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 (1)
| Label | Occurrences |
|---|---|
| Thom transversality theorem canonical | 2 |
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 \(\mathbb{R}^{2n}\)).
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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.
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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.
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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.
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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
NERFINISHED
ⓘ
multi-jet transversality theorem NERFINISHED ⓘ parametric transversality theorem NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| relatesTo |
Morse theory
NERFINISHED
ⓘ
Sard's theorem NERFINISHED ⓘ Sard–Smale theorem NERFINISHED ⓘ Whitney embedding theorem NERFINISHED ⓘ |
| 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
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Instruction
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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 (2)
Full triples — surface form annotated when it differs from this entity's canonical label.