Thom transversality theorem
E627197
The Thom transversality theorem is a fundamental result in differential topology that guarantees generic smooth maps are transverse to given submanifolds, underpinning the study of stable phenomena and cobordism.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Thom transversality theorem canonical | 2 |
| Thom’s transversality theorem | 1 |
| jet transversality theorem | 1 |
| multi-jet transversality theorem | 1 |
| parametric transversality theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6901169 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Thom transversality theorem Context triple: [René Thom, knownFor, Thom transversality theorem]
-
A.
Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
-
B.
Thom–Mather stratification
Thom–Mather stratification is a refined notion of stratification in differential topology that imposes strong regularity and control conditions on how smooth strata fit together, generalizing and strengthening Whitney stratifications.
-
C.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
-
D.
Whitney approximation theorem
The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
-
E.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Thom transversality theorem Target entity description: The Thom transversality theorem is a fundamental result in differential topology that guarantees generic smooth maps are transverse to given submanifolds, underpinning the study of stable phenomena and cobordism.
-
A.
Whitney embedding theorem
The Whitney embedding theorem is a fundamental result in differential topology stating that any smooth n-dimensional manifold can be embedded as a submanifold of Euclidean space of sufficiently high dimension (specifically \(\mathbb{R}^{2n}\)).
-
B.
Thom–Mather stratification
Thom–Mather stratification is a refined notion of stratification in differential topology that imposes strong regularity and control conditions on how smooth strata fit together, generalizing and strengthening Whitney stratifications.
-
C.
Poincaré–Hopf theorem
The Poincaré–Hopf theorem is a fundamental result in differential topology that relates the sum of the indices of a vector field’s isolated zeros on a compact manifold to the manifold’s Euler characteristic.
-
D.
Whitney approximation theorem
The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
-
E.
Nash embedding theorem
The Nash embedding theorem is a fundamental result in differential geometry that shows any Riemannian manifold can be isometrically embedded into some Euclidean space, thereby realizing abstract curved spaces as concrete subsets of standard Euclidean space.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| appearsIn |
René Thom's work on cobordism
ⓘ
standard textbooks on differential topology ⓘ |
| appliesTo |
finite-dimensional smooth manifolds
ⓘ
smooth submanifolds of target manifolds ⓘ |
| assumes | smoothness of manifolds and maps ⓘ |
| concerns |
smooth maps between smooth manifolds
ⓘ
transversality to stratified sets ⓘ transversality to submanifolds ⓘ |
| describes | genericity of transversality for smooth maps ⓘ |
| field |
cobordism theory
ⓘ
differential topology ⓘ singularity theory ⓘ |
| formalism | C^∞-manifolds and smooth maps ⓘ |
| generalizes | basic transversality results for submanifolds in Euclidean space ⓘ |
| guarantees |
existence of smooth maps transverse to a given submanifold
ⓘ
transversality is a generic property in the space of smooth maps ⓘ |
| hasVariant |
jet transversality theorem
ⓘ
linked to:
Thom transversality theorem
multi-jet transversality theorem ⓘ
linked to:
Thom transversality theorem
parametric transversality theorem ⓘ
linked to:
Thom transversality theorem
|
| historicalPeriod | mid 20th century mathematics ⓘ |
| implies |
generic smooth maps are immersions or submersions away from a stratified singular set
ⓘ
generic smooth maps are transverse to given submanifolds ⓘ generic smooth maps have only stable singularities ⓘ |
| influencedBy | work of Hassler Whitney ⓘ |
| influences |
geometric topology
ⓘ
global analysis ⓘ modern differential topology ⓘ |
| namedAfter | René Thom ⓘ |
| relatesTo |
Morse theory
ⓘ
linked to:
Morse Theory
Sard's theorem ⓘ Sard–Smale theorem ⓘ
linked to:
Sard's theorem
Whitney embedding theorem ⓘ |
| states |
for a submanifold N of a manifold Y, the set of smooth maps f : X → Y transverse to N is dense in C^∞(X,Y)
ⓘ
for a submanifold N of a manifold Y, the set of smooth maps f : X → Y transverse to N is residual in C^∞(X,Y) ⓘ |
| toolFor |
classification of singularities of smooth maps
ⓘ
intersection theory on manifolds ⓘ proofs of genericity of Morse functions ⓘ |
| underpins |
cobordism theory
ⓘ
construction of generic position arguments ⓘ the study of stable phenomena in differential topology ⓘ |
| usedFor |
constructing generic stratifications
ⓘ
defining fundamental classes in cobordism ⓘ defining intersection numbers via transverse intersections ⓘ perturbing smooth maps to achieve transversality ⓘ proving stability of singularities ⓘ |
| uses | Whitney C^∞ topology on spaces of smooth maps ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Thom transversality theorem Description of subject: The Thom transversality theorem is a fundamental result in differential topology that guarantees generic smooth maps are transverse to given submanifolds, underpinning the study of stable phenomena and cobordism.
Referenced by (6)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
M. Hirsch, Differential Topology
linked to: Thom transversality theorem
linked to: Thom transversality theorem
linked to: Thom transversality theorem
linked to: Thom transversality theorem