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 (1)
| Label | Occurrences |
|---|---|
| Alexander duality canonical | 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.
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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
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) NERFINISHED ⓘ |
| hasVariant |
Alexander–Spanier cohomology version
ⓘ
Borel–Moore homology version of Alexander duality ⓘ cohomological Alexander duality NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| relatedConcept |
Lefschetz duality
NERFINISHED
ⓘ
Poincaré–Alexander–Lefschetz duality NERFINISHED ⓘ 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 NERFINISHED ⓘ |
| requires |
Mayer–Vietoris sequence
NERFINISHED
ⓘ
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
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Instruction
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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 (1)
Full triples — surface form annotated when it differs from this entity's canonical label.