  B GL_1(A)
  \,,
$$

where $\mathbb{S}$ is the [[sphere spectrum]], $GL_1(-)$ the [[∞-group of units]]-construction and $J$ the [[J-homomorphism]].  (This is what appears as $\mathcal{A}_s$ in [Lurie, middle of p.38](#Lurie)). Notice that by ([Ando-Blumberg-Gepner 10, section 8](#AndoBlumbergGepner10)), for the case $A = $ [[tmf]] this is equivalently the $A$-[[∞-line bundle]] [[associated ∞-bundle|associated]] to the [[universal Chern-Simons line 3-bundle]]

$$
  A(\tfrac{1}{2}p_1)
  \;\colon\;
  B Spin
  \stackrel{\tfrac{1}{2}p_1}{\longrightarrow}
  B^4 \mathbb{Z}
  \stackrel{\tilde \sigma}{\longrightarrow}
  B GL_1(A)
  \,,
$$

where $\tfrac{1}{2}p_1$ is the [[first fractional Pontryagin class]] and $\tilde \sigma$ is an [[adjunct]] of the [[string orientation of tmf]].

In addition, by equivariant elliptic cohomology there is the [[theta line-bundle]]

$$
  \theta \;\colon\; Loc_{Spin}(\Sigma) \longrightarrow \mathbf{B} \mathbb{G}_m
$$

on the derived [[moduli stack of flat connections]] $Loc_{Spin}(\Sigma)$
(where in ([Lurie](#Lurie)) $Loc_{Spin}(\Sigma)$ is denoted $M_{Spin}$). Evaluating this bundle on [[global points]] yields the $A$-[[∞-line bundle]]

$$
  \Gamma_{Spec(A)}(\theta) \;\colon\; \Gamma_{Spec(A)}(Loc_{Spin}(\Sigma))
  \longrightarrow B GL_1(A)
  \,.
$$

So there are a priori two $A$-$\infty$-oine bundles on bare homotopy types here. But (by 2-equivariance, [Lurie, bottom of p. 38](#Lurie)) there is a canonical map between their base spaces

$$
   \phi \;\colon\; B Spin \longrightarrow \Gamma_{Spec(A)}(Loc_{Spin}(\Sigma))
  \,.
$$

Heuristically, this is the map that includes the trivial $Spin$-local system and its [[gauge transformations]] into the (points of the) [[moduli stack]] of all local systems.

Hence the pullback of $\Gamma_{Spec(A)}(Loc_{Spin}(\Sigma))$ yields another $A$-line bundle $\phi^\ast \Gamma_{Spec(A)}(\theta)$ over $B Spin$.

These are equivalent

$$
  J_A \simeq  \phi^\ast \Gamma_{Spec(A)}(\theta)
  \,.
$$

This is ([Lurie, theorem 5.2](#Lurie)).

$$
  \array{
    && Loc_{Spin}(\ast)
    \\
    & \swarrow && \searrow^{\mathrlap{\phi}}
    \\
    Loc_{Spin}(\ast)
    &&
     \swArrow_{\simeq}
    &&
    Loc_{Spin}(\Sigma)
    \\
    & {}_{\mathllap{J_A}}\searrow && \swarrow_{\mathrlap{\theta}}
    \\
    && B GL_1(A)
  }
$$

## Related concepts

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

* [[equivariant elliptic genus]]

* [[rigidity theorem for elliptic genera]]

* [[orbifold cohomology]]

  * [[orbifold K-theory]]

* related but different: [[modular equivariant elliptic cohomology]]

* [[equivariant K-theory]]

* [[equivariant algebraic K-theory]]

## References

[[!include elliptic cohomology -- references]]

### Of quiver varieties

On [[equivariant elliptic cohomology]] of [[quiver varieties]] in relation to the [[AGT correspondence]]:

* [[Mina Aganagic]], [[Andrei Okounkov]], _Elliptic stable envelopes_ ([arXiv:1604.00423](https://arxiv.org/abs/1604.00423))

* [[Andrei Okounkov]], _Inductive construction of stable envelopes and applications, I. Actions of tori. Elliptic cohomology and K-theory_ ([arXiv:2007.09094](https://arxiv.org/abs/2007.09094))

following the analogous non-elliptic discussion in:

* [[Davesh Maulik]], [[Andrei Okounkov]], _Quantum Groups and Quantum Cohomology_, Asterisque 408(2019) ([arXiv:1211.1287](https://arxiv.org/abs/1211.1287), [ISBN: 978-2-85629-900-5](https://bookstore.ams.org/ast-408))

Review in:

* Andrey Smirnov, _Stable envelopes for $A_n$, $\widehat A_n$-quiver varieties_, 2019 ([pdf](https://math.mit.edu/events/quantum/a_smirnov.pdf))

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


equivariant elliptic genus

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=--
#### Cohomology
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#### Manifolds and cobordisms
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The  [[equivariant elliptic cohomology]]-version of the [[elliptic genus]].

## Properties

### As partition function of gauged string sigma-models

See at _[[gauged WZW model]]_.

### Rigidity theorem

See _[[rigidity theorem for elliptic genera]]_

## References

* [[David Gepner]], section 1.1 of _[[Homotopy topoi and equivariant elliptic cohomology]]_, 2005

* Robert Waelder, _Equivariant Elliptic Genera_ ([arXiv:math/0603521](http://arxiv.org/abs/math/0603521))

See also the references at _[[M5-brane elliptic genus]]_.

[[!redirects equivariant elliptic genera]]

[[!redirects equivariant Witten genus]]
[[!redirects equivariant Witten genera]]


equivariant formal group

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#### Algebra
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[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A generalization of the concept of [[formal groups]] to a situation where an additional equivariance group acts.

In generalization of how [[formal groups]] correspond to [[complex oriented cohomology theories]] so equivariant formal groups correspond to [[equivariant complex oriented cohomology theories]].

## Properties

### Universal equivariant formal group

Greenlees shows ([Greenlees 01a](#Greenlees01a), [Greenlees 01b, Theorem 13.1](#Greenlees01b)
) that the coefficient ring of [[equivariant complex cobordism theory]] classifies [[equivariant formal group laws]] over [[Noetherian rings]], but the general conjecture is still open.

## References

* [[Michael Cole]], [[John Greenlees]], [[Igor Kriz]], _Equivariant Formal Group Laws_, Proceedings of the LMS, Volume81, Issue2 (2000)([doi:10.1112/S0024611500012466]( https://doi.org/10.1112/S0024611500012466))

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

* [[John Greenlees]], _Multiplicative equivariant formal group laws_, Journal of Pure and Applied Algebra Volume 165, Issue 2, 7 December 2001, Pages 183-200 (doi:10.1016/S0022-4049(00)00184-5)

* {#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))

* Chun Lung Liu, _Equivariant Algebraic Cobordism and Equivariant Formal Group Laws_ ([arXiv:1305.2053](https://arxiv.org/abs/1305.2053))

[[!redirects equivariant formal groups]]

[[!redirects equivariant formal group law]]
[[!redirects equivariant formal group laws]]


equivariant group

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#### Group Theory
+-- {: .hide}
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=--
#### Representation theory
+-- {: .hide}
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $G$ a fixed [[group]], to be called the _[[equivariance group]]_, then
by a _$G$-equivariant group_ we mean a [[group object]] [[internalization|internal]] to $G$-[[actions]], e.g. [[internalization|internal]] to [[G-sets]], [[G-spaces]], or [[G-manifolds]], etc.

Beware that the term "equivariant group" for this notion is non-standard; or rather: there is no other established term for this notion at all.
The term is meant to rhyme on the established terminology of _[[equivariant principal bundles]]_, in which context it serves to make nicely transparent the full (but often hidden) [[internalization|internal]] nature of the concept: [[equivariant principal bundles]] have, in general, equivariant groups as their [[structure groups]].

## Definition

We discuss  equivariant groups in/as [[topological spaces]], for definiteness and due to their relevance as models in [[equivariant homotopy theory]].  All of the discussion generalizes, say to [[smooth manifolds]] or general [[toposes]] (see at _[category of G-sets -- For internal group actions](category+of+G-sets#InternalGroupActions)_).

### Preliminaries

\begin{defn}
(**convenient category of topological spaces**)
\linebreak
We write
$$
  TopologicalSpaces
  \;\in\;
  Categories
$$
for any [[convenient category of topological spaces]] whose [[mapping space]] serves as an [[internal hom]], such as

* [[D-topological spaces]]

* [[compactly generated topological spaces]].

This means in particular that for $X,Y,A \,\in\, TopologicalSpaces$, we have a [[natural bijection]]

\[
  \label{HomAdjunctInTopologicalSpaces}
  X \times Y
    \longrightarrow
  A
  \;\;\;\;\;\;\;
  \overset{adjuncts}{\leftrightarrow}
  \;\;\;\;\;\;\;
  X
    \longrightarrow
  Maps(Y,A)
\]

between [[maps]] (meaning: [[continuous functions]]) out of the [[product topological space]] with $Y$ and maps into the [[mapping space]].

