nLab
Steenrod-Wockel approximation theorem

Idea

The 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.

Statement

Theorem

Let $X$ be a finite dimensional connected smooth manifold with corners. Let $\pi :E\to X$ be a locally trivial smooth bundle with a locally convex manifold $N$ as typical fiber and $\sigma :X\to E$ a continuous section.

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

1. for each open neighbourhood $O$ of $\sigma (X)$ in $E$ there exists a section $\tau :X\to O$

   • which is smooth in a neighbourhood of $L$;

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

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

   • each $F(t,-)$ is a section of $\pi$;

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

See (Wockel)

$$\begin{array}{ccccccc}& & O& \stackrel{\mathrm{id}}{\to }& O& \stackrel{\mathrm{id}}{\to }& O\\ & {}^{\mathrm{smooth}}\nearrow & {}_{\sigma {\mid }_{X\setminus U}}\uparrow ={\uparrow }_{\tau {\mid }_{X\setminus U}}& & {\uparrow }^{\sigma }& {⇙}_F& {\uparrow }^{\exists \tau }& {\nwarrow }^{\mathrm{smooth}}\\ L\setminus U& \hookrightarrow & X\setminus U& \hookrightarrow & X& \stackrel{\mathrm{id}}{\to }& X& \stackrel{}{\hookleftarrow }& L\end{array}$$\array{ && 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 }

Examples

Smoothing of delayed homotopies

Corollary

Let $f,g:Z\to Y$ be two smooth functions between smooth manifolds. Let $\eta :Z\times [0,1]\to Y$ be a continuous delayed homotopy between them, constant in a neighbourhood $Z\times ([0,ϵ)\coprod (1-ϵ,1])$.

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

Proof

To apply the generalized Steenrod theorem with the notation as stated there, make the following identifications

• set $X:=Z\times [0,1]$;

• set $N=Y$;

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

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

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

• set $L:=Z\times [0,1]$;

• let $U:=Z\times (0,1)$.

Then because by assumption $\eta$ is a continuous delayed homotopy between smooth functions, it follows that $\sigma$ is smooth in a neighbourhood $Z\times ([0,ϵ)\coprod (1-ϵ,1])$ of $L$.

So the theorem applies and provides a smooth homotopy

$$\tau :[0,1]\times Z\to Y$$\tau : [0,1] \times Z \to Y

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

Related theorems

• the Hadamard lemma

• Borel's theorem

• the Tietze extension theorem

References

• Christoph Wockel, A Generalisation of Steenrod’s Approximation Theorem Archivum mathematicum, Volume 45 No. 2 (2009) (arXiv:0610252)