# nLab Steenrod approximation theorem

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# 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 $X$ 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 $X := Z \times [0,1]$;

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

So the theorem applies and provides a smooth homotopy

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

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