TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The transversality homotopy theorem

Statement

Let $f : M \to N$ be smooth and let $Z \subseteq N$ be a closed embedded submanifold. Then $f$ is smoothly homotopic to a smooth map $g : M \to N$ with $g ⋔ Z$ .

Facts & Assumptions

Given: A smooth map $f : M \to N$ and a closed embedded submanifold $Z \subseteq N$ .

[L1]

A smooth map admits a finite-dimensional perturbation family whose evaluation map is a submersion, hence transverse to $Z$ (A tubular target produces a submersive finite-dimensional perturbation family).

[L2]

Parametric transversality says that a smooth family transverse to $Z$ has a dense set of transverse slices (Parametric transversality).

[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).

Proof

technique · direct

1.1

By [L1], choose an open ball $B \subseteq R^{m}$ and a smooth family $F : M \times B \to N$ with $F_{0} = f$ such that the evaluation map $F$ is transverse to $Z$ .

L1givenchoose

2.1

By [L2], the set of parameters $a \in B$ for which the slice $g : = F_{a}$ is transverse to $Z$ is dense in $B$ . Choose such a parameter $a$ .

L2step 1.1choose

3.1

Because $B$ is an open ball containing $0$ , the straight segment $t a$ stays in $B$ for all $t \in I$ . Therefore $H : M \times I \to N, H (x, t) : = F (x, t a),$ is a smooth homotopy from $f$ to $g$ in the sense of [F1]. Thus $f$ is smoothly homotopic to a smooth map transverse to $Z$ .

F1step 1.1step 2.1algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Marco Gualtieri, Topology I: Smooth Manifolds, Part 10, Corollary 3.28 (standard reference, not scraped)