nLab equivariant homotopy theory

Idea

Equivariant homotopy theory is homotopy theory for the case that a group $G$ 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.

(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).

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:

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

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

In topological spaces

Let $G$ be a discrete group.

$G$-Homotopy

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.

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)

Homotopy theory of $G$-spaces

Definition

(models for $G$-equivariant spaces)

Consider the following three 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. .

2. 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).

3. 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.

Theorem

(Elmendorf’s theorem)

The homotopy categories of all three homotopical categories in def. are 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 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).

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

$(\infty,1)$-category of $G$-equivariant spaces

At topological ∞-groupoid it is discussed that the category Top of topological spaces may be understood as the localization of an (∞,1)-category $Sh_{(\infty,1)}(Top)$ 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 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 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).

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).

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

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

2. 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…).

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.

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

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

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.

For $A = \{C(\{U_i\}) \to X\}$ the collection of Cech covers for all 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)-toposes.

• 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 $G$-equivariant cohomology refinement of such cohomology theories, such as motivic cohomology.

  This is Guillou 06, Ex. 4.5.

  (Actually here one localizes moreover at hypercovers and at $\mathbb{A}^1$-homotopies.)

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) $G Top$ is also the base (∞,1)-topos of the cohesion of the global equivariant homotopy theory 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.

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.

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
… sliced over terminal orbispace: $PSh_\infty(Glo)_{/\mathcal{N}}$	$Glo_{/\mathcal{N}}$	orbispaces $PSh_\infty(Orb)$	global orbit category
… 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 $Orb_{/\mathbf{B}G} = Orb_G$

References

Lecture notes:

• Andrew Blumberg, Equivariant homotopy theory, 2017 (pdf, GitHub)

• Bert Guillou, Equivariant Homotopy and Cohomology, lecture notes, 2020 (pdf, pdf)

Textbooks and other accounts

• Tammo tom Dieck, Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)

• Tammo tom Dieck, Transformation Groups and Representation Theory, Lecture Notes in Mathematics 766, Springer 1979 (doi:10.1007/BFb0085965)

• Wolfgang Lück, Chapter I of: Transformation Groups and Algebraic K-Theory, Lecture Notes in Mathematics 1408 (Springer 1989) (doi:10.1007/BFb0083681)

• 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, pdf, pdf]

• Stefan Schwede, appendix A.4 of of Symmetric spectra (2012)

• 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]

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

The case of $\mathbb{Z}/2$-equivariance (such as with KR-theory):

• Michael C. Crabb, $\mathbb{Z}/2$-Homotopy Theory, Lond. Math. Soc. Lecture Notes 44, Cambridge University Press (1980) [ark:/13960/t3jx7bg4w, pdf]

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

• Bert Guillou, A short note on models for equivariant homotopy theory, 2006 (pdf, pdf)

and further discussed in

• Marc Stephan, On equivariant homotopy theory for model categories, Homology Homotopy Appl. 18(2) (2016) 183-208 (arXiv:1308.0856)

See also:

• Graeme Segal, Some results in equivariant homotopy theory (1978) [scan: web, pdf]

  (on equivariant iterated loop spaces and configuration spaces of points)

• 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)

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

• Arthur Wasserman, Equivariant differential topology, Topology Vol. 8, pp. 127-150, 1969 (pdf)

Discussion in the context of global equivariant homotopy theory is in

• Charles Rezk, Global Homotopy Theory and Cohesion (2014)

Discussion via homotopy type theory is in

• Mike Shulman, Univalence for inverse EI diagrams (arXiv: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)