Whitney stratification
E53942
Whitney stratification is a method in differential topology for decomposing singular spaces into smoothly compatible manifolds (strata) that fit together under specific regularity conditions, enabling rigorous analysis of singularities.
Aliases (3)
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical concept
→
stratification → tool in differential topology → |
| appliesTo |
algebraic varieties
→
analytic sets → singular spaces → subsets of smooth manifolds → |
| assumption |
each stratum is a connected smooth submanifold
→
frontier condition: boundary of a stratum is a union of lower-dimensional strata → |
| condition |
Whitney condition A
→
Whitney condition B → |
| dimensionProperty |
dimensions of strata strictly decrease along the frontier
→
|
| ensures |
compatibility of tangent spaces of strata
→
controlled behavior of limits of tangent spaces → good geometric fit between strata → |
| field |
algebraic geometry
→
differential topology → real analytic geometry → singularity theory → |
| hasPart |
strata
→
|
| historicalNote |
introduced by Hassler Whitney in the 1960s
→
|
| namedAfter |
Hassler Whitney
→
|
| property |
compatible with many geometric and analytic constructions
→
refinement of any given decomposition into smooth pieces can often be made Whitney-regular → |
| purpose |
to analyze singularities rigorously
→
to decompose singular spaces into smooth manifolds → |
| regularityCondition |
Whitney conditions
→
|
| relatedConcept |
Thom–Mather stratification
→
Whitney conditions → o-minimal structure → stratified space → subanalytic set → |
| strataAre |
locally finite
→
pairwise disjoint → smooth manifolds → |
| strataUnion |
the whole space
→
|
| usedFor |
constructing stratified spaces
→
defining stratified mappings → intersection homology → stratified Morse theory → stratified transversality → studying topological invariants of singular spaces → |
| usedIn |
equisingularity theory
→
geometric measure theory → microlocal analysis → resolution of singularities → |
Referenced by (7)
| Subject (surface form when different) | Predicate |
|---|---|
|
Whitney stratification
("Whitney condition A")
→
Whitney stratification ("Whitney condition B") → |
condition |
|
Hassler Whitney
→
|
hasConceptNamedAfter |
|
Hassler Whitney
→
|
notableFor |
|
Whitney stratification
("Whitney conditions")
→
|
regularityCondition |
|
Whitney stratification
("Whitney conditions")
→
|
relatedConcept |
|
Whitney embedding theorem
→
|
relatedTo |