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Relative Whitney approximation for manifold-valued maps

Statement

Let $F : M \to N$ be continuous, let $A \subseteq M$ be closed, and suppose $F$ is smooth on a neighbourhood of $A$ . Then there exists a smooth map $\tilde{F} : M \to N$ such that $\tilde{F} = F$ on a neighbourhood of $A$ and $\tilde{F}$ is homotopic to $F$ .

Facts & Assumptions

Given: A continuous map $F : M \to N$ , a closed set $A \subseteq M$ , and the assumption that $F$ is smooth on a neighbourhood of $A$ .

[L1]

Relative Euclidean approximation preserves the map near a closed set (Relative Whitney approximation for Euclidean-valued maps).

[L2]

Fine approximation can be forced into a tubular neighbourhood, and the absolute manifold-valued theorem retracts such an approximation back to the target (A fine Euclidean approximation lands in a prescribed tubular neighbourhood, Whitney approximation for manifold-valued maps).

Proof

technique · direct

1.1

In the proof of the absolute manifold-valued theorem from [L2], fix one Euclidean embedding $j : N ↪ R^{m}$ , one tubular neighbourhood $U$ of $j (N)$ , and one tubular retraction $r : U \to j (N)$ .

L2givenchoose

2.1

Apply the fine-approximation lemma from [L2] to $j \circ F$ and $U$ , then apply the relative Euclidean approximation theorem from [L1] to obtain a smooth map $H : M \to R^{m}$ such that $H = j \circ F$ on a neighbourhood of $A$ and $H (M) \subseteq U$ .

L1L2step 1.1construct

3.1

Define $\tilde{F} : = j^{- 1} \circ r \circ H$ . On the neighbourhood where $H = j \circ F$ , the retraction fixes $j (F)$ pointwise, so $\tilde{F} = F$ there. The same straight-line homotopy inside $U$ as in the absolute theorem gives a homotopy from $F$ to $\tilde{F}$ .

step 1.1step 2.1algebra∎

Depends on

Used by

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Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Smooth Approximation of Maps Between Manifolds (standard reference, not scraped)