TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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Continuously homotopic smooth maps are smoothly homotopic

Statement

Assume countable choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). If two smooth maps $f_{0}, f_{1} : M \to N$ are continuously homotopic, then they are smoothly homotopic.

Facts & Assumptions

Given: Countable choice, smooth maps $f_{0}, f_{1} : M \to N$ , and a continuous homotopy $H : M \times I \to N$ from $f_{0}$ to $f_{1}$ .

[F1]

Homotopies are maps on products with $I = [0, 1]$ (Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints).

[L1]

Products of smooth manifolds carry canonical smooth structures (Products of smooth manifolds have a canonical product smooth structure).

[L2]

Under countable choice, relative manifold-valued approximation can smooth a continuous map while fixing it on closed regions where it is already smooth (Relative Whitney approximation for manifold-valued maps).

Proof

technique · direct

1.1F1L1givenconstruct

Choose a smooth function $λ : R \to [0, 1]$ with $λ (t) = 0$ for $t \leq 1 / 3$ and $λ (t) = 1$ for $t \geq 2 / 3$ . Define $\hat{H} (x, t) : = H (x, λ (t))$ on $M \times R$ . By [F1] and [L1], this is a continuous map on a smooth manifold; it is constant in $t$ on the closed collar regions $A_{0} : = M \times (- \infty, 1 / 3], A_{1} : = M \times [2 / 3, \infty),$ so it is smooth on a neighbourhood of $A_{0} \cup A_{1}$ .

2.1L2step 1.1choose

Apply [L2] to the closed set $A_{0} \cup A_{1} \subseteq M \times R$ . We obtain a smooth map $\tilde{H} : M \times R \to N$ that agrees with $\hat{H}$ on a neighbourhood of those collars.

3.1F1step 2.1algebra∎

Restrict $\tilde{H}$ to $M \times I$ . Near $t = 0$ it equals $f_{0}$ , and near $t = 1$ it equals $f_{1}$ ; after composing with a smooth reparameterization of $I$ that fixes the endpoints, this restriction becomes a smooth homotopy from $f_{0}$ to $f_{1}$ .

Depends on

Used by

Dependency tree · two levels

29 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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