nLab Steenrod 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)

$$\array{ && O &\stackrel{id}{\to} & O & \stackrel{id}{\to}& O \\ & {}^{\mathllap{smooth}}\nearrow & {}_{\mathllap{\sigma|_{X \setminus U}}}\uparrow = \uparrow_{\mathrlap{\tau|_{X \setminus U}}} && \uparrow^{\mathrlap{\sigma}} & \swArrow_F& \uparrow^{\exists \tau} & \nwarrow^{\mathrlap{smooth}} \\ L \setminus U &\hookrightarrow & X \setminus U &\hookrightarrow& X &\stackrel{id}{\to}& X &\stackrel{}{\hookleftarrow}& L }$$

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 $Z \times ([0,\epsilon) \coprod (1-\epsilon,1])$.

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 $X \coloneqq Z \times [0,1]$;

• 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 $L \coloneqq Z \times [0,1]$;

• let $U \coloneqq Z \times (0,1)$.

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

So the theorem applies and provides a smooth homotopy

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

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

• the Whitney extension theorem

• embedding of smooth manifolds into formal duals of R-algebras

• smooth Serre-Swan theorem

• derivations of smooth functions are vector fields

References

• Norman Steenrod, Section 6.7 of: The topology of fibre bundles, Princeton Mathematical Series 14, Princeton Univ. Press (1951) [jstor:j.ctt1bpm9t5]

• Morris Hirsch, Chapter 2.2 (Thm. 2.5) in: Differential topology, Springer Graduate Texts in Mathematics 33 (1976) [doi:10.1007/978-1-4684-9449-5, gBooks]

• Christoph Wockel, A Generalisation of Steenrod’s Approximation Theorem, Archivum mathematicum 45 2 (2009) [arXiv:0610252]