Poincaré duality
E156194
Poincaré duality is a fundamental theorem in algebraic topology that relates the homology and cohomology groups of an oriented closed manifold in complementary dimensions.
All labels observed (4)
| Label | Occurrences |
|---|---|
| Poincaré duality canonical | 4 |
| Poincaré–Lefschetz duality | 2 |
| Poincaré duality spaces | 1 |
| Poincaré duality with local coefficients | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1358653 — 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: Poincaré duality Context triple: [Henri Poincaré, notableWork, Poincaré duality]
-
A.
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.
-
B.
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.
-
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.
Lefschetz operator
The Lefschetz operator is a linear operator in Kähler geometry that acts on differential forms by wedging with the Kähler form, playing a central role in the Hard Lefschetz theorem and Hodge theory.
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E.
Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
- 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: Poincaré duality Target entity description: Poincaré duality is a fundamental theorem in algebraic topology that relates the homology and cohomology groups of an oriented closed manifold in complementary dimensions.
-
A.
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.
-
B.
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.
-
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.
Lefschetz operator
The Lefschetz operator is a linear operator in Kähler geometry that acts on differential forms by wedging with the Kähler form, playing a central role in the Hard Lefschetz theorem and Hodge theory.
-
E.
Gauss–Bonnet theorem (early form)
The Gauss–Bonnet theorem (early form) is an early version of the fundamental result in differential geometry that links the total curvature of a surface to its topological characteristics, originally developed by Carl Friedrich Gauss.
- F. None of above. chosen
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
duality principle
ⓘ
theorem in algebraic topology ⓘ |
| appliesTo |
compact oriented manifolds without boundary
ⓘ
oriented closed manifolds ⓘ |
| characterizes |
Poincaré duality
self-linksurface differs
ⓘ
surface form:
Poincaré duality spaces
|
| cohomologyVersion | isomorphism H^{k}(M;R) ≅ H_{n-k}(M;R) ⓘ |
| context |
de Rham cohomology for smooth manifolds
ⓘ
simplicial homology for triangulated manifolds ⓘ singular homology ⓘ |
| dimensionStatement | for an n-dimensional manifold M, H_k(M) is dual to H^{n-k}(M) ⓘ |
| failsWithout | orientability ⓘ |
| field | algebraic topology ⓘ |
| generalizedBy |
Poincaré duality
self-linksurface differs
ⓘ
surface form:
Poincaré–Lefschetz duality
Verdier duality ⓘ |
| hasVariant |
Poincaré duality for manifolds with boundary
ⓘ
Poincaré duality self-linksurface differs ⓘ
surface form:
Poincaré duality with local coefficients
|
| historicalPeriod | early 20th century mathematics ⓘ |
| holdsWithCoefficients |
field coefficients
ⓘ
orientable local coefficient systems ⓘ |
| implies |
isomorphism between H_k(M;R) and H^{n-k}(M;R) for suitable coefficients R
ⓘ
nondegenerate pairing between homology and cohomology ⓘ symmetry of Betti numbers b_k = b_{n-k} for closed oriented manifolds ⓘ |
| involves |
Kronecker pairing
ⓘ
cup product structure on cohomology ⓘ |
| namedAfter | Henri Poincaré ⓘ |
| relatedConcept |
Alexander duality
ⓘ
Hodge theory ⓘ Lefschetz duality ⓘ intersection pairing ⓘ |
| relates |
cohomology groups
ⓘ
homology and cohomology in complementary degrees ⓘ homology groups ⓘ |
| requires |
finite-dimensional homology groups for compact manifolds
ⓘ
manifold to be closed ⓘ orientation on the manifold ⓘ |
| usedIn |
Morse Theory
ⓘ
surface form:
Morse theory
classification of manifolds ⓘ intersection homology ⓘ study of manifold invariants ⓘ surgery theory ⓘ topological quantum field theory ⓘ |
| usedToShow |
0th cohomology of a connected closed manifold is isomorphic to the coefficient ring
ⓘ
top-dimensional homology of a closed oriented n-manifold is isomorphic to the coefficient ring ⓘ |
| uses |
cap product
ⓘ
fundamental class of a manifold ⓘ |
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: Poincaré duality Description of subject: Poincaré duality is a fundamental theorem in algebraic topology that relates the homology and cohomology groups of an oriented closed manifold in complementary dimensions.
Referenced by (8)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Poincaré–Lefschetz duality
this entity surface form:
Poincaré–Lefschetz duality
this entity surface form:
Poincaré duality spaces
this entity surface form:
Poincaré duality with local coefficients
subject surface form:
Algebraic Topology