PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck 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.

Transversality is stable on a compact source

Statement

Let $M$ be compact, let $Z \subseteq N$ be a closed embedded submanifold, and let $F : M \to N$ be smooth with $F ⋔ Z$ . Then every smooth map $G : M \to N$ sufficiently close to $F$ in the $C^{1}$ topology is also transverse to $Z$ .

Facts & Assumptions

Given: A compact manifold $M$ , a closed embedded submanifold $Z \subseteq N$ , and a smooth map $F : M \to N$ with $F ⋔ Z$ .

[F1]

Transversality at a point is equivalent to surjectivity of the induced map to the normal quotient (Transversality is equivalent to surjectivity on the normal quotient).

[L1]

Embedded submanifolds admit local defining submersions, and the submersion locus is open (Embedded submanifolds admit local defining submersions, The immersion and submersion loci are open).

Proof

technique · direct

1.1

If $F (p) \notin Z$ , closedness of $Z$ gives a neighbourhood $V_{p}$ of $F (p)$ disjoint from $Z$ . Shrink to a relatively compact neighbourhood $U_{p}$ of $p$ with $F (\overline{U_{p}}) \subseteq V_{p}$ . Every map $G$ sufficiently $C^{0}$ -close to $F$ on $\overline{U_{p}}$ still maps $U_{p}$ into $V_{p}$ , so transversality there is vacuous.

givenchoose

1.2

If $F (p) \in Z$ , choose a neighbourhood $V_{p}$ of $F (p)$ and a defining submersion $h_{p} : V_{p} \to R^{c_{p}}$ for $Z$ by [L1]. Because $F ⋔ Z$ , the composite $h_{p} \circ F$ is a submersion at $p$ by [F1]. In source and target coordinates, some $c_{p} \times c_{p}$ minor of $D (h_{p} \circ F)$ is nonzero at $p$ . Shrink to a relatively compact $U_{p}$ so that $F (\overline{U_{p}}) \subseteq V_{p}$ and this minor stays nonzero on $\overline{U_{p}}$ , using the fixed-map openness in [L1]. If $G$ is sufficiently $C^{1}$ -close to $F$ on $\overline{U_{p}}$ , then $G (\overline{U_{p}}) \subseteq V_{p}$ and the corresponding minor of $D (h_{p} \circ G)$ remains nonzero. Thus $h_{p} \circ G$ is a submersion on $U_{p}$ , and $G ⋔ Z$ there.

F1L1givenchoosealgebra

2.1

The sets $U_{p}$ from steps 1.1 and 1.2 cover the compact manifold $M$ , so a finite subcover suffices. Intersect the corresponding finitely many $C^{1}$ neighbourhoods of $F$ . Any $G$ in that intersection is transverse to $Z$ on each $U_{p}$ , hence on all of $M$ .

step 1.1step 1.2givenchoose

3.1

Therefore transversality is stable on a compact source in the $C^{1}$ topology.

step 2.1∎

Depends on

Used by

Uniform C¹ openness of transversality fails on arbitrary noncompact sources False statement

Dependency tree · two levels

12 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)