\end{defn}

\begin{defn}\label{TopologicalGSpaces}
(**topological $G$-spaces**)
\linebreak
For $G$ be a [[topological group]] -- to be called the _[[equivariance group]]_ -- we write
$$
  G Actions(TopologicalSpaces)
  \;\in\;
  Categories
$$
for the [[category]] whose

* [[objects]] $(X,\rho)$ are [[topological spaces]] $X$ equipped with [[continuous function|continuous]] $G$-[[actions]]

  $$
    G \times X \overset{}{\longrightarrow} X
    \;\;\;\;\;\;
    \leftrightarrow
    \;\;\;\;\;\;
    G \overset{\rho}{\longrightarrow} Aut(X) \subset Maps(X,X)
    \,;
  $$

* [[morphisms]] are $G$-[[equivariant function|equivariant]] [[continuous functions]] between them ("maps"); i.e. for the category of "[[G-spaces]]", often denote "$G Spaces$" or even $G Sp$ or similar.

\end{defn}

\begin{remark}
(**further conditions on the equivariance group**)
\linebreak
For purposes of [[equivariant homotopy theory]] one typically assumes the [[topological group|topological]] [[equivariance group]] $G$ in Def. \ref{TopologicalGSpaces} to be that underlying a [[compact Lie group]], such as a [[finite group]] (as that guarantees that [[G-CW-complexes]] are well-behaved and that the [[equivariant Whitehead theorem]] holds). But for the plain [[point-set topology]] of equivariant groups and their [[equivariant bundles]] this condition is not necessary.
\end{remark}

\begin{remark}\label{GSpacesAreCartesianMonoidal}
(**[[topological G-spaces]] are [[Cartesian monoidal category|cartesian monoidal]]**)
  The category of [[topological G-spaces]] (Def. \ref{TopologicalGSpaces}) is a [[Cartesian monoidal category]]: The [[Cartesian product]] of two [[topological G-spaces]] $(X_i, \rho_i)$ is the underlying [[product topological space]] equipped with the [[diagonal action]] by $G$:
$$
  (X_1, \rho_1)
  \times
  (X_2, \rho_2)
  \;=\;
  \Big(
    X_1 \times X_2,
   \,
   \rho_{1,2}(g)(x_1,x_2)
   \,=\,
   \big(
     \rho_1(x_1), \rho_2(x_2)
   \big)
  \Big)
  \,.
$$
\end{remark}
In a [[Cartesian monoidal category]] there is a notion of [[internalization|internal]] [[group objects]]:

### Equivariant groups

\begin{defn}\label{EquivariantTopologicalGroup}
(**equivariant topological groups**)
\linebreak
Given an [[equivariance group]] $G$ (Def. \ref{TopologicalGSpaces}),
a _$G$-[[equivariant topological group]]_ $(\mathcal{G}, \alpha)$ is a [[group object]] [[internalization|internal]] to [[topological G-spaces]] (Def. \ref{TopologicalGSpaces}):
  $$
    \big(
      \mathcal{G},
      \,
      \alpha
    \big)
    \;\in\;
    Groups
    \big(
      G Actions
      (
        TopologicalSpaces
      )
    \big)
    \,.
  $$
\end{defn}
(See Prop. \ref{EquivariantGroupsAsSemidirectProductGroups} below for the choice of notation used here.)

\begin{remark}\label{UnderlyingTopologicalGroupOfAnEquivariantTopologicalGroup}
Since the [[forgetful functor]] from [[topological G-spaces]] (Def. \ref{TopologicalGSpaces}) to underlying [[topological spaces]]
$$
  \array{
    G Actions
    (
      TopologicalSpaces
    )
   &
     \overset{
     }{\longrightarrow}
   &
   TopologicalSpaces
  }
$$
[[preserved limit|preserves]] [[Cartesian products]] (explicitly so by Remark \ref{GSpacesAreCartesianMonoidal}), it preserves [[group objects]] and hence sends $G$-equivariant topological groups (Def. \ref{EquivariantTopologicalGroup}) to underlying plain [[topological groups]]:
\[
  \label{ForgetfulFunctorFromEquivariantTopologicalGroupsToTopologicalGroups}
  \array{
    Groups
    \big(
    G Actions
    (
      TopologicalSpaces
    )
   \big)
   &
     \overset{
     }{\longrightarrow}
   &
   Groups
   (
     TopologicalSpaces
   )
   \\
   \big(
     \mathcal{G}
     ,
     \alpha
   \big)
   &\mapsto&
   \mathcal{G}
   \,.
  }
\]
\end{remark}

## Properties

### Equivalence with semidirect products with $G$

\begin{prop}\label{EquivariantGroupsAsSemidirectProductGroups}
  (**Equivariant groups as semidirect product groups**)
\linebreak
The [[category]] of $G$-equivariant topological groups (Def. \ref{EquivariantTopologicalGroup}) is [[equivalence of categories|equivalent]]

\[
  \label{EquivalenceOfEquivariantTopologicalGroupsToSemidirectProductTopGroups}
  \array{
    Groups
      \big(
        G Actions
        (
          TopologicalSpaces
        )
      \big)
    &
      \overset{
         \;\;
         \;\;
      }{\hookrightarrow}
    &
    Groups^{G/}_{/G}
    \\
    \big(
      \mathcal{G},
      \alpha
    \big)
    &\mapsto&
    \mathcal{G} \rtimes_{\alpha} G
    \,.
  }
\]
to that of [[semidirect product groups]] of the form $(-) \rtimes G$
\[
  \label{ASemidirectProductGroup}
 \mathcal{G} \rtimes_\alpha G
  \;\;
   \coloneqq
  \;\;
  \Big(
    \mathcal{G} \times G
    \,,
    \;\;\;
    (\gamma_1, g_1) \cdot (\gamma_2, g_2)
    \;\coloneqq\;
    \big(
      \gamma_1 \cdot \alpha(g_1)(\gamma_2)
      ,\,
      g_1 \cdot g_2
    \big)
  \Big)
\]
and regarded as [[pointed objects]] in the [[slice category]] of [[Groups]] over $G$ via the canonical homomorphisms
\[
  \label{CoSliceMorphismsInOutOfSemidirectProductGroup}
  \array{
    G
    &\longrightarrow&
    \mathcal{G} \times_\alpha G
    &\longrightarrow&
    G
    \\
    &&
    (\gamma,g) &\mapsto& g
    \\
    g &\mapsto& (e_{\mathcal{G}},g)
    \,.
  }
\]
(which jointly witness the semidirect product as a split group extension of $G$, see [there](group+extension#SplitExtensionsAndSemidirectProductGroups)).
\end{prop}
\begin{proof}
This is a straightforward matter of unwinding the definitions:

First to see that we have a [[functor]] as claimed:

A [[group object]] in $G Actions(TopologicalSpaces)$ is, by definition, a plain [[topological group]] $\mathcal{G}$ whose underlying topological space is equipped with a continuous $G$-[[action]] and such such this $G$-action preserves all its group operations. In other words, this is a group $\mathcal{G}$ and a [[homomorphism]]
$\alpha \;\colon\;G \longrightarrow Aut_{Grp}(\mathcal{G})$ to the group-[[automorphism group]] (whose [[internal hom|hom]]-[[adjunct]] (eq:HomAdjunctInTopologicalSpaces) is [[continuous function|continuous]]).

This is exactly the data that determines the [[semidirect product group]] (eq:ASemidirectProductGroup).

Moreover, a [[homomorphism]] of equivariant groups $(\mathcal{G}_1, \alpha_1) \longrightarrow (\mathcal{G}_2, \alpha_2)$ is a continuous [[group homomorphism]] $\phi \,\colon\, \mathcal{G}_1 \longrightarrow \mathcal{G}_2$ whose underlying map is $G$-[[equivariant function|equivariant]] in that

$$
  \underset{
    g \in G
  }{\forall}
  \;\;
  \;\;
  \phi \circ \alpha_1(g) \;=\; \alpha_2(g) \circ \phi
  \,.
$$

By (eq:ASemidirectProductGroup) this means that $\phi$ induces a [[group homomorphism]] of [[semidirect product groups]] of the form

$$
  \array{
    \mathcal{G}_1 \rtimes_{\alpha_1} G
    &
      \overset{
      }{\longrightarrow}
    &
    \mathcal{G}_2 \rtimes_{\alpha_2} G
    \\
    \mathcal{G}_1 \times G
    &
      \overset{
        \phi \times id_G
      }{\longrightarrow}
    &
    \mathcal{G}_2 \times G
  }
$$

This construction

\[
  \label{FunctorialAssignmentOfSemidirectProductGroupsToEquivariantGroups}
  \array{
    (\mathcal{G}_1, \alpha_1)
    &&
    &&
    \mathcal{G}_1 \rtimes_{\alpha_1} G
    \\
    \big\downarrow
     {}^{_{\mathrlap{\phi}}}
    &&
      \mapsto
    &&
    \big\downarrow
     {}^{_{\mathrlap{\phi \times id_G}}}
    \\
    (\mathcal{G}_2, \alpha_2)
    &&
    &&
    \mathcal{G}_2 \rtimes_{\alpha_2} G
  }
\]

is clearly [[functor|functorial]].

