Stiefel–Whitney classes
E853116
Stiefel–Whitney classes are characteristic classes in algebraic topology that assign cohomology invariants to real vector bundles, capturing their topological and orientability properties.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Stiefel–Whitney classes canonical | 3 |
How this entity was disambiguated
This entity first appeared as the object of triple T10269808 — 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: Stiefel–Whitney classes Context triple: [Chern classes, relatedTo, Stiefel–Whitney classes]
-
A.
Pontryagin classes
Pontryagin classes are characteristic classes associated with real vector bundles that capture topological information about the bundle’s curvature and play a central role in differential topology and geometry.
-
B.
Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
-
C.
Characteristic Classes
Characteristic Classes is a foundational mathematical text in differential topology and geometry that systematically develops the theory of characteristic classes for vector bundles and fiber bundles.
-
D.
Hirzebruch signature theorem
The Hirzebruch signature theorem is a fundamental result in differential topology that expresses the signature of a smooth, compact, oriented 4k-dimensional manifold as a polynomial in its Pontryagin classes.
-
E.
Hirzebruch genera
Hirzebruch genera are topological invariants in algebraic topology and differential geometry that generalize characteristic classes to classify and study manifolds.
- 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: Stiefel–Whitney classes Target entity description: Stiefel–Whitney classes are characteristic classes in algebraic topology that assign cohomology invariants to real vector bundles, capturing their topological and orientability properties.
-
A.
Pontryagin classes
Pontryagin classes are characteristic classes associated with real vector bundles that capture topological information about the bundle’s curvature and play a central role in differential topology and geometry.
-
B.
Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
-
C.
Characteristic Classes
Characteristic Classes is a foundational mathematical text in differential topology and geometry that systematically develops the theory of characteristic classes for vector bundles and fiber bundles.
-
D.
Hirzebruch signature theorem
The Hirzebruch signature theorem is a fundamental result in differential topology that expresses the signature of a smooth, compact, oriented 4k-dimensional manifold as a polynomial in its Pontryagin classes.
-
E.
Hirzebruch genera
Hirzebruch genera are topological invariants in algebraic topology and differential geometry that generalize characteristic classes to classify and study manifolds.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
characteristic class
ⓘ
cohomology class ⓘ topological invariant ⓘ |
| additivity | w(E ⊕ F) = w(E) ∪ w(F) ⓘ |
| appliesTo | real vector bundle ⓘ |
| axiomatizedBy |
Whitney sum formula
NERFINISHED
ⓘ
naturality ⓘ normalization on trivial bundles ⓘ |
| coefficientRing | Z2 ⓘ |
| cohomologyOperationRelation | expressible via Steenrod squares ⓘ |
| component | w_i(E) ⓘ |
| definedFor |
each nonnegative integer i
ⓘ
tangent bundle of a smooth manifold ⓘ |
| definedOn | base space of a vector bundle ⓘ |
| degreeOf | w_i(E) has degree i ⓘ |
| field | algebraic topology ⓘ |
| firstClassInterpretation | w_1 is the obstruction to orientability GENERATED ⓘ |
| functoriality |
natural with respect to bundle maps
ⓘ
natural with respect to continuous maps of base spaces ⓘ |
| introducedBy |
Eduard Stiefel
NERFINISHED
ⓘ
Hassler Whitney NERFINISHED ⓘ |
| invariantOf |
smooth manifold up to homeomorphism
ⓘ
topological manifold ⓘ |
| lowestDegreeClass | w_0(E) ⓘ |
| multiplicativity | total Stiefel–Whitney class is multiplicative under direct sum ⓘ |
| normalizationProperty | all Stiefel–Whitney classes vanish for trivial bundles except w_0 = 1 ⓘ |
| orientabilityCriterion | w_1(E) = 0 if and only if E is orientable ⓘ |
| property | w_0(E) = 1 ⓘ |
| relatedTo |
Chern classes
NERFINISHED
ⓘ
Euler class NERFINISHED ⓘ Pontryagin classes NERFINISHED ⓘ |
| secondClassInterpretation | w_2 is the obstruction to a spin structure on an oriented bundle ⓘ |
| specialCaseOf | general characteristic classes ⓘ |
| spinCriterion | w_2(E) = 0 is necessary for a spin structure ⓘ |
| takesValuesIn |
H^*(X; Z2)
ⓘ
cohomology with Z2 coefficients ⓘ |
| topClass | w_n(E) for rank n bundle ⓘ |
| topClassProperty | w_n(E) is the mod 2 Euler class ⓘ |
| totalClassNotation | w(E) = 1 + w_1(E) + w_2(E) + ⋯ ⓘ |
| usedFor |
classification of real vector bundles up to isomorphism
ⓘ
cobordism theory ⓘ detecting nontriviality of vector bundles ⓘ immersion and embedding problems of manifolds ⓘ obstruction theory NERFINISHED ⓘ surgery theory ⓘ |
| usedIn |
Hopf invariant and related problems
ⓘ
study of vector fields on spheres ⓘ |
| vanishingCondition | w_i(E) = 0 for i greater than rank of E ⓘ |
| yearIntroducedApprox | 1930s ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Stiefel–Whitney classes Description of subject: Stiefel–Whitney classes are characteristic classes in algebraic topology that assign cohomology invariants to real vector bundles, capturing their topological and orientability properties.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.