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

1.1givenconstruct

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

2.1step 1.1construct

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.

3.1L1step 2.1∎

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].

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)