Whitney sum
E285917
The Whitney sum is a construction in differential topology that combines two vector bundles over the same base space into a new vector bundle whose fibers are direct sums of the original fibers.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Whitney sum formula | 2 |
| Whitney sum canonical | 1 |
| Whitney sum of vector bundles | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2652909 — 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: Whitney sum Context triple: [Hassler Whitney, hasConceptNamedAfter, Whitney sum]
-
A.
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.
-
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.
Minkowski sum
The Minkowski sum is a fundamental operation in geometry and convex analysis that combines two sets by adding every vector in one set to every vector in the other, widely used in areas such as optimization, robotics, and computational geometry.
-
D.
Whitney stratification
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.
-
E.
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.
- 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: Whitney sum Target entity description: The Whitney sum is a construction in differential topology that combines two vector bundles over the same base space into a new vector bundle whose fibers are direct sums of the original fibers.
-
A.
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.
-
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.
Minkowski sum
The Minkowski sum is a fundamental operation in geometry and convex analysis that combines two sets by adding every vector in one set to every vector in the other, widely used in areas such as optimization, robotics, and computational geometry.
-
D.
Whitney stratification
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.
-
E.
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.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
operation on vector bundles
ⓘ
vector bundle construction ⓘ |
| alsoKnownAs |
Whitney sum of vector bundles
ⓘ
linked to:
Whitney sum
direct sum of vector bundles ⓘ |
| appearsIn |
construction of stable normal bundles
ⓘ
splitting of exact sequences of vector bundles ⓘ theory of tangent and normal bundles ⓘ |
| appliesTo |
complex vector bundles
ⓘ
real vector bundles ⓘ topological vector bundles ⓘ |
| baseSpacePreserved | true ⓘ |
| compatibility |
compatible with pullback along continuous or smooth maps
ⓘ
compatible with restriction of bundles to subspaces ⓘ |
| definedOn | vector bundles over the same base space ⓘ |
| definesOperationOn | isomorphism classes of vector bundles over a fixed base ⓘ |
| fiberwiseDescription | (E ⊕ F)_x = E_x ⊕ F_x for each x in B ⓘ |
| field |
algebraic topology
ⓘ
differential topology ⓘ geometry ⓘ |
| generalizes | direct sum of vector spaces ⓘ |
| givesMonoidStructureTo | set of isomorphism classes of vector bundles over a base space ⓘ |
| hasIdentityElement | zero vector bundle ⓘ |
| hasInput |
vector bundle E → B
ⓘ
vector bundle F → B ⓘ |
| hasLocalDescription | given local trivializations, transition functions are block-diagonal sums ⓘ |
| hasOutput | vector bundle E ⊕ F → B ⓘ |
| isAssociativeUpToIsomorphism | true ⓘ |
| isCommutativeUpToIsomorphism | true ⓘ |
| isFunctorial | true ⓘ |
| namedAfter | Hassler Whitney ⓘ |
| preservesComplexStructure | true ⓘ |
| preservesSmoothStructure | true ⓘ |
| relatedConcept |
external direct sum of bundles
ⓘ
pullback of vector bundles ⓘ tensor product of vector bundles ⓘ |
| requiresCondition |
bundles are of the same category (e.g. smooth, topological, complex)
ⓘ
bundles share the same base space ⓘ |
| satisfiesProperty |
c(E ⊕ F) = c(E) ∪ c(F) for total Chern classes
ⓘ
p(E ⊕ F) = p(E) ∪ p(F) for Pontryagin classes ⓘ rank(E ⊕ F) = rank(E) + rank(F) ⓘ w(E ⊕ F) = w(E) ∪ w(F) for Stiefel–Whitney classes ⓘ |
| symbol |
⊕
ⓘ
⨁ ⓘ |
| usedIn |
classification of vector bundles
ⓘ
construction of characteristic classes ⓘ construction of topological K-groups ⓘ definition of K-theory ⓘ stable equivalence of vector bundles ⓘ |
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: Whitney sum Description of subject: The Whitney sum is a construction in differential topology that combines two vector bundles over the same base space into a new vector bundle whose fibers are direct sums of the original fibers.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Whitney sum
linked to: Whitney sum
linked to: Whitney sum