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.

Smooth families of maps and their evaluation maps

Definition

Let $M$ , $S$ , and $N$ be smooth manifolds. A smooth family of maps from $M$ to $N$ parametrized by $S$ is a smooth map

$F : M \times S \to N .$

For each $s \in S$ , its slice map is

$F_{s} : M \to N, F_{s} (p) : = F (p, s) .$

The map $F$ itself is the evaluation map of the family.

Depends on

Used by

Geometric cardinality is not homotopy invariant Counterexample
Noncompact intersections can escape during a homotopy Counterexample
Refuted: a homotopy that is immersive at every earlier time is a regular homotopy Counterexample
A regular continuation datum between Morse--Smale pairs Definition
A regular two-parameter continuation datum Definition
Compact parameter pairs and relative families Definition
Regular homotopy of immersions Definition
Smooth isotopies, diffeotopies and ambient isotopies Definition
The mod 2 intersection number Definition
The oriented intersection number Definition
A tubular target produces a submersive finite-dimensional perturbation family Lemma
Algebraic open inclusions surject on fundamental groups Lemma
Joint jet continuity characterises the weak smooth topology Lemma
Metastable approximation of maps by embeddings Lemma
Oriented boundary of an intersection trace has opposite end signs Lemma
Regular homotopy preserves the formal Gauss class Lemma
Smooth families and path components in the weak topology Lemma
The basepoint evaluation of the Stiefel section space is a fibration Lemma
Regular homotopy allows self-intersections but never rank drop Remark
Parametric transversality Theorem
The mod 2 intersection number is homotopy invariant Theorem
The oriented intersection number is homotopy invariant Theorem

Dependency tree · two levels

15 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)