It remains to see that this functor is a [[full subcategory]]-inclusion, hence a [[fully faithful functor]], hence that  it is a [[bijection]] on [[hom-sets]] for any [[pair]] of [[objects]]:

But by (eq:FunctorialAssignmentOfSemidirectProductGroupsToEquivariantGroups) the homomorphisms of [[semidirect product groups]] in its [[image]] are precisely those of the product form $\phi \times id_G$, and this is exactly the form of the homomorphisms between semidirect product groups that is picked out by [[slice category|slicing over]] [[undercategory|and under]] $G$ (by Remark \ref{HomomOutOfSemidirectProductGroupFixedByItsRestrictionToSubgroups}).
\end{proof}

\begin{remark}\label{HomomOutOfSemidirectProductGroupFixedByItsRestrictionToSubgroups}
  Here and in the following we use that a [[group homomorphism]] out of a [[semidirect product group]] (eq:ASemidirectProductGroup) is fixed already by its restriction to the two canonical [[subgroups]]

$$
  \array{
    \mathcal{G}
    \\
    \big\downarrow
    &
    \searrow^{
     \mathrlap{ \gamma \mapsto \phi\big( \gamma, e_G \big) }
    }
    \\
    \mathcal{G}
      \rtimes_{\alpha}
    G
    &
      \overset{\phi}{\longrightarrow}
    &
    \mathcal{G}' \rtimes_\alpha G
    \\
    \big\uparrow
    &
    \nearrow_{
      \mathrlap{ g \mapsto \phi\big(e_{\mathcal{G}},g\big) }
    }
    \\
    G
  }
$$
because every element of the semidirect product is equal to
a product of elements from these subgroups:
\[
  \label{ElementInSemidirectProductAsProductOfElementsInSubgroups}
  (\gamma,g) \;=\; (\gamma, e_G) \cdot ( e_{\mathcal{G}}, g )
  \,.
\]

This implies in particular that

1. $\phi$ being a morphism in $Groups_{/G}$ means equivalently that its restriction to $\mathcal{G}$ factors (via some group homomorphism $\mathcal{G} \to \mathcal{G}'$) through the canonical inclusion of $\mathcal{G}'$ (eq:CoSliceMorphismsInOutOfSemidirectProductGroup);

1. $\phi$ being a morphism in $Groups^{G/}$ means equivalently that its restriction to $G$ is the canonical inclusion of $G$ (eq:CoSliceMorphismsInOutOfSemidirectProductGroup).

\end{remark}

### Equivariant group actions
 {#EquivariantGroupActions}

\begin{prop}\label{ActionsOfEquivariantGroupsAsSemidirectProductGroupActions}
  (**equivariant group actions as semidirect product group actions**) \linebreak
  Under the identification from Prop. \ref{EquivariantGroupsAsSemidirectProductGroups} of $G$-[[equivariant groups]]
  $\big(\mathcal{G}, \alpha \big)$
  with [[semidirect product groups]] $\mathcal{G} \rtimes_\alpha G$,
  we have an [[equivalence of categories|equivalence]] of their
  [[actions]], given by:
  \[
    \label{EquivalenceBetweenEquivariantActionsAndSemidirecProductActions}
    \array{
      \big(
        \mathcal{G},
        \,
        \alpha
      \big)
      Actions
      \big(
        G Actions
        (
          TopologicalSpaces
        )
      \big)
      &
      \overset{\simeq}{
        \longrightarrow
      }
      &
      (
        \mathcal{G} \rtimes G
      )
      Actions( TopologicalSpaces )
      \\
      (
        \mathcal{G}, \alpha
      )
      \overset{
        R
      }{
        \to
      }
      Aut
      \big(
        (X, \rho)
      \big)
      &\mapsto&
      \mathcal{G} \rtimes_\alpha G
      \overset{
        (R,\rho)
      }{\longrightarrow}
      Aut(X)
    }
  \]
  with
  \[
    \label{ActionOfSemidirectProductFromEquivariantComponentActions}
    \underset{\gamma,g,x}{\forall}
    \;\;\;
    (R,\rho)(\gamma, g)(x)
    \;\coloneqq\;
    R(\gamma)\big( \rho(g)(x) \big)
    \,.
  \]
\end{prop}

\begin{proof}

We observe that the given formula in fact establishes a [[bijection]] between the two kinds of actions:

First, notice
by the decomposition (eq:ElementInSemidirectProductAsProductOfElementsInSubgroups) in Remark \ref{HomomOutOfSemidirectProductGroupFixedByItsRestrictionToSubgroups},
that any action of $\mathcal{G} \rtimes_\alpha G$ can be written in the form (eq:ActionOfSemidirectProductFromEquivariantComponentActions) for some actions $R$ and $\rho$ of $\mathcal{G}$ and $G$, respectively, satisfying some compatibility conditions:

1. the [action property](action#eq:ActionPropertyOfGroupActions) of $\rho$  is equivalently the action property of $(R,\rho)$ on elements of the form $(e, g)$;

1. the [action property](action#eq:ActionPropertyOfGroupActions) of $R$ on the underlying topological spaces is equivalently the [action property](action#eq:ActionPropertyOfGroupActions) of $(R,\rho)$ on elements of the form $(\gamma,e)$;

1. the $G$-[[equivariant function|equivariance]] of $R$

   \[
     \label{GEquivarianceOfR}
     \underset{
       \gamma,g,x
     }{\forall}
     \;\;\;
     R
     \big(
       \alpha(g)(\gamma)
     \big)
      \big(
       \rho(g)(x)
     \big)
     \;=\;
      \rho(g)
      \big(
       R(\gamma)(x)
     \big)
   \]

   is equivalent to the [action property](action#eq:ActionPropertyOfGroupActions) of $(R,\phi)$ on mixed pairs of elements of the form $\big( (e_{\mathcal{G}},g), \; (\gamma,e_G) \big)$:

   $$
     \underset{
       \gamma,g,x
     }{\forall}
     \;\;\;
     (R,\rho)
     \big(
       \underset{
         \big(
           \alpha(g)(\gamma),
           g
         \big)
       }{
         \underbrace{
           (e,g) \cdot (\gamma,e)
         }
       }
     \big)
     (x)
     \;=\;
     (R,\rho)(e,g)
     \big(
       (R,\rho)(\gamma,e)
       (x)
     \big)
     \,.
   $$

These three conditions exhaust the conditions on $R$ to be a $G$-equivariant action. Therefore it just remains to see that they also exhaust the conditions on $(R,\rho)$ to be a plain action:

But the remaining mixed [action conditions](action#eq:ActionPropertyOfGroupActions) on $(R, \phi)$

$$
  \underset{
    \gamma_1, \gamma_2, g, x
  }{\forall}
  \;\;\;
  (R,\rho)
  \big(
    \underset{
     = \, (\gamma_1 \cdot \gamma_2,g)
    }{
    \underbrace{
      (\gamma_1,e) \cdot (\gamma_2,g)
    }
    }
  \big)
  (x)
  \;=\;
  (R,\rho)
  \big(
    (\gamma_1,e)
  \big)
  \Big(
    (R,\rho)
    \big(
      (\gamma_2,g)
    \big)
    (x)
  \Big)
$$

and

$$
  \begin{aligned}
  \underset{
    \gamma, g_1, g_2, x
  }{\forall}
  \;\;\;
  (R,\rho)
  \big(
    (e,g_1)
    \cdot
    (\gamma, g_2)
  \big)
  (x)
  & =\;
  (R,\rho)
  \Big(
    \big(
      \alpha(g_1)(\gamma), g_1 \cdot g_2
    \big)
  \Big)
  (x)
  \\
  & =\;
  R
  \big(
    \alpha(g_1)(\gamma)
  \big)
  \big(
    \rho(g_1 \cdot g_2)
    (x)
  \big)
  \\
  & \overset{!}{=} \;
  R
  \big(
    \alpha(g_1)(\gamma)
  \big)
  \Big(
    \rho(g_1)
    \big(
      \rho(g_2)
      (x)
    \big)
  \Big)
  \\
  & \overset{!}{=} \;
  \rho(g_1)
  \Big(
    R
    \big(
      \gamma
    \big)
    \big(
      \rho(g_2)
      (x)
    \big)
  \Big)
  \\
  & =\;
  (R,\rho)
  \big(
    (e,g_1)
  \big)
  \Big(
    (R,\rho)
    \big(
      (\gamma, g_2)
    \big)
    (x)
  \Big)
  \end{aligned}
$$

follow right from the definition (eq:ActionOfSemidirectProductFromEquivariantComponentActions)
and using again (in the highlighted steps "$\overset{!}{=}$"):

* the [action propery](action#eq:ActionPropertyOfGroupActions) of $\rho$ from item 2 above;

* the $G$-[[equivariant function|equivariance]] of $R$ (eq:GEquivarianceOfR) from item 3 above.

