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\section*{equivariant stable homotopy theory}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - 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{basic_definitions}{Basic definitions}\dotfill \pageref*{basic_definitions} \linebreak
\noindent\hyperlink{in_terms_of_looping_by_representation_spheres}{In terms of looping by representation spheres}\dotfill \pageref*{in_terms_of_looping_by_representation_spheres} \linebreak
\noindent\hyperlink{in_terms_of_orthogonal_spectra_with_action}{In terms of orthogonal spectra with $G$-action}\dotfill \pageref*{in_terms_of_orthogonal_spectra_with_action} \linebreak
\noindent\hyperlink{in_terms_of_mackeyfunctors}{In terms of Mackey-functors}\dotfill \pageref*{in_terms_of_mackeyfunctors} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{equivariant_cohomology}{Equivariant cohomology}\dotfill \pageref*{equivariant_cohomology} \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 stable homotopy theory over some [[topological group]] $G$ is the [[stable homotopy theory]] of [[G-spectra]]. This includes the [[naive G-spectra]] which constitute the actual [[stabilization]] of [[equivariant homotopy theory]], but is more general, one speaks of \emph{genuine $G$-spectra}. Notably a genuine $G$-spectrum has [[homotopy groups]] graded not by the group of [[integers]], but by the [[representation ring]] of $G$ (usually called [[RO(G)-grading]]).

The concept of [[cohomology]] in equivariant stable homotopy theory is \emph{[[equivariant cohomology]]}:

[[!include equivariant cohomology -- table]]

\hypertarget{basic_definitions}{}\subsection*{{Basic definitions}}\label{basic_definitions}

\hypertarget{in_terms_of_looping_by_representation_spheres}{}\subsubsection*{{In terms of looping by representation spheres}}\label{in_terms_of_looping_by_representation_spheres}

The definition of \emph{[[G-spectrum]]} is typically given in generalization of the definition of [[coordinate-free spectrum]].

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

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

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

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

A \textbf{$G$-spectrum} $E$ (indexed on the chosen universe $U$) is

\begin{itemize}%
\item for each indexing space $V \subset U$ a $G$-space $E V$;


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

\begin{displaymath}
E V \stackrel{\simeq}{\to} \Omega^{W-V} E W
  \,.
\end{displaymath}


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

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

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

This may be expressed directly in terms of the notion of \textbf{homotopy group of a $G$-spectrum}: this is \ldots{}

\hypertarget{in_terms_of_orthogonal_spectra_with_action}{}\subsubsection*{{In terms of orthogonal spectra with $G$-action}}\label{in_terms_of_orthogonal_spectra_with_action}

\ldots{} (\hyperlink{Schwede15}{Schwede 15})\ldots{}

\hypertarget{in_terms_of_mackeyfunctors}{}\subsubsection*{{In terms of Mackey-functors}}\label{in_terms_of_mackeyfunctors}

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

\begin{displaymath}
S \;\colon\; Corr_1^{eff}(\mathcal{C}) \longrightarrow Spectra
\end{displaymath}
For $G$ a [[finite group]] and $\mathcal{C}= G Set$ its category of [[permutation representations]], we have that $S$ is a genuine $G$-[[equivariant spectrum]] (\hyperlink{GuillouMay11}{Guillou-May 11}). So in this case the [[homotopy theory]] of spectral Mackey functors is a presentation for [[equivariant stable homotopy theory]] (\hyperlink{GuillouMay11}{Guillou-May 11}, \hyperlink{Barwick14}{Barwick 14}).

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

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item [[equivariant sphere spectrum]], [[equivariant stable cohomotopy]]


\item [[equivariant bordism homology theory]], [[equivariant cobordism cohomology theory]]


\item [[equivariant connective topological K-theory]]



\end{itemize}
[[!include Segal completion -- table]]

\begin{itemize}%
\item [[KR cohomology theory]]


\item [[equivariant complex cobordism cohomology theory]]


\item [[equivariant Morava K-theory]]



\end{itemize}
\hypertarget{equivariant_cohomology}{}\subsection*{{Equivariant cohomology}}\label{equivariant_cohomology}

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

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[equivariant differential topology]]


\item [[global equivariant homotopy theory]]


\item [[rational equivariant stable homotopy theory]]


\item [[equivariant motivic homotopy theory]]


\item [[equivariant spectrum]],

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


\item [[G-spectra]], [[naive G-spectra]], [[spectra with G-action]]

[[G-spaces]]


\item [[Burnside category]], [[Burnside ring]]

[[Mackey functor]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Original articles include

\begin{itemize}%
\item [[Graeme Segal]], \emph{Equivariant stable homotopy theory}, Actes du Congr\`e{}s International des Math\'e{}maticiens (Nice, 1970), Tome 2, pp. 59--63. Gauthier-Villars, Paris, 1971. ([[SegalEquivariantStableHomotopyTheory.pdf:file]])

