nLab
Steenrod-Wockel approximation theorem

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

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 XX be a finite dimensional connected smooth manifold with corners. Let π:EX\pi : E \to X be a locally trivial smooth bundle with a locally convex manifold NN as typical fiber and σ:XE\sigma : X \to E a continuous section.

If LXL \subset X is a closed subset and UXU \subset X is an open subset such that σ\sigma is smooth in a neighbourhood of LUL \setminus U, then:

  1. for each open neighbourhood OO of σ(X)\sigma(X) in EE there exists a section τ:XO\tau : X \to O

    • which is smooth in a neighbourhood of LL;

    • and which equals σ\sigma on XUX \setminus U;

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

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

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

See (Wockel)

O id O id O smooth σ XU= τ XU σ F τ smooth LU XU X id X L \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:ZYf,g : Z \to Y be two smooth functions between smooth manifolds. Let η:Z×[0,1]Y\eta : Z \times [0,1] \to Y be a continuous delayed homotopy between them, constant in a neighbourhood Z×([0,ϵ)(1ϵ,1])Z \times ([0,\epsilon) \coprod (1-\epsilon,1]).

Then there exists also smooth homotopy between ff and gg which is itself continuously homotopic to η\eta.

Proof

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

  • set X:=Z×[0,1]X := Z \times [0,1];

  • set N=YN = Y;

  • let E=Z×[0,1]×YE = Z \times [0,1] \times Y be the trivial ZZ-bundle over XX

    (so that sections of EE are equivalently functions Z×[0,1]YZ \times[0,1] \to Y)

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

  • set L:=Z×[0,1]L := Z \times [0,1];

  • let U:=Z×(0,1)U := Z \times (0,1).

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

So the theorem applies and provides a smooth homotopy

τ:[0,1]×ZY \tau : [0,1] \times Z \to Y

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

References

Revised on September 11, 2013 10:40:00 by Urs Schreiber (82.169.114.243)