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\begin{document}

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\section*{fixed point space}

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

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

[[!include topology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{for_topological_spaces}{For topological $G$-spaces}\dotfill \pageref*{for_topological_spaces} \linebreak
\noindent\hyperlink{in_equivariant_homotopy_theory}{In equivariant homotopy theory}\dotfill \pageref*{in_equivariant_homotopy_theory} \linebreak
\noindent\hyperlink{in_equivariant_stable_homotopy_theory}{In equivariant stable homotopy theory}\dotfill \pageref*{in_equivariant_stable_homotopy_theory} \linebreak
\noindent\hyperlink{in_equivariant_differential_topology}{In equivariant differential topology}\dotfill \pageref*{in_equivariant_differential_topology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

Generally, given some kind of [[space]] equipped with the [[action]] of a [[group]], the locus of \emph{[[fixed points]]} of the action may form a suitable sub-space: the \emph{fixed point space}.

\hypertarget{for_topological_spaces}{}\subsubsection*{{For topological $G$-spaces}}\label{for_topological_spaces}

Specifically, given a [[topological group]] $G$ and a [[topological G-space]], its \emph{fixed point space} is the set of the set-theoretic [[fixed points]] of the $G$-[[action]], equipped with the [[subspace topology]].

\hypertarget{in_equivariant_homotopy_theory}{}\subsubsection*{{In equivariant homotopy theory}}\label{in_equivariant_homotopy_theory}

The statement of [[Elmendorf's theorem]] is essentially that the [[equivariant homotopy theory]] of [[G-space|topological $G$-spaces]] is equivalently encoded in their systems of $H$-fixed point spaces, as $H$ varies over [[closed subspace|closed]] [[subgroups]] of $G$.

\hypertarget{in_equivariant_stable_homotopy_theory}{}\subsubsection*{{In equivariant stable homotopy theory}}\label{in_equivariant_stable_homotopy_theory}

In [[equivariant stable homotopy theory]] the concept of fixed point spaces branches into various closely related, but different concepts:

\begin{itemize}%
\item [[fixed point spectra]],


\item [[geometric fixed point spectra]]


\item [[categorical fixed point spectra]]



\end{itemize}
\hypertarget{in_equivariant_differential_topology}{}\subsubsection*{{In equivariant differential topology}}\label{in_equivariant_differential_topology}

In [[equivariant differential topology]]:

\begin{prop}
\label{ExistenceOfGInvariantTubularNeighbourhoods}\hypertarget{ExistenceOfGInvariantTubularNeighbourhoods}{}
\textbf{(existence of $G$-invariant [[tubular neighbourhoods]])}

Let $X$ be a [[smooth manifold]], $G$ a [[Lie group]] and $\rho \;\colon\; G \times X \to X$ a \emph{[[proper action|proper]]} [[action]] by [[diffeomorphisms]].

If $\Sigma \overset{\iota}{\hookrightarrow} X$ is a [[closed manifold|closed]] [[smooth manifold|smooth]] [[submanifold]] inside the $G$-[[fixed locus]]

$\backslash$begin\{center\} $\backslash$begin\{xymatrix\} $\backslash$Sigma $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}rr{\tt \symbol{94}}-\{$\backslash$iota{\tt \symbol{94}}G\} $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}dr\emph{\{$\backslash$iota\} \&\& X{\tt \symbol{94}}G $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}dl $\backslash$ \& X $\backslash$end\{xymatrix\} $\backslash$end\{center\}}

then $\Sigma$ admits a $G$-invariant [[tubular neighbourhood]] $\Sigma \subset U \subset X$.

Moreover, any two choices of such $G$-invariant tubular neighbourhoods are $G$-equivariantly [[isotopy|isotopic]].

\end{prop}
(\href{equivariant+differential+topology#Kankaanrinta07}{Kankaanrinta 07, theorem 4.4, theorem 4.6})

\begin{prop}
\label{FixedLociOfSmoothProperActionsAreSubmanifolds}\hypertarget{FixedLociOfSmoothProperActionsAreSubmanifolds}{}
\textbf{([[fixed loci]] of [[smooth function|smooth]] [[proper actions]] are [[submanifolds]])}

Let $X$ be a [[smooth manifold]], $G$ a [[Lie group]] and $\rho \;\colon\; G \times X \to X$ a \emph{[[proper action|proper]]} [[action]] by [[diffeomorphisms]].

Then the $G$-[[fixed locus]] $X^G \hookrightarrow X$ is a [[smooth manifold|smooth]] [[submanifold]].

\end{prop}
(see also \href{https://math.stackexchange.com/a/1739784/58526}{this MO discussion})

\begin{proof}
Let $x \in X^G \subset X$ be any [[fixed point]]. Since this is in particular a closed invariant [[submanifold]], Prop. \ref{ExistenceOfGInvariantTubularNeighbourhoods} applies and shows that an [[open neighbourhood]] of $x$ in $X$ is $G$-equivariantly [[diffeomorphism|diffeomorphic]] to a [[linear representation]] $V \in RO(G)$. The [[fixed locus]] $V^G \subset V$ of that is hence [[diffeomorphism|diffeomorphic]] to an [[open neighbourhood]] of $x$ in $\Sigma$.

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
Without the assumption of [[proper action]] in Prop. \ref{FixedLociOfSmoothProperActionsAreSubmanifolds} the conclusion generally fails. See \href{https://math.stackexchange.com/a/1739768/58526}{this MO comment} for a counter-example.

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

\begin{itemize}%
\item [[homotopy fixed points]]


\item [[Elmendorf's theorem]]


\item [[equivariant differential topology]]



\end{itemize}
[[!redirects fixed point spaces]]

[[!redirects fixed point set]] [[!redirects fixed point sets]]

[[!redirects fixed locus]] [[!redirects fixed loci]]

[[!redirects fixed point subspace]] [[!redirects fixed point subspaces]]



\end{document}