In conclusion, $(R,\rho)$ is an action of the semidirect product group $\mathcal{G} \rtimes_\alpha G$ on $X$ iff $R$ is a $G$-equivariant action of $\mathcal{G}$ on $X$.

This implies immediately that the condition for a map $X \to X$ to be an action homomorphisms on both sides are the same.

And so the functor (eq:EquivalenceBetweenEquivariantActionsAndSemidirecProductActions) is in fact an [[isomorphism]] on both objects as well as morphisms, hence in particular is an [[equivalence of categories]].
\end{proof}

## Related concepts

* [[equivariant bundle]]

* [[equivariance group]], [[structure group]], [[gauge group]]

\linebreak

[[!redirects equivariant groups]]

[[!redirects equivariant topological group]]
[[!redirects equivariant topological groups]]


equivariant homology

There is also [[equivariant cohomology]] entry.

A comparison of several aproaches to (generalized) equivariant homology is given in

* Jason Hanson, _Generalized equivariant homology on simplicial complexes_, [arxiv/1103.1555](http://arxiv.org/abs/1103.1555)


equivariant homotopy group

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

#Contents#
* table of contents
{:toc}

## Idea

The generalization of the concept of [[homotopy group]] from [[homotopy theory]] and [[stable homotopy theory]] to [[equivariant homotopy theory]] and [[equivariant stable homotopy theory]].

## Definition

### Abstractly

For $X$ a [[pointed topological space|pointed]] [[topological G-space]] and $H \subset G$ a closed [[subgroup]], the $n$th _unstable_ $H$-equivariant homotopy group of $X$ is simply the ordinary $n$-th [[homotopy group]] of the $H$-[[fixed point]] space $X^H$:

$$
  \pi_n^H(X) \coloneqq \pi_n(X^H)
  \,.
$$

With $G/H$ denoting the [[quotient space]], this is equivalently the $G$-[[homotopy classes]] of $G$-equivariant [[continuous functions]] from the [[smash product]] $S^n \wedge G/H_+$ to $X$:

$$
  \pi_n^H(X) \simeq [G/H_+ \wedge S^n, X]^G
  \,.
$$

In this form the definition directly generalizes to [[G-spectra]] and hence to stable equivariant homotopy groups: for $E$ a [[G-spectrum]], then

$$
  \pi_n^H(X) \simeq [G/H_+ \wedge \Sigma^\infty S^n, X]^G
  \,.
$$

where now $\Sigma^\infty S^n \simeq \Sigma^n \mathbb{S}$ is the [[suspension spectrum]] of the [[n-sphere]] and $[-,-]^G$ now denotes the [[hom functor]] in the [[equivariant stable homotopy category]].

### Via genuine $G$-spectra

Consider [[genuine G-spectra]] modeled on a [[G-universe]] $U$.

For a _finite_ based [[G-CW complex]] $X$ and base [[topological G-space]] $Y$, write

$$
  \{X,Y\}_G = [\Sigma^\infty_G X, \Sigma^\infty_G Y] \coloneqq \underset{\longrightarrow}{\lim}_{V \subset U} [\Sigma^V X, \Sigma^V Y]_G
$$

for the [[colimit]] over $G$-[[homotopy classes]] of maps between [[suspensions]] $\Sigma^V X \coloneqq S^V \wedge X$, where $V$ runs through the indexing spaces in the universe and $S^V$ denotes its [[representation sphere]].

([May 96, IX.2 def. 2.1](#May96))

The equivariant stable homotopy groups of $X$ are

$$
  \pi_V^G(\Sigma^\infty_G X) \coloneqq \{S^V,X\}_G
  \,.
$$

([May 96, IX.2 remark 2.4](#May96))

And for subgroups $H \subset G$

$$
  \pi_V^H(\Sigma^\infty_G X) \coloneqq \{G/H_+ \wedge S^V,X\}_G
$$

([Greenlees-May 95, p. 11](#GreenleesMay95))

### Via orthogonal spectra and $G$-equivariant maps

Let $G$ be a [[finite group]]. For $X$ a $G$-[[equivariant spectrum]] modeled as an [[orthogonal spectrum]] with $G$-[[action]], then for $k \in \mathbb{N}$ the $k$th equivariant homotopy group of $X$ is the [[colimit]]

$$
  \pi_k^G(X)
  \coloneqq
  \underset{\longrightarrow_{\mathrlap{n}}}{\lim} [S^{n \rho_G}, (\Omega^k X)(n \rho_G)]_H
  \,,
$$

where

* $\rho_G$ denotes the [[fundamental representation]] of $G$

* $n \rho_G = (\rho_G)^{\oplus_n}$;

* $S^{n \rho_G}$ is its [[representation sphere]];

* $X(n \rho_G)$ is the value of $X$ in [[RO(G)-degree]] $n\rho_G$;

* $[-,-]_G$ is the set of [[homotopy classes]] of $G$-equivariant maps of pointed [[topological G-spaces]].

(e.g. [Schwede 15, section 3](#Schwede15))

More generally for $H \hookrightarrow G$ a [[subgroup]] then one writes $\pi_\bullet^H(X)$ for the $H$-equivariant subgroups of $X$ with $X$ regarded now as an $H$-[[equivariant spectrum]], via restriction of the action.

(e.g. [Schwede 15, p. 16](#Schwede15))

### Via fixed point spectra

Equivalently, the $k$th $G$-equivariant homotopy group of a $G$-equivariant spectrum $E$ is the plain $k$th [[stable homotopy group]] of its [[fixed point spectrum]] $F^G E$

\[
  \label{ViaFixedPointSpectra}
  \pi_k^G(E) \simeq \pi_k(F^G E)
  \,.
\]

(e.g. [Schwede 15, prop. 7.2](#Schwede15))

### Via equivariant cohomology of the point
 {#ViaEquivariantCohomologyOfThePoint}

The identification (eq:ViaFixedPointSpectra) in turn is the [[equivariant cohomology]] of the point,

$$
  E^{-k}_G(\ast)
  \;\coloneqq\;
  \left[ \epsilon^\sharp \Sigma^k \mathbb{S} , E\right]_G
  \;\simeq\;
  \left[ \Sigma^k \mathbb{S}, F^G E \right]
  \;\simeq\;
  \pi_k(F^G E)
$$

due to the [[base change]] [[adjunction]]

$$
  G Spectra
    \underoverset
      { \underset{ F^G }{\longrightarrow} }
      {\overset{ \epsilon^\sharp }{\longleftarrow}}
      { \phantom{AA} \bot \phantom{AA} }
  Spectra
$$

## Examples

### Of equivariant suspension spectra

For $X$ a [[pointed topological space|pointed]] [[topological G-space]], then by the discussion [there](equivariant+suspension+spectrum#ROGDegrees)) the formula for the equivariant homotopy groups of its [[equivariant suspension spectrum]] $\Sigma^\infty_G X$ reduces to

$$
  \pi_k^G(\Sigma^\infty_G X)
  \coloneqq
  \underset{\longrightarrow_n}{\lim} [S^{n \rho_g}, (\Omega^k X)\wedge S^{n \rho_G}]_G
$$

which in turn decomposes as a [[direct sum]] of ordinary [[homotopy groups]] of [[Weyl group]]-[[homotopy quotients]] of naive [[fixed point]] spaces -- see at _[[tom Dieck splitting]]_.

### Of the equivariant sphere spectrum

For the [[equivariant sphere spectrum]] $\mathbb{S} = \Sigma^\infty_G S^0$ the [[tom Dieck splitting]] gives that its 0th [[equivariant homotopy group]]
is the [[free abelian group]] on the set of [[conjugacy classes]] of [[subgroups]] of $G$:

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

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

## Properties

### Relation to Mackey functors

As $H$-varies over the [[subgroups]] of a $G$-[[equivariant spectrum]] $E$, the $H$-equivariant homotopy groups organize into a [[contravariant functor|contravariant]] [[additive functor]] from the [[full subcategory]] of the [[equivariant stable homotopy category]] (called a [[Mackey functor]])

$$
  \underline{\pi}_\bullet(E)
  \colon
  G/H \mapsto \pi^H_\bullet(X)
  \,.
$$

(e.g. [Schwede 15, p. 16 and section 4](#Schwede15))

(...)

### Equivariant Whitehead theorem

See at *[[equivariant Whitehead theorem]]*.

