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\begin{document}

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\section*{equivariant homotopy group}

\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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{abstractly}{Abstractly}\dotfill \pageref*{abstractly} \linebreak
\noindent\hyperlink{via_genuine_spectra}{Via genuine $G$-spectra}\dotfill \pageref*{via_genuine_spectra} \linebreak
\noindent\hyperlink{via_orthogonal_spectra_and_equivariant_maps}{Via orthogonal spectra and $G$-equivariant maps}\dotfill \pageref*{via_orthogonal_spectra_and_equivariant_maps} \linebreak
\noindent\hyperlink{via_fixed_point_spectra}{Via fixed point spectra}\dotfill \pageref*{via_fixed_point_spectra} \linebreak
\noindent\hyperlink{ViaEquivariantCohomologyOfThePoint}{Via equivariant cohomology of the point}\dotfill \pageref*{ViaEquivariantCohomologyOfThePoint} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{of_equivariant_suspension_spectra}{Of equivariant suspension spectra}\dotfill \pageref*{of_equivariant_suspension_spectra} \linebreak
\noindent\hyperlink{of_the_equivariant_sphere_spectrum}{Of the equivariant sphere spectrum}\dotfill \pageref*{of_the_equivariant_sphere_spectrum} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{relation_to_mackey_functors}{Relation to Mackey functors}\dotfill \pageref*{relation_to_mackey_functors} \linebreak
\noindent\hyperlink{equivariant_stable_whitehead_theorem}{Equivariant stable Whitehead theorem}\dotfill \pageref*{equivariant_stable_whitehead_theorem} \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}

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

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{abstractly}{}\subsubsection*{{Abstractly}}\label{abstractly}

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

\begin{displaymath}
\pi_n^H(X) \coloneqq \pi_n(X^H)
  \,.
\end{displaymath}
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 $X$:

\begin{displaymath}
\pi_n^H(X) \simeq [G/H_+ \wedge S^n, X]^G
  \,.
\end{displaymath}
In this form the definition directly generalizes to [[G-spectra]] and hence to stable equivariant homotopy groups: for $E$ a [[G-spectrum]], then

\begin{displaymath}
\pi_n^H(X) \simeq [G/H_+ \wedge \Sigma^\infty S^n, X]^G
  \,.
\end{displaymath}
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]].

\hypertarget{via_genuine_spectra}{}\subsubsection*{{Via genuine $G$-spectra}}\label{via_genuine_spectra}

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

For a \emph{finite} based [[G-CW complex]] $X$ and base [[topological G-space]] $Y$, write

\begin{displaymath}
\{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
\end{displaymath}
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]].

(\hyperlink{May96}{May 96, IX.2 def. 2.1})

The equivariant stable homotopy groups of $X$ are

\begin{displaymath}
\pi_V^G(\Sigma^\infty_G X) \coloneqq \{S^V,X\}_G
  \,.
\end{displaymath}
(\hyperlink{May96}{May 96, IX.2 remark 2.4})

And for subgroups $H \subset G$

\begin{displaymath}
\pi_V^H(\Sigma^\infty_G X) \coloneqq \{G/H_+ \wedge S^V,X\}_G
\end{displaymath}
(\hyperlink{GreenleesMay95}{Greenlees-May 95, p. 11})

\hypertarget{via_orthogonal_spectra_and_equivariant_maps}{}\subsubsection*{{Via orthogonal spectra and $G$-equivariant maps}}\label{via_orthogonal_spectra_and_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]]

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

\begin{itemize}%
\item $\rho_G$ denotes the [[fundamental representation]] of $G$


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


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


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


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



\end{itemize}
(e.g. \hyperlink{Schwede15}{Schwede 15, section 3})

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. \hyperlink{Schwede15}{Schwede 15, p. 16})

\hypertarget{via_fixed_point_spectra}{}\subsubsection*{{Via fixed point spectra}}\label{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]] $F^G E$

\begin{equation}
\pi_k^G(E) \simeq \pi_k(F^G E)
  \,.
\label{ViaFixedPointSpectra}\end{equation}
(e.g. \hyperlink{Schwede15}{Schwede 15, prop. 7.2})

\hypertarget{ViaEquivariantCohomologyOfThePoint}{}\subsubsection*{{Via equivariant cohomology of the point}}\label{ViaEquivariantCohomologyOfThePoint}

The identification \eqref{ViaFixedPointSpectra} in turn is the [[equivariant cohomology]] of the point,

\begin{displaymath}
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)
\end{displaymath}
due to the [[base change]] [[adjunction]]

\begin{displaymath}
G Spectra
    \underoverset
      { \underset{ F^G }{\longrightarrow} }
      {\overset{ \epsilon^\sharp }{\longleftarrow}}
      { \phantom{AA} \bot \phantom{AA} }
  Spectra
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{of_equivariant_suspension_spectra}{}\subsubsection*{{Of equivariant suspension spectra}}\label{of_equivariant_suspension_spectra}

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

\begin{displaymath}
\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
\end{displaymath}
which in turn decomposes as a [[direct sum]] of ordinary [[homotopy groups]] of [[Weyl group]]-[[homotopy quotients]] of naive [[fixed point]] spaces -- see at \emph{[[tom Dieck splitting]]}.

\hypertarget{of_the_equivariant_sphere_spectrum}{}\subsubsection*{{Of the equivariant sphere spectrum}}\label{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 $G$:

\begin{displaymath}
\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]
\end{displaymath}
(e.g. \hyperlink{Schwede15}{Schwede 15, p. 64})

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{relation_to_mackey_functors}{}\subsubsection*{{Relation to Mackey functors}}\label{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]])

\begin{displaymath}
\underline{\pi}_\bullet(E)
  \colon
  G/H \mapsto \pi^H_\bullet(X)
  \,.
\end{displaymath}
(e.g. \hyperlink{Schwede15}{Schwede 15, p. 16 and section 4})

(\ldots{})

\hypertarget{equivariant_stable_whitehead_theorem}{}\subsubsection*{{Equivariant stable Whitehead theorem}}\label{equivariant_stable_whitehead_theorem}

The equivariant version of the stable [[Whitehead theorem]] holds:

a mpa of $G$-spectra $f \colon E \longrightarrow F$ is a [[weak equivalence]] (e.g. an $RO(G)$-degree-wise [[weak homotopy equivalence]] of [[topological G-spaces]] in the \hyperlink{IndexedOnAllRepresentations}{model via indexing on all representations}) precisely it if induces [[isomorphisms]] $\pi_\bullet(f) \colon \pi_\bullet(E) \longrightarrow \pi_{\bullet}(F)$ on all [[equivariant homotopy group]] [[Mackey functors]].

(\hyperlink{GreenleesMay}{Greenlees-May, theorem 2.3})

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

\begin{itemize}%
\item [[homotopy group]]


\item [[stable homotopy group]]



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

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


\item [[Peter May]], section IX.4 of \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. Comeza\textasciitilde{}na, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (\href{http://www.math.rochester.edu/u/faculty/doug/otherpapers/alaska1.pdf}{pdf})


\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}
[[!redirects equivariant homotopy groups]]

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



\end{document}