DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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 embedded submanifolds

Definition

Let $S, T \subseteq M$ be embedded submanifolds. They are transverse, written $S ⋔ T$ , when their inclusion maps into $M$ are transverse. Equivalently, for every $p \in S \cap T$ ,

$T_{p} S + T_{p} T = T_{p} M .$

Depends on

Used by

A cycle has zero algebraic intersection with a bounding cycle Corollary
Negative expected dimension forces empty generic intersections Corollary
Transverse embedded submanifolds intersect in the expected codimension Corollary
Geometrically cancelling adjacent handle pair Definition
Neat submanifolds of a manifold with boundary Definition
The geometric intersection pairing on a closed oriented manifold Definition
The local oriented intersection sign Definition
The mod 2 intersection number Definition
Transverse complementary-dimensional intersection sets Definition
Whitney circle for a pair of intersection points Definition
Transverse and tangent intersections of plane curves Example
Intersecting submanifolds need not be transverse False statement
Algebraic cancellation does not yet give geometric cancellation Lemma
Graph-diagonal transversality is exactly fixed-point nondegeneracy Lemma
Normal push-off zeros are the self-intersection points Lemma
Transverse submanifolds have product charts Lemma
An m-dimensional submanifold transverse to vertical fibres is locally a graph Proposition
The index of a zero is its zero-section intersection number Proposition
The geometric intersection number is the Poincare-dual cup pairing Theorem

Dependency tree · two levels

6 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)