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.
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.