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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{equivariant homotopy theory} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory} [[!include homotopy - contents]] \hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory} [[!include representation theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{in_topological_spaces}{In topological spaces}\dotfill \pageref*{in_topological_spaces} \linebreak \noindent\hyperlink{homotopy}{$G$-Homotopy}\dotfill \pageref*{homotopy} \linebreak \noindent\hyperlink{HomotopyTheory}{Homotopy theory of $G$-spaces}\dotfill \pageref*{HomotopyTheory} \linebreak \noindent\hyperlink{InfGTop}{$(\infty,1)$-category of $G$-equivariant spaces}\dotfill \pageref*{InfGTop} \linebreak \noindent\hyperlink{global_equivariant_homotopy_theory}{Global equivariant homotopy theory}\dotfill \pageref*{global_equivariant_homotopy_theory} \linebreak \noindent\hyperlink{in_more_general_model_categories}{In more general model categories}\dotfill \pageref*{in_more_general_model_categories} \linebreak \noindent\hyperlink{in_stack_toposes}{In $\infty$-stack $(\infty,1)$-toposes}\dotfill \pageref*{in_stack_toposes} \linebreak \noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak \noindent\hyperlink{equivariant_hopf_degree_theorem}{Equivariant Hopf degree theorem}\dotfill \pageref*{equivariant_hopf_degree_theorem} \linebreak \noindent\hyperlink{topos}{$(\infty,1)$-topos}\dotfill \pageref*{topos} \linebreak \noindent\hyperlink{stabilization}{Stabilization}\dotfill \pageref*{stabilization} \linebreak \noindent\hyperlink{RelationToInfinityActions}{Relation to ∞-Actions (fine and coarse equivariance)}\dotfill \pageref*{RelationToInfinityActions} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{equivariance}{$S^1$-Equivariance}\dotfill \pageref*{equivariance} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{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 \emph{\hyperlink{HomotopyTheory}{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 \hyperlink{RelationToInfinityActions}{below}). The union of $G$-equivariant homotopy theories as $G$ is allowed to vary is \emph{[[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 \emph{[[equivariant cohomology]]}: [[!include equivariant cohomology -- table]] \hypertarget{in_topological_spaces}{}\subsection*{{In topological spaces}}\label{in_topological_spaces} Let $G$ be a [[discrete group]]. \hypertarget{homotopy}{}\subsubsection*{{$G$-Homotopy}}\label{homotopy} \begin{defn} \label{GSpace}\hypertarget{GSpace}{} A \textbf{[[topological G-space]]} is a [[topological space]] equipped with a $G$-[[action]]. \end{defn} 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]]. \begin{defn} \label{GHomotopy}\hypertarget{GHomotopy}{} A \textbf{$G$-[[homotopy]]} $\eta$ between $G$-maps, $f, g : X \to Y$, is a [[left homotopy]] with respect to this $I$ \begin{displaymath} \itexarray{ 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 } \,. \end{displaymath} \end{defn} (e.g. \hyperlink{May96}{May 96, p. 15}) \hypertarget{HomotopyTheory}{}\subsubsection*{{Homotopy theory of $G$-spaces}}\label{HomotopyTheory} \begin{defn} \label{HomotopicalStructures}\hypertarget{HomotopicalStructures}{} \textbf{(models for $G$-equivariant spaces)} Consider the following three [[homotopical category|homotopical categories]] that model $G$-spaces: \begin{enumerate}% \item Write \begin{displaymath} G Top_{cof} \subset G Top \end{displaymath} 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}. \item Write \begin{displaymath} G Top_{loc} \end{displaymath} 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]]). \item Write \begin{displaymath} [Orb_G^{op}, Top_{loc}]_{proj} \end{displaymath} 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]]). \end{enumerate} \end{defn} The following theorem (the [[equivariant Whitehead theorem]] together with [[Elmendorf's theorem]]) says that these models all present the same [[homotopy theory]]. \begin{theorem} \label{EquivariantWhiteheadAndElmendorfTheorem}\hypertarget{EquivariantWhiteheadAndElmendorfTheorem}{} \textbf{(Elmendorf's theorem)} The [[homotopy category|homotopy categories]] of all three [[homotopical categories]] in def. \ref{HomotopicalStructures} are [[equivalence of categories|equivalent]]: \begin{displaymath} Ho(G Top_{cof}) \overset{\simeq}{\longrightarrow} Ho(G Top_{loc}) \overset{\simeq}{\longrightarrow} Ho([Orb_G^{op}, Top]) \,, \end{displaymath} 