## Related concepts

* [[equivariant weak homotopy equivalence]]

* [[equivariant Whitehead theorem]]

* [[homotopy group]]

* [[stable homotopy group]]

## References

* {#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))

* {#May96} [[Peter May]], section IX.4 of _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. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. ([pdf](http://www.math.rochester.edu/u/faculty/doug/otherpapers/alaska1.pdf))

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

[[!redirects equivariant homotopy groups]]

[[!redirects equivariant stable homotopy group]]
[[!redirects equivariant stable homotopy groups]]


equivariant homotopy theory

+-- {: .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

Equivariant homotopy theory is [[homotopy theory]] for the case that a [[group]] $G$ [[action|acts]] on all the [[topological spaces]]  or other objects involved, hence the homotopy theory of [[topological G-spaces]].

The canonical [[homomorphisms]] of topological $G$-spaces are $G$-equivariant [[continuous functions]], and the canonical choice of [[homotopies]] between these are $G$-equivariant continuous homotopies (for trivial $G$-action on the interval). A $G$-equivariant version of the [[Whitehead theorem]] says that on [[G-CW complexes]] these $G$-equivariant [[homotopy equivalences]] are equivalently those maps that induce [[weak homotopy equivalences]] on all [[fixed point]] spaces for all [[subgroups]] of $G$ (compact subgroups, if $G$ is allowed to be a [[Lie group]]). By [[Elmendorf's theorem]], this, in turn, is equivalent to the [[(∞,1)-presheaves]] over the [[orbit category]] of $G$. See below at _[In topological spaces -- Homotopy theory](#HomotopyTheory)_.

(Beware that $G$-homotopy theory is crucially different from (namely "finer" and "more geometric" than) the homotopy theory of the [[∞-actions]] of the underlying [[homotopy type]] [[∞-group]] of $G$, and this is so even when $G$ is a [[discrete group]], see [below](#RelationToInfinityActions)).

The union of $G$-equivariant homotopy theories as $G$ is allowed to vary is _[[global equivariant homotopy theory]]_.

The direct [[stabilization]] of equivariant homotopy theory is the theory of [[spectra with G-action]]. More generally there is a concept of [[G-spectra]] and they are the subject of [[equivariant stable homotopy theory]].

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

[[!include equivariant cohomology -- table]]

## In topological spaces

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

### $G$-Homotopy

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

A **[[topological G-space]]** is a [[topological space]] equipped with a $G$-[[action]].

=--

Let $I = \mathbb{R}$ be the [[interval object]] $({*} \stackrel{0}{\to} I \stackrel{1}{\leftarrow} {*})$ regarded as a $G$-space by equipping it with the trivial $G$-[[action]].

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

A **$G$-[[homotopy]]**  $\eta$ between $G$-maps, $f, g  : X \to Y$, is a [[left homotopy]] with respect to this $I$

$$
  \array{
    X \times {*} = X
    \\
    {}^{\mathllap{Id \times 0}}\downarrow & \searrow^{f}
    \\
    X \times I &\stackrel{\eta}{\to}& Y
    \\
    {}^{\mathllap{1}}\uparrow & \nearrow_{g}
    \\
    X\times {*} = X
  }
  \,.
$$

=--

(e.g. [May 96, p. 15](#May96))

### Homotopy theory of $G$-spaces
 {#HomotopyTheory}

+-- {: .num_defn #HomotopicalStructures}
###### Definition
**(models for $G$-equivariant spaces)**

Consider the following three [[homotopical category|homotopical categories]] that model $G$-spaces:

1. Write

   $$
     G Top_{cof} \subset G Top
   $$

   for the full [[subcategory]] of [[G-CW-complexes]], regarded as equipped with the structure of a [[category with weak equivalences]] by taking the weak equivalences to be the $G$- [[homotopy equivalences]] according to def. \ref{GHomotopy}.

1. Write

   $$
     G Top_{loc}
   $$

   for all of $G Top$ equipped with weak equivalences given by those morphisms $(f : X \to Y) \in G Top$ that induce for all [[subgroups]] $H \subset G$ [[weak homotopy equivalences]] $f^H : X^H \to Y^H$ on the $H$-[[fixed point]] spaces, in the standard [[Quillen model structure on topological spaces]] (i.e. inducing [[isomorphism]] on [[homotopy groups]]).

1. Write

   $$
     [Orb_G^{op}, Top_{loc}]_{proj}
   $$

   for the projective [[global model structure on functors]]
   from the [[opposite category]] of the [[orbit category]] $O_G$ of $G$ to the [[Top]] (with its [[classical model structure on topological spaces]]).

=--

The following theorem (the [[equivariant Whitehead theorem]] together with [[Elmendorf's theorem]]) says that these models all present the same [[homotopy theory]].

+-- {: .num_theorem #EquivariantWhiteheadAndElmendorfTheorem}
###### Theorem
**(Elmendorf's theorem)**

The [[homotopy category|homotopy categories]] of all three [[homotopical categories]] in def. \ref{HomotopicalStructures} are [[equivalence of categories|equivalent]]:

$$
  Ho(G Top_{cof})
    \overset{\simeq}{\longrightarrow}
  Ho(G Top_{loc})
    \overset{\simeq}{\longrightarrow}
  Ho([Orb_G^{op}, Top])
  \,,
$$

where the equivalence is induced by the [[functor]] that sends a $G$-space to the [[presheaf]] that it [[representable functor|represents]].

The first of these equivalences is the _[[equivariant Whitehead theorem]]_, the second is _[[Elmendorf's theorem]]_.

=--

This is stated as ([May 96, theorem VI.6.3 ](#May96)).

$$
  \array{
    Ho(G Top_{cof})
      &\underset{}{\longrightarrow}&
    Ho(Top_{cof})
    \\
    {\mathllap{\text{equivariant} \atop \text{Whitehead}}}\big\downarrow{\mathrlap{\simeq}}
    &&
    {\mathllap{\simeq}}\big\downarrow{\mathrlap{\text{Whitehead}}}
    \\
    Ho(G Top_{loc})
    &\overset{}{\longrightarrow}&
    Ho(Top_{loc})
    \\
    {\mathllap{Elmendorf}}\big\downarrow{\mathrlap{\simeq}}
    &&
    \downarrow^{\mathrlap{=}}
    \\
    Ho( PSh( Orb_G, Top_{loc} ) )_{proj}
    &\longrightarrow&
    Ho( \ast, Top_{loc} )_{proj}
  }
$$

### $(\infty,1)$-category of $G$-equivariant spaces
 {#InfGTop}

At [[topological ∞-groupoid]] it is discussed that the category [[Top]] of [[topological space]]s may be understood as the [[localization of an (∞,1)-category]] $Sh_{(\infty,1)}(Top)$ [[(∞,1)-category of (∞,1)-sheaves|of (∞,1)-sheaves]] on $Top$, at the collection of morphisms of the form $\{X \times I \to X\}$ with $I$ the real line.

The analogous statement is true for $G$-spaces: the equivariant homotopy category is the [[homotopy localization]] of the category of $\infty$-stacks on $G Top$.

More in detail: let $G Top$ be the [[site]] whose objects are $G$-spaces that admit $G$-equivariant open covers, morphisms are $G$-equivariant maps and morphism $Y \to X$ is in the [[coverage]] if it admits a $G$-equivariant splitting over such $G$-equivariant open covers.

Write

$$
  sSh(G Top)_{loc}
$$

for the corresponding [[hypercompletion|hypercomplete]] local [[model structure on simplicial sheaves]].

Let $I$ be the unit interval, the standard [[interval object]] in [[Top]], equipped with the trivial $G$-action, regarded as an object of $G Top$ and hence in $sSh(G Top)$.

Write

$$
  sSh(G Top)_{loc}^I  \stackrel{\leftarrow}{\to}
  sSh(G Top)_{loc}
$$

for the left [[Bousfield localization of model categories|Bousfield localization]] at thecollection of morphisms $\{X \stackrel{Id \times 0}{\to} X \times I\}$.

Then the [[homotopy category]] of $sSh(G Top)_{loc}^I$ is the equivariant homotopy category described above

$$
  Ho(sSh(G Top)_{loc}^{I}) \simeq G Top_{loc}
  \,.
$$

This is ([Morel-Voevodsky 03, example 3, p. 50](#MorelVoevodsky03)).

### Global equivariant homotopy theory

The above constructions may be unified to apply "for all groups at once",
this is the content of _[[global equivariant homotopy theory]]_.

See at *[[cohesion of global- over G-equivariant homotopy theory]]*.

## In more general model categories

Let $G$ be a finite [[group]] as above. We describe the generalizaton of the above story as [[Top]] is replaced by a more general [[model category]] $C$ ([Guillou 2006](#Guillou06)).

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

1. Let $C$ be a [[cofibrantly generated model category]] with generating cofibrations $I$ and generating acyclic cofibrations $J$.

