DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 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.

Transverse smooth maps

Definition

Let $F : M \to N$ and $G : P \to N$ be smooth maps. They are transverse, written $F ⋔ G$ , when for every pair $(p, q) \in M \times P$ with $F (p) = G (q) = y$ ,

$d F_{p} (T_{p} M) + d G_{q} (T_{q} P) = T_{y} N .$

Depends on

Used by

Negative expected dimension forces empty generic intersections Corollary
Framed regular preimages of a map to a sphere Definition
Morse--Smale pairs Definition
Regular oriented link diagrams Definition
Smooth maps transverse to a regular foliation Definition
The mod 2 intersection number Definition
Transverse complementary-dimensional intersection sets Definition
Transverse embedded submanifolds Definition
A fibre product of submersions Example
The double point locus has the expected dimension 2m-n Lemma
Unstable orientations induce orientations of the trajectory moduli spaces Lemma
Transversality is invariant under diffeomorphic change of source and target Proposition
Intersection number under factor interchange Theorem
Transverse fibre products are embedded submanifolds Theorem

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)