Alexander–Spanier cohomology
E679316
Alexander–Spanier cohomology is a cohomology theory in algebraic topology defined using cochains on all finite subsets of a space, notable for its generality and close relationship to Čech and singular cohomology.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Alexander–Spanier cohomology canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7648173 — 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–Spanier cohomology Context triple: [Alexandrov–Čech cohomology, relatedTo, Alexander–Spanier cohomology]
-
A.
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.
-
B.
Mayer–Vietoris sequence in de Rham cohomology
The Mayer–Vietoris sequence in de Rham cohomology is a long exact sequence that computes the de Rham cohomology of a manifold by relating it to the cohomology of an open cover and their intersection.
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C.
Pontryagin classes
Pontryagin classes are characteristic classes associated with real vector bundles that capture topological information about the bundle’s curvature and play a central role in differential topology and geometry.
-
D.
Lefschetz duality
Lefschetz duality is a generalization of Poincaré duality that relates the homology of a compact manifold with boundary to the cohomology of the manifold relative to its boundary.
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E.
Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
- 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–Spanier cohomology Target entity description: Alexander–Spanier cohomology is a cohomology theory in algebraic topology defined using cochains on all finite subsets of a space, notable for its generality and close relationship to Čech and singular cohomology.
-
A.
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.
-
B.
Mayer–Vietoris sequence in de Rham cohomology
The Mayer–Vietoris sequence in de Rham cohomology is a long exact sequence that computes the de Rham cohomology of a manifold by relating it to the cohomology of an open cover and their intersection.
-
C.
Pontryagin classes
Pontryagin classes are characteristic classes associated with real vector bundles that capture topological information about the bundle’s curvature and play a central role in differential topology and geometry.
-
D.
Lefschetz duality
Lefschetz duality is a generalization of Poincaré duality that relates the homology of a compact manifold with boundary to the cohomology of the manifold relative to its boundary.
-
E.
Atiyah–Hirzebruch spectral sequence
The Atiyah–Hirzebruch spectral sequence is a fundamental computational tool in algebraic topology that relates generalized cohomology theories, such as K-theory, to ordinary cohomology, enabling the step-by-step calculation of these invariants from simpler data.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic topology concept
ⓘ
cohomology theory ⓘ |
| agreesWith |
sheaf cohomology for suitable sheaves on nice spaces
ⓘ
singular cohomology for reasonable spaces ⓘ Čech cohomology for paracompact Hausdorff spaces ⓘ |
| appearsIn | Edwin Spanier's book "Algebraic Topology" NERFINISHED ⓘ |
| appliesTo |
locally compact spaces
ⓘ
metric spaces ⓘ paracompact Hausdorff spaces ⓘ |
| canBeDefinedWithCoefficientsIn |
abelian groups
ⓘ
modules ⓘ rings ⓘ |
| cochainComplexType | Alexander–Spanier cochain complex ⓘ |
| cochainDefinition |
cochains defined on all finite subsets of a space
ⓘ
functions on ordered tuples of points from a space ⓘ |
| coefficientVariable | abelian group G ⓘ |
| definedOn | topological spaces ⓘ |
| degreeNComponent | group of functions on (n+1)-tuples of points ⓘ |
| differential | alternating sum of restriction maps omitting one coordinate ⓘ |
| domainVariable | topological space X ⓘ |
| field | algebraic topology ⓘ |
| generalizes |
singular cohomology
ⓘ
Čech cohomology NERFINISHED ⓘ |
| hasCharacteristic |
closely related to singular cohomology
ⓘ
closely related to Čech cohomology ⓘ very general definition ⓘ |
| hasProperty |
Mayer–Vietoris sequence
NERFINISHED
ⓘ
dimension axiom for CW complexes and manifolds ⓘ excision for suitable pairs ⓘ homotopy invariance ⓘ long exact sequence of a pair ⓘ |
| hasVersion |
reduced Alexander–Spanier cohomology
ⓘ
relative Alexander–Spanier cohomology ⓘ |
| historicalDevelopment | introduced in the mid 20th century ⓘ |
| isContravariantIn | topological spaces ⓘ |
| isEquivalentTo |
singular cohomology on CW complexes
ⓘ
Čech cohomology on compact metric spaces ⓘ |
| namedAfter |
Edwin H. Spanier
NERFINISHED
ⓘ
James Waddell Alexander II NERFINISHED ⓘ |
| notation | H^n_{AS}(X;G) ⓘ |
| relatedConcept |
Alexander–Spanier cochains
ⓘ
sheaf cohomology ⓘ singular cohomology ⓘ Čech cohomology NERFINISHED ⓘ |
| satisfies | Eilenberg–Steenrod axioms on suitable categories of spaces ⓘ |
| targetCategory | graded abelian groups ⓘ |
| usedFor |
foundations of sheaf-theoretic cohomology
ⓘ
relating different cohomology theories ⓘ studying invariants of topological spaces ⓘ |
| uses | cochains on all finite subsets of a topological space ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Alexander–Spanier cohomology Description of subject: Alexander–Spanier cohomology is a cohomology theory in algebraic topology defined using cochains on all finite subsets of a space, notable for its generality and close relationship to Čech and singular cohomology.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.