   There is a cofibrantly generated model category

   $$
     [O_G^{op}, C]_{loc}
   $$

   on the [[functor category]] from the [[orbit category]] of $G$ to $C$ by taking the generating cofibrations to be

   $$
     I_{O_G} := \{G/H \times i\}_{i \in I, H \subset G}
   $$

   and the generating acyclic cofibrations to be

   $$
     J_{O_G} := \{G/H \times j\}_{j \in J, H \subset G}
     \,.
   $$

1. Let $\mathbf{B}G$ be the [[delooping]] [[groupoid]] of $G$ and let

   $$
     [\mathbf{B}G^{op}, C]_{loc}
   $$

   be the [[functor category]] from $\mathbf{B}G$ to $C$ -- the category of objects in $C$ equipped with a $G$-[[action]] equipped with a set of generating (acyclic) cofibrations

   $$
     I_{\mathbf{B}G} := \{G/H \times i\}_{i \in I, H \subset G}
   $$

   and the generating acyclic cofibrations to be

   $$
     J_{\mathbf{B}G} := \{G/H \times j\}_{j \in J, H \subset G}
     \,.
   $$

   This defines a cofibrantly generated model category if $[\mathbf{B}G^{op}, C]$ has a _cellular fixed point functor_ (see...).

=--

+-- {: .num_defn}
###### Definition and proposition
**(generalized Elmendorf's theorem)**

There is a [[Quillen adjunction]]

$$
  G/e \times (-) : C \stackrel{\leftarrow}{\to} [\mathbf{B}G^{op},C]_{loc} : (-)^e
$$

and a [[Quillen equivalence]]

$$
  \Theta : [O_G^{op}, C]_{loc} \stackrel{\leftarrow}{\to} [\mathbf{B}G^{op},C]_{loc}
  : \Phi
  \,.
$$

=--

This is  [Guillou 2006, Prop. 3.1.5](#Guillou06).

### In $\infty$-stack $(\infty,1)$-toposes

The assumption on the [[model category]] $C$ entering the
generalized Elmendorf theorem above is satisfied in particular by every left [[Bousfield localization of model categories|Bousfield localization]]

$$
  C := L_A SPSh(D)
$$

of the global projective [[model structure on simplicial presheaves]] onany [[small category]] $C$ at any [[set]] $A$ of morphisms, i.e. for every [[combinatorial model category]] $C$.

This is [Guillou 2006, Ex. 4.4](#Guillou06).

For $A = \{C(\{U_i\}) \to X\}$ the collection of [[Cech cover]]s for all [[sieve|covering families]] of a [[Grothendieck topology]] on $D$, this are the standard [[models for ∞-stack (∞,1)-toposes]] $\mathbf{H}$.

This way the above theorem provides a model for $G$-equivariant refinements of [[∞-stack]] [[(∞,1)-topos]]es.

* For instance, in [[motivic homotopy theory]] one considers cohomology in a [[homotopy localization]] of the [[∞-stack]] [[(∞,1)-topos]] on the [[Nisnevich site]], presented by $C := L_{Cech} SPSh(Nis)$ . Its $G$-equivariant version as above should be the right context for the [[Bredon cohomology|Bredon]] $G$-[[equivariant cohomology]] refinement of such cohomology theories, such as [[motivic cohomology]].

  This is [Guillou 06, Ex. 4.5](#Guillou06).

  (Actually here one localizes moreover at [[hypercovers]] and at [[A1-homotopy theory|$\mathbb{A}^1$-homotopies]].)

## Properties
 {#Properties}

### Basic facts

* [[G-CW approximation]]

* [[equivariant Whitehead theorem]]

### Elmendorf's theorem

By [[Elmendorf's theorem]] the $G$-[[equivariant homotopy theory]] is an [[(∞,1)-topos]].

By ([Rezk 14](#Rezk14)) $G Top$ is also the [[base (∞,1)-topos]] of the [[cohesion]] of the [[global equivariant homotopy theory]] [[slice (∞,1)-topos|sliced]] over $\mathbf{B}G$. See at *[[cohesion of global- over G-equivariant homotopy theory]]*.

### Equivariant Hopf degree theorem

See at _[equivariant Hopf degree theorem](Hopf+degree+theorem#InEquivariantHomotopyTheory)_.

### Stabilization

The [[stabilization]] of the [[(∞,1)-topos]] $G Top \simeq PSh_\infty(Orb_G)$ is the [[equivariant stable homotopy theory]] of [[spectra with G-action]] ("[[naive G-spectra]]").

## Examples

### $S^1$-Equivariance

[[circle group]]-equivariant homotopy theory may be presented by [[cyclic sets]].

## Related concepts

* [[equivariance]]

* [[equivariant structure]]

* [[Elmendorf's theorem]]

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

* [[equivariant homotopy group]]

* [[equivariant differential topology]]

* [[proper equivariant homotopy theory]]

* [[equivariant stable homotopy theory]]

* [[equivariant cobordism theory]]

* [[equivariant rational homotopy theory]]

  [[Borel-equivariant rational homotopy theory]]

* [[equivariant motivic homotopy theory]]

* [[Sullivan conjecture]]

* [[Arf-Kervaire invariant problem]]

* [[equivariant symmetric monoidal category]]

* [[Parametrized Higher Category Theory and Higher Algebra]]

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

Equivariant homotopy theory is to [[equivariant stable homotopy theory]] as [[homotopy theory]] is to [[stable homotopy theory]].

[[!include equivariant homotopy theory -- table]]

## References

Lecture notes:

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

* {#Guillou20} [[Bert Guillou]], *Equivariant Homotopy and Cohomology*, lecture notes, 2020 ([pdf](http://www.ms.uky.edu/~guillou/F20/751Notes.pdf), [[GuillouEquivariantHomotopyAndCohomology.pdf:file]])

Textbooks and other accounts

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

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

* {#Lueck89} [[Wolfgang Lück]], Chapter I of: _Transformation Groups and Algebraic K-Theory_, Lecture Notes in Mathematics **1408** (Springer 1989) ([doi:10.1007/BFb0083681](https://doi.org/10.1007/BFb0083681))

* {#May96} [[Peter May]] et al., _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) [[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]]]

* {#Schwede12} [[Stefan Schwede]], appendix A.4 of of _[[Symmetric spectra]]_ (2012)

* {#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)]

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

The case of [[cyclic group of order 2|$\mathbb{Z}/2$]]-equivariance (such as with [[KR-theory]]):

* {#Crabb80} [[Michael C. Crabb]], *$\mathbb{Z}/2$-Homotopy Theory*, Lond. Math. Soc. Lecture Notes **44**, Cambridge University Press (1980) [[ark:/13960/t3jx7bg4w](https://archive.org/details/zz2homotopytheor0000crab), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/crabb.pdf)]

The generalization of the homotopy theory of $G$-spaces and of [[Elmendorf's theorem]] to that of $G$-objects in more general [[model category|model categories]] is in

* {#Guillou06} [[Bert Guillou]], _A short note on models for equivariant homotopy theory_, 2006 ([pdf](https://faculty.math.illinois.edu/~bertg/EquivModels.pdf), [[GuillouModelsForEquivariantHomotopyTheory.pdf:file]])

and further discussed in

* {#Stephan13} [[Marc Stephan]], _On equivariant homotopy theory for model categories_, Homology Homotopy Appl. 18(2) (2016) 183-208 ([arXiv:1308.0856](http://arxiv.org/abs/1308.0856))

See also

* {#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))

Specifically with an eye towards equivariant [[differential topology]] (such as [[Pontryagin-Thom construction]] for [[equivariant cohomotopy]]):

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

Discussion in the context of [[global equivariant homotopy theory]] is in

* {#Rezk14} [[Charles Rezk]], _[[Global Homotopy Theory and Cohesion]]_ (2014)

Discussion via [[homotopy type theory]] is in

* {#Shulman15} [[Mike Shulman]], _Univalence for inverse EI diagrams_ ([arXiv:1508.02410](http://arxiv.org/abs/1508.02410))

An alternative [[model category]]-structure:

* Mehmet Akif Erdal, Aslı Güçlükan İlhan, _A model structure via orbit spaces for equivariant homotopy_ ([arXiv:1903.03152](https://arxiv.org/abs/1903.03152))

[[!redirects equivariant algebraic topology]]

[[!redirects equivariant homotopy]]
[[!redirects equivariant homotopy]]

[[!redirects equivariant homotopy class]]
[[!redirects equivariant homotopy classes]]


equivariant homotopy theory -- table

**[[Global Homotopy Theory and Cohesion|Rezk]]-[[global equivariant homotopy theory]]:**

| [[cohesive (∞,1)-topos]]  |  its [[(∞,1)-site]]  |  [[base (∞,1)-topos]] | its [[(∞,1)-site]] |
|---|----|----|----|
| [[global equivariant homotopy theory]] $PSh_\infty(Glo)$ |  [[global equivariant indexing category]] $Glo$ | [[∞Grpd]] $ \simeq PSh_\infty(\ast)$ | point |
| ... [[slice (∞,1)-topos|sliced]] over terminal [[orbispace]]: $PSh_\infty(Glo)_{/\mathcal{N}}$ | $Glo_{/\mathcal{N}}$ | [[orbispaces]] $PSh_\infty(Orb)$ | [[global orbit category]] |
| ... [[slice (∞,1)-topos|sliced]] over $\mathbf{B}G$: $PSh_\infty(Glo)_{/\mathbf{B}G}$ | $Glo_{/\mathbf{B}G}$ | $G$-[[equivariant homotopy theory]] of [[G-spaces]] $L_{we} G Top \simeq PSh_\infty(Orb_G)$ | $G$-[[orbit category]] $Orb_{/\mathbf{B}G} = Orb_G$ |


equivariant homotopy type

#Contents#
* table of contents
{:toc}

## Idea

A [[homotopy type]] in [[equivariant homotopy theory]].

