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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.