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\begin{document}

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\section*{equivariant differential topology}

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

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

[[!include topology - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - 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{properties}{Properties}\dotfill \pageref*{properties} \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 subject of \emph{equivariant differential topology} is the enhancement of results of [[differential topology]] from plain [[manifolds]]/[[topological spaces]] to those equipped with [[actions]] of some [[group]] ([[G-spaces]]).

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

\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]].

If in addition $X$ is equipped with a [[Riemannian metric]] and $G$ acts by [[isometries]] then the [[submanifold]] $X^G$ is a [[totally geodesic submanifold]].

\end{prop}
(e.g. \hyperlink{Ziller13}{Ziller 13, theorem 3.5.2}, 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}
\begin{prop}
\label{GActionOnNormalBundleToFixedLocus}\hypertarget{GActionOnNormalBundleToFixedLocus}{}
\textbf{($G$-action on [[normal bundle]] to [[fixed locus]])}

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 linearization of the $G$-action aroujnd the [[fixed locus]] $X^G \subset X$ equips the [[normal bundle]] $N_X\left( X^G\right)$ with [[smooth function|smooth]] and [[fiber]]-wise [[linear map|linear]] $G$-[[action]].

\end{prop}
(e.g. \hyperlink{CrainicStruchiner13}{Crainic-Struchiner 13, Example 1.7})

\begin{prop}
\label{ExistenceOfGInvariantRiemannianMetrics}\hypertarget{ExistenceOfGInvariantRiemannianMetrics}{}
\textbf{(existence of $G$-invariant [[Riemannian metric]])}

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

Then there exists a [[Riemannian metric]] on $X$ with its [[invariant]] with respect to the $G$-[[action]], hence such that all elements of $G$ act by [[isometries]].

\end{prop}
(\hyperlink{Bredon72}{Bredon 72, VI Theorem 2.1}, see also \hyperlink{Ziller13}{Ziller 13, Theorem 3.0.2})

\begin{defn}
\label{GInvariantTubularNeighbourhood}\hypertarget{GInvariantTubularNeighbourhood}{}
\textbf{($G$-invariant [[tubular neighbourhood]])}

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]].

For $\Sigma \subset X^G \subset X$ a [[closed manifold|closed]] [[smooth manifold|smooth]] [[submanifold]] inside the [[fixed locus]], a \emph{$G$-invariant [[tubular neighbourhood]]} $\mathcal{N}(\Sigma \subset X)$ of $\Sigma$ in $X$ is

\begin{enumerate}%
\item a [[smooth vector bundle]] $E \to \Sigma$ equipped with a [[fiber]]-wise [[linear map|linear]] $G$-[[action]];


\item an equivariant [[diffeomorphism]] $E \overset{}{\longrightarrow} X$ onto an [[open neighbourhood]] of $\Sigma$ in $X$ which takes the [[zero section]] identically to $\Sigma$.



\end{enumerate}
\end{defn}
\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

\begin{enumerate}%
\item $\Sigma$ admits a $G$-invariant [[tubular neighbourhood]] $\Sigma \subset U \subset X$ (Def. \ref{GInvariantTubularNeighbourhood});


\item any two choices of such $G$-invariant tubular neighbourhoods are $G$-equivariantly [[isotopy|isotopic]];


\item there always exists an $G$-invariant tubular neighbourhood parametrized specifically by the [[normal bundle]] $N(\Sigma \subset X)$ of $Sigma$ in $X$, equipped with its induced $G$-[[action]] from Def. \ref{GActionOnNormalBundleToFixedLocus}, and such that the $G$-equivariant [[diffeomorphism]] is given by the [[exponential map]]

\begin{displaymath}
\exp_\epsilon
    \;\colon\;
  N(\Sigma \subset X)
    \overset{\simeq}{\longrightarrow}
  \mathcal{N}(\Sigma \subset X)
\end{displaymath}


\end{enumerate}
with respect to a $G$-invariant [[Riemannian metric]] (which exists according to Prop. \ref{ExistenceOfGInvariantRiemannianMetrics}):

\end{prop}
The existence of the $G$-invariant tubular neighbourhoods is for instance in \hyperlink{Bredon72}{Bredon 72 VI Theorem 2.2}, \hyperlink{Kankaanrinta07}{Kankaanrinta 07, theorem 4.4}. The uniqueness up to equivariant isotopy is in \hyperlink{Kankaanrinta07}{Kankaanrinta 07, theorem 4.4, theorem 4.6}. The fact that one may always use the [[normal bundle]] appears at the end of the proof of \hyperlink{Bredon72}{Bredon 72 VI Theorem 2.2}, and as a special case of a more general statement about invariant tubular neighbourhoods in [[Lie groupoids]] it follows from \hyperlink{PflaumPosthumaTang11}{Pflaum-Posthuma-Tang 11, Theorem 4.1} by applying the construction there to each point in $\Sigma$ for one and the same choice of background metric. See also for instance \hyperlink{PflaumWilkin17}{Pflaum-Wilkin 17, Example 2.5}.

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

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


\item [[equivariant stable homotopy theory]]


\item [[equivariant rational homotopy theory]]


\item [[equivariant cohomology]]



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

\begin{itemize}%
\item [[Arthur Wasserman]], section 3 of \emph{Equivariant differential topology}, Topology Vol. 8, pp. 127-150, 1969 (\href{https://web.math.rochester.edu/people/faculty/doug/otherpapers/wasserman.pdf}{pdf})


\item [[Glen Bredon]], \emph{Introduction to compact transformation groups}, Academic Press 1972 (\href{http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf}{pdf})


\item [[Marja Kankaanrinta]], \emph{Equivariant collaring, tubular neighbourhood and gluing theorems for proper Lie group actions}, Algebr. Geom. Topol. Volume 7, Number 1 (2007), 1-27 (\href{https://projecteuclid.org/euclid.agt/1513796653}{euclid:agt/1513796653})


\item [[Markus Pflaum]], Hessel Posthuma, X. Tang, \emph{Geometry of orbit spaces of proper Lie groupoids}, Journal für die reine und angewandte Mathematik (Crelles Journal) 2014.694 (\href{https://arxiv.org/abs/1101.0180}{arXiv:1101.0180}, \href{https://doi.org/10.1515/crelle-2012-0092}{doi:10.1515/crelle-2012-0092})


\item Wolfgang Ziller, \emph{Group actions}, 2013 (\href{https://www.math.upenn.edu/~wziller/math661/LectureNotesLee.pdf}{pdf})


\item Marius Crainic, Ivan Struchiner, \emph{On the linearization theorem for proper Lie groupoids}, Annales scientifiques de l'École Normale Supérieure, Série 4, Volume 46 (2013) no. 5, p. 723-746 (\href{http://www.numdam.org/item/ASENS_2013_4_46_5_723_0/}{numdam:ASENS\_2013\_4\_46\_5\_723\_0} \href{https://doi.org/10.24033/asens.2200}{doi:10.24033/asens.2200})


\item [[Markus Pflaum]], Graeme Wilkin, \emph{Equivariant control data and neighborhood deformation retractions}, Methods and Applications of Analysis, 2019 (\href{https://arxiv.org/abs/1706.09539}{arXiv:1706.09539})



\end{itemize}


\end{document}