de Rham cohomology
E551975
de Rham cohomology is a cohomology theory for smooth manifolds that uses differential forms to capture their global topological and geometric properties.
All labels observed (2)
| Label | Occurrences |
|---|---|
| de Rham cohomology canonical | 16 |
| de Rham theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T5837399 — 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: de Rham cohomology Context triple: [Hodge theory, relatedTo, de Rham cohomology]
-
A.
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.
-
B.
Poincaré duality
Poincaré duality is a fundamental theorem in algebraic topology that relates the homology and cohomology groups of an oriented closed manifold in complementary dimensions.
-
C.
Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
-
D.
Deligne cohomology
Deligne cohomology is a refined cohomology theory in algebraic geometry that combines singular cohomology and differential forms to capture both topological and arithmetic information about complex algebraic varieties.
-
E.
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.
- 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: de Rham cohomology Target entity description: de Rham cohomology is a cohomology theory for smooth manifolds that uses differential forms to capture their global topological and geometric properties.
-
A.
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.
-
B.
Poincaré duality
Poincaré duality is a fundamental theorem in algebraic topology that relates the homology and cohomology groups of an oriented closed manifold in complementary dimensions.
-
C.
Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
-
D.
Deligne cohomology
Deligne cohomology is a refined cohomology theory in algebraic geometry that combines singular cohomology and differential forms to capture both topological and arithmetic information about complex algebraic varieties.
-
E.
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.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
cohomology theory
ⓘ
functor ⓘ topological invariant ⓘ |
| appliesTo |
non-compact smooth manifolds
ⓘ
oriented smooth manifolds ⓘ |
| captures |
global geometric properties of manifolds
ⓘ
global topological properties of manifolds ⓘ |
| cochainComplex | (Ω^*(M), d) ⓘ |
| coefficientsCanBe | complex numbers ⓘ |
| coefficientsTypically | real numbers ⓘ |
| cohomologyGroupNotation | H^k_{dR}(M) ⓘ |
| constructedFrom |
complex of differential forms
ⓘ
exterior derivative ⓘ |
| definedAs | cohomology of the de Rham complex ⓘ |
| domain | smooth manifolds ⓘ |
| field |
algebraic topology
ⓘ
differential geometry ⓘ global analysis ⓘ |
| generalizes | notion of closed and exact differential forms ⓘ |
| H0Interpretation | space of locally constant functions on a connected manifold ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| HnInterpretation | top-degree cohomology related to orientation and volume forms ⓘ |
| invariantUnder | diffeomorphisms of manifolds ⓘ |
| isFunctorFrom | category of smooth manifolds with smooth maps ⓘ |
| isFunctorTo | category of graded real vector spaces ⓘ |
| isomorphicTo | singular cohomology with real coefficients for smooth manifolds ⓘ |
| isomorphismType | H^k_{dR}(M) ≅ H^k_{sing}(M;ℝ) ⓘ |
| kthGroupDefinition | kernel of d:Ω^k→Ω^{k+1} modulo image of d:Ω^{k-1}→Ω^k ⓘ |
| namedAfter | Georges de Rham ⓘ |
| relatedConcept |
Hodge theory
ⓘ
Mayer–Vietoris sequence ⓘ Poincaré duality ⓘ Poincaré lemma ⓘ elliptic complexes ⓘ sheaf cohomology ⓘ Čech–de Rham complex ⓘ |
| relatedTo | singular cohomology ⓘ |
| satisfies | de Rham theorem ⓘ |
| toolFor |
classification of smooth manifolds up to homotopy type
ⓘ
formulation of Stokes theorem in cohomological terms ⓘ study of integration of differential forms ⓘ |
| usedIn |
gauge theory
ⓘ
index theory ⓘ string theory ⓘ theory of characteristic classes ⓘ |
| uses | differential forms ⓘ |
| variant |
compactly supported de Rham cohomology
ⓘ
equivariant de Rham cohomology ⓘ relative de Rham cohomology ⓘ |
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: de Rham cohomology Description of subject: de Rham cohomology is a cohomology theory for smooth manifolds that uses differential forms to capture their global topological and geometric properties.
Referenced by (17)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: de Rham cohomology