False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-09-01

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Uniform $C^{1}$ openness of transversality fails on arbitrary noncompact sources

Statement

False claim: if $M$ is noncompact and $F : M \to N$ is transverse to an embedded submanifold $Z \subseteq N$ , then every map uniformly $C^{1}$ -close to $F$ is still transverse to $Z$ .

Facts & Assumptions

Given: The map $F : R \to R$ , $F (x) = e^{- x}$ , the point submanifold $Z = {0}$ , a smooth bump $β$ with support in $[- 1, 1]$ , $β (0) = 1$ , and $β^{'} (0) = 0$ .

[L1]

Compact-source $C^{1}$ openness is the honest theorem (Transversality is stable on a compact source).

Refutation

technique · direct

1.1

The map $F$ never meets $0$ , so it is vacuously transverse to $Z$ . For each integer $n \geq 1$ , define $G_{n} (x) : = e^{- x} - e^{- n} β (x - n) + e^{- n} (x - n) β (x - n) .$ Then $G_{n} (n) = 0$ and $G_{n}^{'} (n) = - e^{- n} - e^{- n} β^{'} (0) + e^{- n} β (0) = 0$ .

givenalgebra

2.1

The differences $G_{n} - F$ and $G_{n}^{'} - F^{'}$ are supported in $[n - 1, n + 1]$ , and their sup norms are bounded by constants times $e^{- n}$ . Hence $G_{n} \to F$ in the uniform $C^{1}$ topology.

givenstep 1.1algebra

3.1

But step 1.1 gives a critical zero of $G_{n}$ at $x = n$ , so $G_{n}$ is not transverse to ${0}$ by the regular-value criterion. Therefore no uniform $C^{1}$ neighbourhood of $F$ consists entirely of transverse maps. This agrees with [L1], which required compact source.

L1step 1.1step 2.1∎

Depends on

Transversality is stable on a compact source

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

Marco Gualtieri, Topology I: Smooth Manifolds, cumulative notes (standard reference, not scraped)

Uniform C1C1 openness of transversality fails on arbitrary noncompact sources