False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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FALSE: the tubular-neighbourhood retraction is canonical

Statement

False claim: the tubular-neighbourhood retraction of an embedded submanifold is canonical.

Facts & Assumptions

Given: The annulus

A : = {(r \cos θ, r \sin θ) : 1 / 2 < r < 3 / 2}

around the unit circle $S^{1} \subseteq R^{2}$ .

[L1]

Two tubular neighbourhoods are unique only up to shrinking and germ isomorphism near the zero section (Two tubular neighbourhood germs are isomorphic near the zero section).

Refutation

technique · direct

1.1

The radial map $r_{0} (r \cos θ, r \sin θ) : = (\cos θ, \sin θ)$ is a smooth retraction $A \to S^{1}$ .

givenconstruct

2.1

Choose a smooth function $α : (1 / 2, 3 / 2) \to R$ with $α (1) = 0$ and $α (r) \neq 0$ for some $r \neq 1$ . The tubular chart $Φ_{1} (e^{i θ}, t) : = (1 + t) e^{i (θ - α (1 + t))}$ is a diffeomorphism from $S^{1} \times (- 1 / 2, 1 / 2)$ onto $A$ and agrees with the inclusion at $t = 0$ . Its induced tubular retraction is $r_{1} (r e^{i θ}) = e^{i (θ + α (r))} .$ Thus $r_{1}$ is genuinely a tubular-neighbourhood retraction, and $r_{1} \neq r_{0}$ away from the circle.

step 1.1construct

3.1

Therefore the retraction depends on the chosen tubular chart and is not canonical. The correct uniqueness statement is the weaker germ statement recorded in [L1].

L1step 2.1∎

Depends on

Two tubular neighbourhood germs are isomorphic near the zero section

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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., Tubular Neighborhoods (standard reference, not scraped)