Kronecker pairing
E620672
The Kronecker pairing is a fundamental bilinear map in algebraic topology that evaluates cohomology classes on homology classes, linking the two via an integer (or field) value.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Kronecker pairing canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6801432 — 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: Kronecker pairing Context triple: [Poincaré duality, involves, Kronecker pairing]
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A.
Kronecker product
The Kronecker product is a matrix operation that forms a large block matrix from two smaller matrices and is widely used in linear algebra, quantum computing, and signal processing.
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B.
Weil pairing
The Weil pairing is a bilinear, alternating, non-degenerate pairing on the torsion points of an elliptic curve, fundamental in number theory and modern cryptography.
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C.
Cartan–Killing form
The Cartan–Killing form is a canonical symmetric bilinear form on a Lie algebra that plays a central role in classifying and studying the structure of Lie algebras and Lie groups.
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D.
Kronecker delta
The Kronecker delta is a function of two variables that equals 1 when the variables are equal and 0 otherwise, widely used in linear algebra, tensor calculus, and discrete mathematics to represent identity relations.
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E.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
- 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: Kronecker pairing Target entity description: The Kronecker pairing is a fundamental bilinear map in algebraic topology that evaluates cohomology classes on homology classes, linking the two via an integer (or field) value.
-
A.
Kronecker product
The Kronecker product is a matrix operation that forms a large block matrix from two smaller matrices and is widely used in linear algebra, quantum computing, and signal processing.
-
B.
Weil pairing
The Weil pairing is a bilinear, alternating, non-degenerate pairing on the torsion points of an elliptic curve, fundamental in number theory and modern cryptography.
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C.
Cartan–Killing form
The Cartan–Killing form is a canonical symmetric bilinear form on a Lie algebra that plays a central role in classifying and studying the structure of Lie algebras and Lie groups.
-
D.
Kronecker delta
The Kronecker delta is a function of two variables that equals 1 when the variables are equal and 0 otherwise, widely used in linear algebra, tensor calculus, and discrete mathematics to represent identity relations.
-
E.
Kronecker’s lemma
Kronecker’s lemma is a result in real analysis and summability theory that links the convergence of series with weighted averages of their partial sums, often used in the study of Fourier series and ergodic theorems.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
bilinear map
ⓘ
construction in algebraic topology ⓘ evaluation pairing ⓘ |
| alsoKnownAs | Kronecker evaluation pairing NERFINISHED ⓘ |
| appearsIn |
algebraic topology
ⓘ
differential topology ⓘ homological algebra ⓘ |
| canBeDefinedFor |
reduced homology and cohomology
ⓘ
relative homology and cohomology ⓘ |
| codomain |
coefficient field
ⓘ
coefficient ring ⓘ integers ⓘ |
| definedOn |
H^n(X;R)
ⓘ
H_n(X;R) ⓘ |
| domain |
singular cohomology
ⓘ
singular homology ⓘ |
| generalizes | evaluation of linear functionals on vectors ⓘ |
| hasProperty |
R-bilinearity for real coefficients
ⓘ
Z-bilinearity for integer coefficients ⓘ |
| isBilinearIn |
cohomology class
ⓘ
homology class ⓘ |
| isCentralTo | duality theories in topology ⓘ |
| isCompatibleWith |
change of coefficients
ⓘ
long exact sequences in homology and cohomology ⓘ universal coefficient theorem decomposition ⓘ |
| isDefinedAtChainLevelBy | evaluation of cochains on chains ⓘ |
| isDefinedFor | any topological space X with defined singular (co)homology ⓘ |
| isFunctorialIn | topological space X ⓘ |
| isNaturalWithRespectTo | continuous maps of spaces ⓘ |
| isNondegenerate |
for closed oriented manifolds in top degree
ⓘ
for finite type homology over a field ⓘ |
| isRelatedTo |
cap product
ⓘ
cup product via duality ⓘ |
| isStandardNotation | ⟨·,·⟩ ⓘ |
| isUsedTo |
identify fundamental class with orientation via cohomology
ⓘ
pair de Rham cohomology with singular homology via integration ⓘ show that H^n(X;R) ≅ Hom(H_n(X;R),R) under finiteness conditions ⓘ |
| maps | H^n(X;R) × H_n(X;R) → R ⓘ |
| namedAfter | Leopold Kronecker NERFINISHED ⓘ |
| relates |
cohomology
ⓘ
homology ⓘ |
| requires | choice of coefficient ring ⓘ |
| usedFor |
defining Poincaré duality
ⓘ
defining cap product ⓘ defining universal coefficient theorem isomorphisms ⓘ evaluating cohomology classes on homology classes ⓘ identifying homology with dual of cohomology in finite type cases ⓘ |
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: Kronecker pairing Description of subject: The Kronecker pairing is a fundamental bilinear map in algebraic topology that evaluates cohomology classes on homology classes, linking the two via an integer (or field) value.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.