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\begin{document}

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\section*{Steenrod approximation theorem}

[[!redirects Steenrod-Wockel approximation theorem]]

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

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

[[!include synthetic differential geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{SmoothingOfDelayedHomotopies}{Smoothing of delayed homotopies}\dotfill \pageref*{SmoothingOfDelayedHomotopies} \linebreak
\noindent\hyperlink{related_theorems}{Related theorems}\dotfill \pageref*{related_theorems} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The \emph{Steenrod-approximation theorem} states mild conditions under which an extension of a [[smooth function]] on a [[closed subset]] by a [[continuous function]] may itself be improved to an extension by a smooth function.

This is a smooth enhancement of the [[Tietze extension theorem]].

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\begin{theorem}
\label{}\hypertarget{}{}
Let $X$ be a finite [[dimension]]al [[connected]] [[smooth manifold]] [[manifold with corners|with corners]]. Let $\pi : E \to X$ be a locally trivial [[Diff|smooth]] [[bundle]] with a [[locally convex]] [[manifold]] $N$ as typical [[fiber]] and $\sigma : X \to E$ a [[continuous function|continuous]] [[section]].

If $L \subset X$ is a [[closed subset]] and $U \subset X$ is an [[open subset]] such that $\sigma$ is [[smooth function|smooth]] in a [[neighbourhood]] of $L \setminus U$, then:

\begin{enumerate}%
\item for each [[open neighbourhood]] $O$ of $\sigma(X)$ in $E$ there exists a [[section]] $\tau : X \to O$

\begin{itemize}%
\item which is [[smooth function|smooth]] in a neighbourhood of $L$;


\item and which equals $\sigma$ on $X \setminus U$;



\end{itemize}

\item there exists a [[homotopy]] $F : [0,1] \times X \to O$ between $\sigma$ and $\tau$ such that

\begin{itemize}%
\item each $F(t,-)$ is a section of $\pi$;


\item for $(t,x) \in [0,1] \times (X \setminus U)$ we have $F(t,x) = \sigma(x) = \tau(x)$.



\end{itemize}


\end{enumerate}
\end{theorem}
See (\hyperlink{Wockel}{Wockel})

\begin{displaymath}
\itexarray{
    && O &\stackrel{id}{\to} & O &  \stackrel{id}{\to}& O
    \\
    & {}^{\mathllap{smooth}}\nearrow
    &
    {}_{\mathllap{\sigma|_{X \setminus U}}}\uparrow
     = \uparrow_{\mathrlap{\tau|_{X \setminus U}}} 
    && 
   \uparrow^{\mathrlap{\sigma}}
    & \swArrow_F& 
    \uparrow^{\exists \tau}
    & \nwarrow^{\mathrlap{smooth}}
    \\
    L \setminus U &\hookrightarrow & X \setminus U 
    &\hookrightarrow& X 
    &\stackrel{id}{\to}&
   X
   &\stackrel{}{\hookleftarrow}& L
  }
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{SmoothingOfDelayedHomotopies}{}\subsubsection*{{Smoothing of delayed homotopies}}\label{SmoothingOfDelayedHomotopies}

\begin{ucorollary}
Let $f,g : Z \to Y$ be two [[smooth function]]s between [[smooth manifold]]s. Let $\eta : Z \times [0,1] \to Y$ be a continuous [[delayed homotopy]] between them, constant in a [[neighbourhood]] $Z \times ([0,\epsilon) \coprod (1-\epsilon,1])$.

Then there exists also smooth homotopy between $f$ and $g$ which is itself continuously homotopic to $\eta$.

\end{ucorollary}
\begin{proof}
To apply the \hyperlink{GeneralizedSteenrodTheorem}{generalized Steenrod theorem} with the notation as stated there, make the following identifications

\begin{itemize}%
\item set $X := Z \times [0,1]$;


\item set $N = Y$;


\item let $E = Z \times [0,1] \times Y$ be the trivial $Z$-bundle over $X$

(so that sections of $E$ are equivalently functions $Z \times[0,1] \to Y$)


\item let $(\sigma : X \to E) := (\eta : Z \times [0,1] \to Y)$ be the given continuous homotopy;


\item set $L := Z \times [0,1]$;


\item let $U := Z \times (0,1)$.



\end{itemize}
Then because by assumption $\eta$ is a continuous [[delayed homotopy]] between [[smooth function]]s, it follows that $\sigma$ is smooth in a neighbourhood $Z \times ([0,\epsilon) \coprod (1-\epsilon,1])$ of $L$.

So the theorem applies and provides a smooth homotopy

\begin{displaymath}
\tau : [0,1] \times Z \to Y
\end{displaymath}
which moroever is itself (continuously) homotopic to $\eta$ via some continuous $F : [0,1] \times [0,1] \times Z \to Y$.

\end{proof}
\hypertarget{related_theorems}{}\subsection*{{Related theorems}}\label{related_theorems}

\begin{itemize}%
\item the [[Hadamard lemma]]


\item [[Borel's theorem]]


\item the [[Tietze extension theorem]]


\item the [[Whitney extension theorem]]


\item [[embedding of smooth manifolds into formal duals of R-algebras]]


\item [[smooth Serre-Swan theorem]]


\item [[derivations of smooth functions are vector fields]]



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

\begin{itemize}%
\item [[Christoph Wockel]], \emph{A Generalisation of Steenrod's Approximation Theorem}, Archivum mathematicum, Volume 45 No. 2 (2009) (\href{http://arxiv.org/abs/math/0610252}{arXiv:0610252})

\end{itemize}
[[!redirects Steenrod approximation theorem]]



\end{document}