## Related concepts

* [[stable homotopy type]]

* [[modal homotopy type]]

* [[geometric homotopy type]]

* [[cohesive homotopy type]]

[[!redirects equivariant homotopy types]]


equivariant Hopf degree theorem

+-- {: .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

The _equivariant Hopf degree theorem_ -- Theorem \ref{EquivariantHopfDegreeTheorem} below -- generalizes the plain [[Hopf degree theorem]] from plain [[homotopy theory]] to [[equivariant homotopy theory]], due to [tomDieck 79, 8.4](#tomDieck79).

## Preliminaries: Matching $G$-Spaces

We need the following list of ingredients and assumptions:

Let $G$ be a [[finite group]]. For $H \subset G$ a [[subgroup]], write

\[
  \label{WeylGroup}
  W_G H \coloneqq (N_G H) / H
\]

for its [Weyl group](Weyl+group#in_equivariant_homotopy_theory).

For $X$ a [[G-space]], we write

\[
  \label{IsotropySubgroups}
  Isotr_X(G) \subset Sub(G)
\]

for the [[subset]] of the [[subgroup lattice]] on the [[isotropy groups]] of $X$, hence those [[subgroups]] which appear as [[stabilizer subgroups]] $Stab_G(x)$ of some point $x \in X$. This means  that if $H_1, H_2 \in Iso_X(G)$ and $H_1 \lt H_2$ is a strict inclusion, then the [[fixed loci]] differ $X^{H_1} \gt X^{H_2}$.

+-- {: .num_defn #MatchingGSpaces}
###### Definition
**(matching pair of $G$-spaces)**

For $G$ a [[finite group]], we say that a [[pair]] $X,Y \in G Spaces$ of [[topological G-spaces]], where $X$ is a [[G-CW-complex]], is a _matching pair_ if the following conditions are satisfied for all [[isotropy groups]] $H \in Isotr_X(G)$ (eq:IsotropySubgroups) of the $G$-action on $X$.

1. The [[fixed point space]] $X^H$ is a $W_G H = (N_G H) / H $[[G-CW-complex|-complex]] of [[finite number|finite]] [[dimension of a cell complex|dimension]] $dim\big( X^H\big) \in \mathbb{N}$;

1. $H^{dim(X^H)}\Big( X^H , \mathbb{Z}\Big) \simeq \mathbb{Z}$ ([[integral cohomology]] of the [[fixed point space]]),

   this implies that the [[action]] of $W_G(H)$ on cohomology induces a [[group homomorphism]]

   \[
     \label{OrientationBehaviour}
     e_{H,X}
     \;\colon\;
     W_G(H)
       \longrightarrow
     Aut_{Ab}(\mathbb{Z})
       \simeq
     \mathbb{Z}^\times
   \]

   to be called the _orientation behaviour_ of the action of $W_G(H)$ on $X^H$;

and

1. $Y^H$ is $(dim(X^H)-1)$-[[n-connected topological space|connected]]

   $$
     \pi_{ \lt \mathrm{dim}\big( X^H \big) }
     \big(
       Y^H
     \big)
     \;=\;
     0
   $$

   (hence [[connected topological space|connected]] if $dim\left(X^H\right) = 1$, [[simply connected topological space|simply connected]] if $dim\left(X^H\right) = 2$, etc.);

1. $\pi_{dim(X^H)}\big( Y^H\big) \simeq \mathbb{Z}$ ([[homotopy groups]] of [[fixed point space]]),

   with the previous point this implies (by the [[Hurewicz theorem]]) that  $H^{dim(X^H)}\Big( Y^H , \mathbb{Z}\Big) \simeq \mathbb{Z}$ and hence orientation behaviour (eq:OrientationBehaviour) $e_{H,Y} \;\colon\; W_G(H) \to \mathbb{Z}^\times$

1. $e_{H,X} = e_{H,Y}$, the orientation behaviour (eq:OrientationBehaviour) of $X$ and $Y$ agrees at all [[isotropy groups]].

For simplicity we also demand that

* $dim(X^H) \geq 1$.

Given a matching pair of $G$-spaces, we say that a choice of generators in the [[cohomology groups]] of all the fixed strata

\[
  \label{ChoiceOfStratumWiseOrientations}
  \begin{aligned}
    or_{X,H}
    & =
    \pm 1 \in
    \mathbb{Z}
    \simeq
    H^{dim(X^H)}\big(X^H, \mathbb{Z} \big)
    \\
    or_{Y,H}
    & =
    \pm 1 \in
    \mathbb{Z}
    \simeq
    H^{dim(X^H)}\big(Y^H, \mathbb{Z} \big)
  \end{aligned}
\]

is a _choice of singularity-wise [[orientations]]_.  Given such a choice and an equivariant continuous function $f \colon X \to Y$, we have for each [[isotropy group]] $H \in Isotr_X(G)$ that the continuous function $f^H \;\colon\; X^H \to Y^H$ has a well-defined [[integer]] [[degree of a continuous function|degree]]

\[
  \label{IntegerDegreeOfEquivariantFunctionOnFixedStrata}
  deg(f^H)
  \;
    \in
  \;\mathbb{Z}
  \,.
\]

=--

([tom Dieck 79, p. 212 and p. 213](#tomDieck79))

+-- {: .num_example #RepresentationSphereToRepresentationSphereIsMatchingPair}
###### Example
**([[representation spheres]] match [[representation spheres]])**

Let $G$ be a [[finite group]] and $V \in RO(G)$ a [[finite-dimensional vector space|finite-dimensional]] [[orthogonal group|orthogonal]] [[linear representation]] of $G$. Then the pair

$$
  (X \coloneqq S^V,\; Y \coloneqq S^V)
$$

consisting of two copies of the [[representation sphere]] of $V$ is a matching pair of $G$-spaces, according to Def. \ref{MatchingGSpaces}.

=--

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

First notice that an $H$-[[fixed locus]] of any $G$-representation sphere is itself a $W_G(H)$-representation sphere, since $\left(S^V\right)^H \simeq S^{\left( V^H\right)}$. That these are all [[G-CW-complex|WH-CW-complexes]] follows because generally [[G-representation spheres are G-CW-complexes]]. Moreover, since the [[topological spaces]] underlying all these fixed loci are [[n-spheres]], and since we have the same spheres for $X$ and $Y$, the connectivity and orientability conditions in Def. \ref{MatchingGSpaces} are evidently satisfied.

=--

+-- {: .num_example #RepresentationTorusToRepresentationSphereIsMatchingPair}
###### Example
**([[representation tori]] match [[representation spheres]])**

Let $G$ be a [[finite group]] which arises as the [[point group]] $G \simeq S/N$ of a [[crystallographic group]] $S \subset Iso(E)$ of some [[Euclidean space]] $E$. Then the pair

$$
  \big(
    X \coloneqq E/N
    \,;
    Y \coloneqq S^E
  \big)
$$

consisting of

1. the [[torus]] $E/N$ which is the [[quotient space]] of $E$ by the given crystallographic sub-lattice $N \subset E$ and equipped with the $G$-[[action]] descending from that on $E$ ([this Prop.](crystallographic+group#InducedPointGroupActionOnTorus));

1. the [[representation sphere]] of the [[linear representation|linear action]] of the point group $G$ on $E$

is a matching pair of $G$-spaces, according to Def. \ref{MatchingGSpaces}.

=--

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

The [[torus]] $E/N$ carries the [[structure]] of a [[smooth manifold]] for which the $G$-[[action]] is [[smooth]]. Since $G$ is [[finite group|finite]], also all its [[fixed loci]] $(E/N)^H$ are smooth manifolds ([this Prop.](equivariant+differential+topology#FixedLociOfSmoothProperActionsAreSubmanifolds)). By the [[equivariant triangulation theorem]], all these are [[G-CW-complexes|WH-CW-complexes]].

Moreover, the orientability and connectivity assumptions in Def. \ref{MatchingGSpaces} are evidently satisfied, using the fact that both $E/N$ as well as $S^E$ are modeled on the same linear $G$-representation space $E$.

=--

> graphics grabbed from [SS 19](#SatiSchreiber19)

## Equivariant Hopf degree theorem

We first state the equivariant Hopf degree theorem for full (i.e. unstable/[[non-abelian cohomology|non-abelian]])[[equivariant Cohomotopy]], and then its [[stabilization]] to [[stable equivariant Cohomotopy]].

