CorollaryStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-23 (gpt-6-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Generic translations of a Euclidean-valued map are transverse

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). Let $f : M \to R^{n}$ be smooth and let $Z \subseteq R^{n}$ be an embedded submanifold. Then the set of $a \in R^{n}$ for which the translated map

$f_{a} (p) : = f (p) + a$

fails to be transverse to $Z$ is a null subset of $R^{n}$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , a smooth map $f : M \to R^{n}$ and an embedded submanifold $Z \subseteq R^{n}$ .

[L1]

Under ${A C}_{ω}$ , parametric transversality applies to a smooth family whose evaluation map is transverse to $Z$ (Parametric transversality).

[L2]

A submersion is transverse to every embedded submanifold (A submersion is transverse to every embedded submanifold).

Proof

technique · direct

1.1L1givenconstruct

Define the smooth family $F : M \times R^{n} \to R^{n}$ by $F (p, a) = f (p) + a$ . Its slice at $a$ is exactly $f_{a}$ .

2.1L2step 1.1algebra

The differential of $F$ in the parameter direction is the identity on $R^{n}$ , so $F$ is a submersion. Therefore [L2] gives $F ⋔ Z$ .

3.1L1step 2.1∎

Applying [L1] to this family shows that the nontransverse parameters form a null subset of $R^{n}$ .

Depends on

Used by

Outside a null set every translation makes a chosen value a transverse zero Corollary

Dependency tree · two levels

13 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)