\end{itemize}
A textbook account in terms of [[G-spectra]] modeled on a complete [[G-universe]] is in

\begin{itemize}%
\item [[L. Gaunce Lewis]], [[Peter May]], and Mark Steinberger (with contributions by J.E. McClure), \emph{Equivariant stable homotopy theory}, Springer Lecture Notes in Mathematics Vol.1213. 1986 (\href{http://www.math.uchicago.edu/~may/BOOKS/equi.pdf}{pdf}, \href{https://link.springer.com/book/10.1007/BFb0075778}{doi:10.1007/BFb0075778})

\end{itemize}
and a more modern version taking into account the theory of [[symmetric monoidal categories of spectra]] is in

\begin{itemize}%
\item [[Michael Mandell]], [[Peter May]], \emph{Equivariant orthogonal spectra and S-modules}, Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108. MR 2003i:55012 (\href{http://www.math.uiuc.edu/K-theory/0408/MMM.pdf}{pdf}, \href{http://www.math.uiuc.edu/K-theory/0408/}{K-theory archive})

\end{itemize}
See also

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

\end{itemize}
Lecture notes are in

\begin{itemize}%
\item [[Andrew Blumberg]], \emph{The Burnside category}, 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}
Further introductions and surveys include the following

\begin{itemize}%
\item [[Gunnar Carlsson]], \emph{A survey of equivariant stable homotopy theory},Topology, Vol 31, No. 1, pp. 1-27, 1992 (\href{http://www.maths.ed.ac.uk/~aar/papers/carlsson1.pdf}{pdf})


\item [[Anna Marie Bohmann]], \emph{Basic notions of equivariant stable homotopy theory} (\href{http://math.northwestern.edu/~bohmann/basicequivnotions.pdf}{pdf})


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


\item [[Brooke Shipley]], \emph{An introduction to equivariant homotopy theory} (\href{http://homepages.math.uic.edu/~bshipley/bonn.2.pdf}{pdf})


\item \href{http://math.mit.edu/conferences/talbot/index.php?year=2016}{Talbot workshop 2016}



\end{itemize}
Lecture notes on [[G-spectra]] modeled as [[orthogonal spectra]] with $G$-actions are

\begin{itemize}%
\item [[Stefan Schwede]], \emph{[[Lectures on Equivariant Stable Homotopy Theory]]}, 2015 (\href{http://www.math.uni-bonn.de/people/schwede/equivariant.pdf}{pdf})

\end{itemize}
An alternative perspective on this is in

\begin{itemize}%
\item [[Mark Hovey]], [[David White]], \emph{An alternative approach to equivariant stable homotopy theory} (\href{http://arxiv.org/abs/1312.3846}{arXiv:1312.3846})

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

\begin{itemize}%
\item [[Halvard Fausk]], \emph{Equivariant homotopy theory for pro-spectra} (\href{http://arxiv.org/abs/math/0609635}{arXiv:math/0609635})

\end{itemize}
The [[May recognition theorem]] for [[G-spaces]] and [[genuine G-spectra]] is discussed in

\begin{itemize}%
\item Costenoble and Warner, \emph{Fixed set systems of equivariant infinite loop spaces} Transactions of the American mathematical society, volume 326, Number 2 (1991) (\href{http://www.jstor.org/pss/2001770}{JSTOR})

\end{itemize}
Characterization of [[G-spectra]] \href{spectrum+object#ViaExcisiveFunctors}{via excisive functors} on [[G-spaces]] is in

\begin{itemize}%
\item [[Andrew Blumberg]], \emph{Continuous functors as a model for the equivariant stable homotopy category} (\href{http://arxiv.org/abs/math.AT/0505512}{arXiv:math.AT/0505512})

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

\begin{itemize}%
\item [[Stefan Schwede]], [[Brooke Shipley]], \emph{Classification of stable model categories} (\href{http://hopf.math.purdue.edu/Schwede-Shipley/class.final.pdf}{pdf})

\end{itemize}
A construction of equivariant stable homotopy theory in terms of [[spectral Mackey functors]] is due to

\begin{itemize}%
\item [[Bert Guillou]], [[Peter May]], \emph{Models of $G$-spectra as presheaves of spectra, (\href{http://arxiv.org/abs/1110.3571}{arXiv:1110.3571})}

\emph{Permutative $G$-categories in equivariant infinite loop space theory (\href{http://arxiv.org/abs/1207.3459}{arXiv:1207.3459})}



\end{itemize}
see at \emph{[[spectral Mackey functor]]} for more references.

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

\begin{itemize}%
\item [[Clark Barwick]], \emph{Spectral Mackey functors and equivariant algebraic K-theory (I)}, Adv. Math., 304:646–727 (\href{http://arxiv.org/abs/1404.0108}{arXiv:1404.0108})

\end{itemize}


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