### In unstable equivariant Cohomotopy

+-- {: .num_theorem #EquivariantHopfDegreeTheorem}
###### Theorem
**([[equivariant Hopf degree theorem]])**

Given a matching pair of $G$-spaces $X, Y$ (Def. \ref{MatchingGSpaces}) the function
(from $G$-[[equivariant homotopy theory|equivariant]] [[homotopy classes]] to [[tuples]] of [[degree of a continuous function|degrees]] labeled by [[isotropy groups]])
which sends any equivariant homotopy class $[f]$ of an equivariant continuous function $f \colon X \to Y$ to its $H$-degrees (eq:IntegerDegreeOfEquivariantFunctionOnFixedStrata)

$$
  \array{
    deg
    &\colon&
    \pi_0 \mathrm{Maps}\big( X,Y \big)^G
    &
    \overset{
      \phantom{AAAA}
    }{\hookrightarrow}
    &
    \underset{
      { H \in Isotr_X(G) }
    }{\prod} \mathbb{Z}
    \\
    &&
    [f]
      &\mapsto &
    \big(
      H \mapsto \mathrm{deg}\big( f^H\big)
    \big)
  }
$$

is an [[injective function|injection]].

Moreover, for each $[f] \in \pi_0 \mathrm{Maps}\big( X,Y \big)^G$ and for each $H \in Isotropy_X(G)$

1. the $H$-degree (eq:IntegerDegreeOfEquivariantFunctionOnFixedStrata) modulo the [[order of a group|order]] of the [[Weyl group]]

   \[
     \label{DegreeConstraintsInEquivariantHopfDegreeTheorem}
     deg\left( f^H\right)
     \;mod\;
     {\vert W_G(H)\vert}
     \;\in\;
     \mathbb{Z}/{\vert W_G(H)\vert}
   \]

   is fixed by the degrees $deg\big( f^K\big)$ for all $K \gt H$;

1. there exists $f'$ with specified degrees $deg\big( (f')^K\big) = deg\big( f^K\big)$ for $K \gt H$ and realizing any of the degrees $deg\big( (f')^H\big) \in \mathbb{Z}$ allowed by the  constraint  (eq:DegreeConstraintsInEquivariantHopfDegreeTheorem).

=--

([tom Dieck 79, 8.4](#tomDieck79))

### In stable equivariant Cohomotopy

[[stabilization]] to [[stable equivariant Cohomotopy]]

(...)

> graphics grabbed from [SS19](equivariant+Hopf+degree+theorem#SatiSchreiber19)

## Examples

### Equivariant Cohomotopy of representation spheres
 {#EquivariantCohomotopyOfRepresentationSpheres}

As a special case of the [[equivariant Hopf degree theorem]] (Theorem \ref{EquivariantHopfDegreeTheorem}), we obtain the following:

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

Let $G \in \mathrm{Grp}_{\mathrm{fin}}$ and $V \in \mathrm{RO}(G)$. Then the pointed [[equivariant cohomotopy]]  of the [[representation sphere]] $S^V$  in [[RO(G)-degree]] $V$ is the [[Cartesian product]] of one copy of the [[integers]] for each _proper_ [[isotropy group|isotropy]] subgroup (eq:IsotropySubgroups) $H \underset{\neq}{\subset} G$ in $S^V$, and a copy of $\mathbb{Z}_2$ or $\mathbb{Z}$ depending on whether $V^G = 0$ or not:

$$
  \array{
      \pi^V\left( S^V\right)^{\{\infty\}/}
      &
        \overset{\simeq}{\longrightarrow}
      &
      \underset{
        {
          {
              {H \in \mathrm{Isotr}_{S^V}(G)}
          }
        }
      }{\prod}
      \;\;
      {\vert W_G(H)\vert } \cdot
      \pi^{\left( V^H\right)}
      \left(
        S^{\left( V^H \right)}
      \right)
      \\
      && =
      \left\{
        \array{
          \mathbb{Z}_2 &\vert& V^G = 0
          \\
          \mathbb{Z} &\vert& \text{otherwise}
        }
      \right\}
      \times
      \underset{
        {
          {
              {H \in \mathrm{Isotr}_{S^V}(G)}
              \atop
              {H \neq G}
          }
        }
      }{\prod}
      \;\;
      {\vert W_G(H)\vert } \cdot \mathbb{Z}
      \\
      \big[
        S^V
          \overset{c}{\longrightarrow}
        S^V
      \big]
      &\mapsto&
      \Big(
        H
          \mapsto
        \mathrm{deg}
        \big(
          c^H
        \big)
        -
        \mathrm{offs}(c,H)
      \Big)
  }
$$

where on the right

$$
  \mathrm{deg}
  \Big(
    \big(
      S^V
    \big)^H
    \overset{
      c^H
    }{\longrightarrow}
    \big(
      S^V
    \big)^H
  \Big)
  \in
  \left\{
    \array{
      \mathbb{Z} &\vert& dim\left(V^H\right) \gt 0
      \\
      \mathbb{Z}_2 &\vert& dim\left( V^H\right) = 0
    }
  \right\}
$$

is the [[winding number]] of the underlying [[continuous function]] of $c$ ([[corestriction|co]])[[restriction|restricted]] to $H$-[[fixed points]], and part of the claim is  that in the cases with $dim\left( V^H\right) \gt 0$ this is an integer multiple of the order of the [[Weyl group]] $W_G(H)$ (eq:WeylGroup) up to an offset

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

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

=--

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

This follows as a special case of the equivariant Hopf degree theorem (Theorem \ref{EquivariantHopfDegreeTheorem}).

Here $(S^V, S^V)$ is a matching pair of $G$-spaces according to Example \ref{RepresentationSphereToRepresentationSphereIsMatchingPair}.

This equivariant Hopf degree theorem is stated above under the simplifying assumption that the dimension of all fixed loci is positive. But the proof from [tomDieck 79, 8.4](#tomDieck79) immediately applies to our situation where the dimension of the fixed locus at the full subgroup $H = G$ may be 0, with $\left( S^V\right)^G = S^0$. This gives a choice in $\mathbb{Z}_2$ in the first step of the inductive argument in [tomDieck 79, 8.4](#tomDieck79), and from there on the proof applies verbatim.

Alternatively, if $V^G = 0$ we may consider maps $S^{1+V} \to S^{1+V}$ which restrict on $S^1 \to S^1$ to degree zero or one.

=--

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

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

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

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

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

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

Equivalently, if we identify

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

then the involution action is

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

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

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

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

Notice that the map

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

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

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

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

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

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

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

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

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

=--

> graphics grabbed form [Sati-Schreiber 19](#SatiSchreiber19)

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

This result

$$
  \pi^{ \mathbb{R}_{sgn} }\big( \mathbb{R}_{sgn}\big)^{\{\infty\}/} \simeq \mathbb{Z}_2 \times \mathbb{Z}
$$

from Example \ref{EquivariantCohomotopyOfRepresentationSphereOfSignRepresentationInThatDegree}

becomes, after [[stabilization]] to [[equivariant stable homotopy theory]], the stable homotopy groups of the [[equivariant sphere spectrum]] in [[RO(G)-grading]] given by

$$
  \pi^{ \mathbb{R}_{sgn} }_{stab}\big( \mathbb{R}_{sgn}\big)^{\{\infty\}/} \simeq \mathbb{Z} \times \mathbb{Z}
$$

see [there](equivariant+sphere+spectrum#Z2equivariance).

=--

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

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

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

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

=--

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

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

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

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

=--

\linebreak

### Equivariant Cohomotopy of representation tori
 {#EquivariantCohomotopyOfRepresentationTori}

For the case of [[representation tori]]

(...)

> graphics grabbed form [Sati-Schreiber 19](#SatiSchreiber19)

\linebreak

## Related concepts

* [[Hopf degree theorem]]

* [[Poincaré–Hopf theorem]]

* [[Pontrjagin theorem]], [[equivariant Pontrjagin theorem]]

## References

Discussion of equivariant maps between suitable matching pairs of $G$-spaces is in

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

* {#tomDieck87} [[Tammo tom Dieck]], Section II.4 of: _[[Transformation Groups]]_, de Gruyter 1987 ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

Discussion specifically for the [[codomain]] a [[representation sphere]] (i.e.  for [[equivariant cohomotopy]]):

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

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

* [[Markus Szymik]], _A stable approach to the equivariant Hopf theorem_ ([pdf](https://folk.ntnu.no/markussz/papers/szymik.hopf.pdf), [[StableApproachToEquivariantHopfTheorem.pdf:file]])

The above discussion follows

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

which applies equivariant Cohomotopy to [[RR-field tadpole cancellation]].

[[!redirects equivariant Hopf degree theorems]]

[[!redirects equivariant Hopf theorem]]
[[!redirects equivariant Hopf theorems]]