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 \emph{[[equivariant Whitehead theorem]]}, the second is \emph{[[Elmendorf's theorem]]}. \end{theorem} This is stated as (\hyperlink{May96}{May 96, theorem VI.6.3}). \begin{displaymath} \itexarray{ 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} } \end{displaymath} \hypertarget{InfGTop}{}\subsubsection*{{$(\infty,1)$-category of $G$-equivariant spaces}}\label{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 \begin{displaymath} sSh(G Top)_{loc} \end{displaymath} 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 \begin{displaymath} sSh(G Top)_{loc}^I \stackrel{\leftarrow}{\to} sSh(G Top)_{loc} \end{displaymath} 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 \begin{displaymath} Ho(sSh(G Top)_{loc}^{I}) \simeq G Top_{loc} \,. \end{displaymath} This is (\hyperlink{MorelVoevodsky03}{Morel-Voevodsky 03, example 3, p. 50}). \hypertarget{global_equivariant_homotopy_theory}{}\subsubsection*{{Global equivariant homotopy theory}}\label{global_equivariant_homotopy_theory} The above constructions may be unified to apply ``for all groups at once'', this is the content of \emph{[[global equivariant homotopy theory]]}. \hypertarget{in_more_general_model_categories}{}\subsection*{{In more general model categories}}\label{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]] $C$ (\hyperlink{Guillou}{Guillou}). \begin{defn} \label{}\hypertarget{}{} \begin{enumerate}% \item Let $C$ be a [[cofibrantly generated model category]] with generating cofibrations $I$ and generating acyclic cofibrations $J$. There is a cofibrantly generated model category \begin{displaymath} [O_G^{op}, C]_{loc} \end{displaymath} on the [[functor category]] from the [[orbit category]] of $G$ to $C$ by taking the generating cofibrations to be \begin{displaymath} I_{O_G} := \{G/H \times i\}_{i \in I, H \subset G} \end{displaymath} and the generating acyclic cofibrations to be \begin{displaymath} J_{O_G} := \{G/H \times j\}_{j \in I, H \subset G} \,. \end{displaymath} \item Let $\mathbf{B}G$ be the [[delooping]] [[groupoid]] of $G$ and let \begin{displaymath} [\mathbf{B}G^{op}, C]_{loc} \end{displaymath} 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 \begin{displaymath} I_{\mathbf{B}G} := \{G/H \times i\}_{i \in I, H \subset G} \end{displaymath} and the generating acyclic cofibrations to be \begin{displaymath} J_{\mathbf{B}G} := \{G/H \times j\}_{j \in I, H \subset G} \,. \end{displaymath} This defines a cofibrantly generated model category if $[\mathbf{B}G^{op}, C]$ has a \emph{cellular fixed point functor} (see\ldots{}). \end{enumerate} \end{defn} \begin{defn} \label{}\hypertarget{}{} \textbf{(generalized Elmendorf's theorem)} There is a [[Quillen adjunction]] \begin{displaymath} G/e \times (-) : C \stackrel{\leftarrow}{\to} [\mathbf{B}G^{op},C]_{loc} : (-)^e \end{displaymath} and a [[Quillen equivalence]] \begin{displaymath} \Theta : [O_G^{op}, C]_{loc} \stackrel{\leftarrow}{\to} [\mathbf{B}G^{op},C]_{loc} : \Phi \,. \end{displaymath} \end{defn} \begin{proof} This is proposition 3.1.5 in \href{http://www.math.uiuc.edu/~bertg/EquivModels.pdf#page=6}{Guillou}. \end{proof} \hypertarget{in_stack_toposes}{}\subsubsection*{{In $\infty$-stack $(\infty,1)$-toposes}}\label{in_stack_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]] \begin{displaymath} C := L_A SPSh(D) \end{displaymath} 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 example 4.4 in \href{http://www.math.uiuc.edu/~bertg/EquivModels.pdf#page=7}{Guillou}. 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. \begin{itemize}% \item 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 example 4.5 in \href{http://www.math.uiuc.edu/~bertg/EquivModels.pdf#page=8}{Guillou}. (Actually here one localizes moreover at [[hypercovers]] and at [[A1-homotopy theory|A1-homotopies]].) \end{itemize} \hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties} \hypertarget{equivariant_hopf_degree_theorem}{}\subsubsection*{{Equivariant Hopf degree theorem}}\label{equivariant_hopf_degree_theorem} \begin{itemize}% \item \href{Hopf+degree+theorem#InEquivariantHomotopyTheory}{equivariant Hopf degree theorem} \end{itemize} \hypertarget{topos}{}\subsubsection*{{$(\infty,1)$-topos}}\label{topos} By [[Elmendorf's theorem]] the $G$-[[equivariant homotopy theory]] is an [[(∞,1)-topos]]. By (\hyperlink{Rezk14}{Rezk 14}) $G Top$ is also the [[base (∞,1)-topos]] of the [[cohesion]] of the [[global equivariant homotopy theory]] [[slice (∞,1)-topos|sliced]] over $\mathbf{B}G$. \hypertarget{stabilization}{}\subsubsection*{{Stabilization}}\label{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]]''). \hypertarget{RelationToInfinityActions}{}\subsubsection*{{Relation to ∞-Actions (fine and coarse equivariance)}}\label{RelationToInfinityActions} For $G$ a [[discrete group]] ([[geometrically discrete ∞-groupoid|geometrically discrete]]) the homotopy theory of [[G-spaces]] which enters [[Elmendorf's theorem]] is different (finer) than the standard homotopy theory of $G$-[[∞-actions]], which is presented by the [[Borel model structure]] (see there for more, and see (\hyperlink{Guillou}{Guillou})). \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{equivariance}{}\subsubsection*{{$S^1$-Equivariance}}\label{equivariance} [[circle group]]-equivariant homotopy theory may be presented by [[cyclic sets]]. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[equivariance]] \item [[equivariant structure]] \item [[equivariant homotopy group]] \item [[equivariant differential topology]] \item [[equivariant stable homotopy theory]] \item [[equivariant motivic homotopy theory]] \item [[Sullivan conjecture]] \item [[Arf-Kervaire invariant problem]] \item [[equivariant symmetric monoidal category]] \item [[Parametrized Higher Category Theory and Higher Algebra]] \item [[Burnside category]], [[Burnside ring]] \end{itemize} Equivariant homotopy theory is to [[equivariant stable homotopy theory]] as [[homotopy theory]] is to [[stable homotopy theory]]. [[!include equivariant homotopy theory -- table]] \hypertarget{references}{}\subsection*{{References}}\label{references} Detailed lecture notes include \begin{itemize}% \item [[Andrew Blumberg]], \emph{Equivariant homotopy theory}, 2017 (\href{https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf}{pdf}, \href{https://github.com/adebray/equivariant_homotopy_theory}{GitHub}) \end{itemize} A standard text is \begin{itemize}% \item [[Peter May]], \emph{Equivariant homotopy and cohomology theory}, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (\href{https://web.math.rochester.edu/people/faculty/doug/otherpapers/alaska1.pdf}{pdf}, [[MayEtAlEquivariant96.pdf:file]]) \end{itemize} Other accounts include \begin{itemize}% \item [[Tammo tom Dieck]], \emph{[[Transformation Groups and Representation Theory]]}, Lecture Notes in Mathematics 766, Springer 1979 \item [[Stefan Schwede]], appendix A.4 of of \emph{[[Symmetric spectra]]} (2012) \item [[Michael Hill]], [[Michael Hopkins]], [[Douglas Ravenel]], section 1.3 of \emph{The Arf-Kervaire problem in algebraic topology: Sketch of the proof} ([[HHRKervaire.pdf:file]]) (with an eye towards application to the [[Arf-Kervaire invariant problem]]) \end{itemize} 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 \begin{itemize}% \item [[Bert Guillou]], \emph{A short note on models for equivariant homotopy theory} (\href{http://www.math.uiuc.edu/~bertg/EquivModels.pdf}{pdf}) \end{itemize} and further discussed in \begin{itemize}% \item [[Marc Stephan]], \emph{On equivariant homotopy theory for model categories}, Homology Homotopy Appl. 18(2) (2016) 183-208 (\href{http://arxiv.org/abs/1308.0856}{arXiv:1308.0856}) \end{itemize} See also \begin{itemize}% \item [[Stefan Waner]], \emph{Equivariant Homotopy Theory and Milnor's Theorem}, Transactions of the American Mathematical Society Vol. 258, No. 2 (Apr., 1980), pp. 351-368 (\href{http://www.jstor.org/stable/1998061}{JSTOR}) \end{itemize} Specifically with an eye towards equivariant [[differential topology]] (such as [[Pontryagin-Thom construction]] for [[equivariant cohomotopy]]): \begin{itemize}% \item [[Arthur Wasserman]], \emph{Equivariant differential topology}, Topology Vol. 8, pp. 127-150, 1969 (\href{https://web.math.rochester.edu/people/faculty/doug/otherpapers/wasserman.pdf}{pdf}) \end{itemize} Discussion in the context of [[global equivariant homotopy theory]] is in \begin{itemize}% \item [[Charles Rezk]], \emph{[[Global Homotopy Theory and Cohesion]]} (2014) \end{itemize} Discussion via [[homotopy type theory]] is in \begin{itemize}% \item [[Mike Shulman]], \emph{Univalence for inverse EI diagrams} (\href{http://arxiv.org/abs/1508.02410}{arXiv:1508.02410}) \end{itemize} An alternative [[model category]]-structure: \begin{itemize}% \item Mehmet Akif Erdal, Aslı Güçlükan İlhan, \emph{A model structure via orbit spaces for equivariant homotopy} (\href{https://arxiv.org/abs/1903.03152}{arXiv:1903.03152}) \end{itemize} \end{document}