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.
-
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"
ⓘ
linked to:
"Algebraic Topology"
|
| 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 ⓘ
linked to:
Alexandrov–Čech cohomology
|
| hasCharacteristic |
closely related to singular cohomology
ⓘ
closely related to Čech cohomology ⓘ very general definition ⓘ |
| hasProperty |
Mayer–Vietoris sequence
ⓘ
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
ⓘ
linked to:
Edwin Spanier
James Waddell Alexander II ⓘ |
| notation | H^n_{AS}(X;G) ⓘ |
| relatedConcept |
Alexander–Spanier cochains
ⓘ
sheaf cohomology ⓘ singular cohomology ⓘ Čech cohomology ⓘ
linked to:
Alexandrov–Čech cohomology
|
| 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
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–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.