Euler class
E627991
The Euler class is a topological characteristic class associated with oriented real vector bundles, capturing obstruction information such as the existence of nowhere-vanishing sections.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Euler class canonical | 6 |
| Euler class in de Rham cohomology | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6908860 — 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: Euler class Context triple: [Chern–Weil theory, constructs, Euler class]
-
A.
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.
-
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.
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.
-
D.
Chern–Weil theory
Chern–Weil theory is a framework in differential geometry that constructs characteristic classes of vector bundles from curvature forms, linking topology and geometry through invariant polynomials.
-
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: Euler class Target entity description: The Euler class is a topological characteristic class associated with oriented real vector bundles, capturing obstruction information such as the existence of nowhere-vanishing sections.
-
A.
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.
-
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.
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.
-
D.
Chern–Weil theory
Chern–Weil theory is a framework in differential geometry that constructs characteristic classes of vector bundles from curvature forms, linking topology and geometry through invariant polynomials.
-
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 (44)
| Predicate | Object |
|---|---|
| instanceOf |
characteristic class
ⓘ
cohomology class ⓘ topological invariant ⓘ |
| appearsIn |
Gysin sequence
ⓘ
Leray–Hirsch theorem for sphere bundles ⓘ |
| associatedWith |
orientation
ⓘ
vector bundle ⓘ |
| canBeRepresentedBy |
Pfaffian of curvature form
ⓘ
closed differential form ⓘ |
| captures |
obstruction to existence of nonzero section
ⓘ
obstruction to existence of nowhere-vanishing section ⓘ |
| constructedVia |
Thom class
ⓘ
zero section of the bundle ⓘ |
| definedAs | pullback of Thom class along zero section ⓘ |
| definedFor |
oriented rank n real vector bundle
ⓘ
oriented sphere bundles via associated vector bundle ⓘ |
| definedOn | oriented real vector bundle ⓘ |
| degree | rank of the vector bundle ⓘ |
| dependsOn | isomorphism class of the bundle ⓘ |
| functorialUnder | pullback of bundles ⓘ |
| generalizes | Euler characteristic of a closed oriented manifold ⓘ |
| hasVariant |
Euler class in de Rham cohomology
ⓘ
linked to:
Euler class
|
| independentOf | choice of connection ⓘ |
| is |
primary obstruction to a nonvanishing section of an oriented bundle
ⓘ
top Stiefel–Whitney class modulo 2 ⓘ |
| isMultiplicativeFor | direct sum of oriented bundles of odd rank under suitable conditions ⓘ |
| isNonzeroIf | no global nowhere-vanishing section exists ⓘ |
| isZeroIf | bundle admits a nowhere-vanishing section ⓘ |
| livesIn |
even-degree cohomology
ⓘ
top-degree cohomology of the base space ⓘ |
| mod2ReductionIs | top Stiefel–Whitney class ⓘ |
| namedAfter | Leonhard Euler ⓘ |
| naturalWithRespectTo | bundle pullback ⓘ |
| relatedTo |
Euler characteristic via integration over the fundamental class
ⓘ
Gauss–Bonnet theorem ⓘ
linked to:
Gauss–Bonnet theorem (early form)
Poincaré–Hopf theorem ⓘ |
| requires | orientation of the vector bundle ⓘ |
| takesValuesIn |
H^{n}(B;\mathbb{Z})
ⓘ
integral cohomology ⓘ |
| usedIn |
classification of vector bundles
ⓘ
differential topology ⓘ index theory ⓘ obstruction theory ⓘ |
| vanishesFor | trivial oriented real vector bundle ⓘ |
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: Euler class Description of subject: The Euler class is a topological characteristic class associated with oriented real vector bundles, capturing obstruction information such as the existence of nowhere-vanishing sections.
Referenced by (7)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Euler class