Alexander duality
E620671
Alexander duality is a theorem in algebraic topology that relates the homology (or cohomology) of a subspace of a sphere to the reduced cohomology of its complement.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Alexander duality canonical | 2 |
| cohomological Alexander duality | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6801416 — 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: Alexander duality Context triple: [Poincaré duality, relatedConcept, Alexander duality]
-
A.
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.
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B.
Duality
Duality is a 1998 collaborative album by Lisa Gerrard and Pieter Bourke that blends ethereal vocals with atmospheric, world-influenced soundscapes.
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C.
Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
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D.
Hodge decomposition
Hodge decomposition is a fundamental result in differential geometry and Hodge theory that expresses differential forms on a Riemannian manifold uniquely as sums of exact, co-exact, and harmonic components.
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E.
Grothendieck duality
Grothendieck duality is a foundational theory in algebraic geometry that generalizes classical Serre duality to a broad categorical and sheaf-theoretic framework for studying duality on schemes and morphisms.
- 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: Alexander duality Target entity description: Alexander duality is a theorem in algebraic topology that relates the homology (or cohomology) of a subspace of a sphere to the reduced cohomology of its complement.
-
A.
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.
-
B.
Duality
Duality is a 1998 collaborative album by Lisa Gerrard and Pieter Bourke that blends ethereal vocals with atmospheric, world-influenced soundscapes.
-
C.
Alexandrov–Čech cohomology
Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.
-
D.
Hodge decomposition
Hodge decomposition is a fundamental result in differential geometry and Hodge theory that expresses differential forms on a Riemannian manifold uniquely as sums of exact, co-exact, and harmonic components.
-
E.
Grothendieck duality
Grothendieck duality is a foundational theory in algebraic geometry that generalizes classical Serre duality to a broad categorical and sheaf-theoretic framework for studying duality on schemes and morphisms.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
theorem in algebraic topology
ⓘ
topological duality theorem ⓘ |
| appearsIn | standard graduate textbooks on algebraic topology ⓘ |
| appliesTo |
locally contractible subsets of spheres
ⓘ
subspaces of spheres ⓘ |
| assumes |
A is a nonempty closed subset of S^n
ⓘ
n \ge 1 ⓘ |
| codomain |
cohomology groups
ⓘ
homology groups ⓘ |
| domain |
spheres
ⓘ
topological spaces ⓘ |
| expresses | isomorphism between homology of a subspace and reduced cohomology of its complement ⓘ |
| field |
algebraic topology
ⓘ
cohomology theory ⓘ homology theory ⓘ |
| generalizationOf |
Jordan curve theorem (via homological methods)
ⓘ
linked to:
Jordan curve theorem
|
| hasVariant |
Alexander–Spanier cohomology version
ⓘ
Borel–Moore homology version of Alexander duality ⓘ cohomological Alexander duality ⓘ
linked to:
Alexander duality
|
| historicalPeriod | early 20th century mathematics ⓘ |
| holdsFor |
finite CW-complexes embedded in spheres
ⓘ
polyhedra embedded in spheres ⓘ |
| involves |
complements in spheres
ⓘ
reduced cohomology ⓘ reduced homology ⓘ singular cohomology ⓘ singular homology ⓘ |
| isPartOf | classical results of algebraic topology ⓘ |
| namedAfter | James Waddell Alexander II ⓘ |
| relatedConcept |
Lefschetz duality
ⓘ
Poincaré–Alexander–Lefschetz duality ⓘ
linked to:
Lefschetz duality
knot complement ⓘ link complement ⓘ |
| relates |
homology of a subspace of a sphere
ⓘ
reduced cohomology of the complement of a subspace in a sphere ⓘ |
| relatesTo | Poincaré duality ⓘ |
| requires |
Mayer–Vietoris sequence
ⓘ
excision in homology ⓘ long exact sequence of a pair ⓘ |
| typicalAssumption |
A is locally contractible
ⓘ
coefficients in a principal ideal domain ⓘ |
| typicalForm | \tilde H_i(S^n \setminus A) \cong \tilde H^{n-i-1}(A) ⓘ |
| usedFor |
computing homology of complements of subsets in spheres
ⓘ
knot theory ⓘ linking phenomena in topology ⓘ studying embeddings of complexes in spheres ⓘ |
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: Alexander duality Description of subject: Alexander duality is a theorem in algebraic topology that relates the homology (or cohomology) of a subspace of a sphere to the reduced cohomology of its complement.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Alexander duality