TheoremStatement: AI-adaptedProof: AI-generatedverified 2026-09-23 (gpt-6-sol)

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The transversality homotopy theorem

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )). 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: ${A C}_{ω}$ , 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]

Under ${A C}_{ω}$ , parametric transversality makes the bad-slice parameter set null (Parametric transversality). In a positive-dimensional ball its complement is dense; in a zero-dimensional ball a null subset is empty.

[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

1.1L1givenchoose

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 $Z$ .

2.1L2step 1.1

By [L2], the bad-slice parameter set is null in $B$ . If $m > 0$ , its complement is dense and nonempty; if $m = 0$ , the null set is empty and $a = 0$ is good. Choose a good parameter $a$ .

3.1F1step 1.1step 2.1algebra∎

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 $Z$ .

Depends on

Used by

Negative expected dimension forces empty generic intersections Corollary
The oriented intersection number reduces to the mod 2 number Corollary
The mod 2 intersection number Definition
The oriented intersection number Definition
Bordant cycles have equal intersection numbers Lemma
General-position isotopies of links Lemma
Intersection number under factor interchange Theorem
The geometric intersection number is the Poincare-dual cup pairing Theorem
The mod 2 intersection number is homotopy invariant Theorem
The oriented intersection number is homotopy invariant Theorem
The universal Pontryagin-Thom correspondence for unoriented and oriented bordism Theorem

Dependency tree · two levels

22 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, Part 10, Corollary 3.28 (standard reference, not scraped)