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 |
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 ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.