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\begin{document}

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\section*{topological G-space}

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

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{equivariant_tietze_extension_theorem}{Equivariant Tietze extension theorem}\dotfill \pageref*{equivariant_tietze_extension_theorem} \linebreak
\noindent\hyperlink{ModelStructureAndHomotopyTheory}{Model structure and homotopy theory}\dotfill \pageref*{ModelStructureAndHomotopyTheory} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{EuclideanGSpace}{Euclidean $G$-spaces}\dotfill \pageref*{EuclideanGSpace} \linebreak
\noindent\hyperlink{representation_spheres}{Representation spheres}\dotfill \pageref*{representation_spheres} \linebreak
\noindent\hyperlink{representation_tori}{Representation tori}\dotfill \pageref*{representation_tori} \linebreak
\noindent\hyperlink{gcw_complexes}{G-CW complexes}\dotfill \pageref*{gcw_complexes} \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}

In the context of [[topology]], a \emph{topological $G$-space} (traditionally just \emph{$G$-space}, for short, if the context is clear) is a [[topological space]] equipped with an [[action]] of a [[topological group]] $G$ (often, but crucially not always, taken to be a [[finite group]]).

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 Whitehead theorem|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}}.

See (\hyperlink{HenriquesGepner07}{Henriques-Gepner 07}) for expression in terms of [[topological groupoids]]/[[orbispaces]].

In the context of [[stable homotopy theory]] the [[stabilization]] of $G$-spaces is given by [[spectra with G-action]]; these lead to [[equivariant stable homotopy theory]]. See there for more details. (But beware that in this context one considers the richer concept of [[G-spectra]], which have a [[forgetful functor]] to [[spectra with G-action]] but better homotopy theoretic properties. ) The union of this as $G$ is allowed to vary is the [[global equivariant stable homotopy theory]].

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

\hypertarget{equivariant_tietze_extension_theorem}{}\subsubsection*{{Equivariant Tietze extension theorem}}\label{equivariant_tietze_extension_theorem}

See at \emph{[[equivariant Tietze extension theorem]]}

\hypertarget{ModelStructureAndHomotopyTheory}{}\subsubsection*{{Model structure and homotopy theory}}\label{ModelStructureAndHomotopyTheory}

The standard [[homotopy theory]] on $G$-spaces used in [[equivariant homotopy theory]] considers [[weak equivalences]] which are [[weak homotopy equivalence]] on all (ordinary) [[fixed point]] spaces for all suitable [[subgroups]]. By [[Elmendorf's theorem]], this is equivalent to [[(∞,1)-presheaves]] over the [[orbit category]] of $G$.

On the other hand there is also the standard homotopy theory of [[infinity-actions]], presented by the [[Borel model structure]], in this context also called the ``coarse'' or ``naive'' equivariant model structure (\hyperlink{Guillou}{Guillou}).

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

We discuss some classes of examples of [[G-spaces]].

\hypertarget{EuclideanGSpace}{}\subsubsection*{{Euclidean $G$-spaces}}\label{EuclideanGSpace}

Let $V \in RO(G)$ be an [[orthogonal group|orthogonal]] [[linear representation]] of a [[finite group]] $G$ on a [[real vector space]] $V$. Then the underlying [[Euclidean space]] $\mathbb{R}^V$ inherits the [[mathematical structure|structure]] of a [[G-space]]

$\backslash$begin\{xymatrix\} $\backslash$mathbb\{R\}{\tt \symbol{94}}V$\backslash$ar@(ul,ur){\tt \symbol{94}}G $\backslash$end\{xymatrix\}

We may call this the \emph{[[Euclidean G-space]]} associated with the linear representation $V$.

\hypertarget{representation_spheres}{}\subsubsection*{{Representation spheres}}\label{representation_spheres}

Let $V \in RO(G)$ be an [[orthogonal group|orthogonal]] [[linear representation]] of a [[finite group]] $G$ on a [[real vector space]] $V$. Then the [[one-point compactification]] of the underlying [[Euclidean space]] $\mathbb{R}^V$ inherits the [[mathematical structure|structure]] of a [[G-space]] with the [[one-point compactification|point at infinity]] a [[fixed point]]. This is called the \emph{$V$-[[representation sphere]]}

$\backslash$begin\{xymatrix\} S{\tt \symbol{94}}V$\backslash$ar@(ul,ur){\tt \symbol{94}}G $\backslash$ar@\{\}r|- \& $\backslash$big( $\backslash$mathbb\{R\}{\tt \symbol{94}}V$\backslash$ar@(ul,ur){\tt \symbol{94}}G$\backslash$big){\tt \symbol{94}}\{$\backslash$mathrm\{cpt\}\} $\backslash$end\{xymatrix\}

\hypertarget{representation_tori}{}\subsubsection*{{Representation tori}}\label{representation_tori}

Let $V \in RO(G)$ be an [[orthogonal group|orthogonal]] [[linear representation]] of a [[finite group]] $G$ on a [[real vector space]] $V$.

If $G$ is the [[point group]] of a [[crystallographic group]] inside the [[Euclidean group]]

\begin{displaymath}
N \rtimes G
  \hookrightarrow
  Iso(\mathbb{R}^V)
\end{displaymath}
then the $G$-[[action]] on the Euclidean space $\mathbb{R}^V$ descends to the [[quotient]] by the [[action]] of the translational [[normal subgroup]] [[lattice in a vector space|lattice]] $N$ (\href{crystallographic+group#InducedPointGroupActionOnTorus}{this Prop.}). The resulting $G$-space is an [[n-torus]] with $G$-action, which might be called the \emph{[[representation torus]]} of $V$

$\backslash$begin\{xymatrix\} $\backslash$mathbb\{R\}{\tt \symbol{94}}V$\backslash$ar@(ul,ur){\tt \symbol{94}}G $\backslash$ar@\{\}r|- \& $\backslash$big( $\backslash$mathbb\{R\}{\tt \symbol{94}}V$\backslash$ar@(ul,ur){\tt \symbol{94}}G$\backslash$big)/N $\backslash$end\{xymatrix\}

\begin{quote}%
graphics grabbed from \href{equivariant+Hopf+degree+theorem#SatiSchreiber19}{SS 19}


\end{quote}
\hypertarget{gcw_complexes}{}\subsubsection*{{G-CW complexes}}\label{gcw_complexes}

See at \emph{[[G-CW complex]]}.

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

\begin{itemize}%
\item [[G-set]]


\item [[fixed point space]]


\item [[cyclotomic space]]



\end{itemize}
[[!include equivariant homotopy theory -- table]]

See also

\begin{itemize}%
\item \emph{[[equivariant stable homotopy theory]]}

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

\begin{itemize}%
\item [[Glen Bredon]], chapter II of \emph{[[Introduction to compact transformation groups]]}, Academic Press 1972


\item [[Bert Guillou]], \emph{A short note on models for equivariant homotopy theory} (\href{http://www.math.uiuc.edu/~bertg/EquivModels.pdf}{pdf})


\item [[André Henriques]], [[David Gepner]], \emph{Homotopy Theory of Orbispaces} (\href{http://arxiv.org/abs/math/0701916}{arXiv:math/0701916})



\end{itemize}
See also the references at \emph{[[equivariant homotopy theory]]}.

[[!redirects topological G-spaces]] [[!redirects topological G-spaces]]

[[!redirects G-space]] [[!redirects G-spaces]]



\end{document}