Chern character
E391904
The Chern character is a fundamental homomorphism from K-theory to cohomology that translates vector bundles into characteristic classes, playing a central role in index theory and algebraic topology.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Chern character canonical | 3 |
| Chern character in cyclic cohomology | 1 |
| Connes–Chern character | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3821387 — 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: Chern character Context triple: [Atiyah–Singer index theorem, usesConcept, Chern character]
-
A.
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.
-
B.
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.
-
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–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
-
E.
Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
- 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: Chern character Target entity description: The Chern character is a fundamental homomorphism from K-theory to cohomology that translates vector bundles into characteristic classes, playing a central role in index theory and algebraic topology.
-
A.
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.
-
B.
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.
-
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–Riemann–Roch theorem
The Hirzebruch–Riemann–Roch theorem is a fundamental result in algebraic geometry and topology that expresses the holomorphic Euler characteristic of a complex manifold in terms of characteristic classes, unifying and extending classical Riemann–Roch type formulas.
-
E.
Atiyah–Singer index theorem
The Atiyah–Singer index theorem is a fundamental result in mathematics that links the analytical properties of elliptic differential operators to topological invariants of manifolds, unifying analysis, topology, and geometry.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
characteristic class
ⓘ
mathematical concept ⓘ natural transformation ⓘ ring homomorphism ⓘ |
| appearsIn |
Chern–Weil theory
ⓘ
algebraic index theorems ⓘ topological index formulas ⓘ |
| associatedWith |
Todd class
ⓘ
total Chern class ⓘ Â-genus ⓘ |
| codomain |
cohomology
ⓘ
rational cohomology ⓘ |
| definedFor |
algebraic vector bundles
ⓘ
complex vector bundles ⓘ perfect complexes in derived categories ⓘ |
| domain |
algebraic K-theory
ⓘ
topological K-theory ⓘ |
| expressesInTermsOf | Chern classes ⓘ |
| field |
K-theory
ⓘ
algebraic geometry ⓘ algebraic topology ⓘ differential geometry ⓘ |
| generalizedTo |
cyclic homology
ⓘ
higher K-theory ⓘ noncommutative geometry ⓘ |
| isHomomorphismOf |
graded rings
ⓘ
rings ⓘ |
| isNaturalIn |
topological space X
ⓘ
vector bundle ⓘ |
| mapsFrom | K^0(X) ⓘ |
| mapsTo | H^{even}(X;\mathbb{Q}) ⓘ |
| namedAfter | Shiing-Shen Chern ⓘ |
| property |
additive on direct sums of vector bundles
ⓘ
becomes an isomorphism after tensoring K-theory with \mathbb{Q} ⓘ compatible with pullbacks ⓘ determines K-theory with rational coefficients ⓘ multiplicative on tensor products of vector bundles ⓘ |
| relates |
K-theory
ⓘ
cohomology theory ⓘ |
| relatesTo |
Chern–Simons forms
ⓘ
Hodge theory ⓘ de Rham cohomology ⓘ |
| usedIn |
Atiyah–Singer index theorem
ⓘ
Grothendieck–Riemann–Roch theorem ⓘ Hirzebruch–Riemann–Roch theorem ⓘ Riemann–Roch theorem ⓘ
surface form:
Riemann–Roch type formulas
algebraic cycles and Chow groups ⓘ classification of vector bundles ⓘ index theory ⓘ |
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: Chern character Description of subject: The Chern character is a fundamental homomorphism from K-theory to cohomology that translates vector bundles into characteristic classes, playing a central role in index theory and algebraic topology.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Connes–Chern character
this entity surface form:
Chern character in cyclic cohomology