Bockstein homomorphism
E911365
The Bockstein homomorphism is a connecting homomorphism in cohomology arising from a short exact sequence of coefficient groups, used to relate and detect characteristic classes and torsion phenomena in topology.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
cohomology operation
ⓘ
connecting homomorphism ⓘ homomorphism ⓘ natural transformation ⓘ |
| appearsIn | Bockstein spectral sequence NERFINISHED ⓘ |
| arisesFrom |
short exact sequence of coefficient groups
ⓘ
short exact sequence of coefficient modules ⓘ |
| associatedTo | short exact sequence 0 → A → B → C → 0 ⓘ |
| codomain | cohomology group ⓘ |
| definedFromExactSequence | 0 → A → B → C → 0 ⓘ |
| definedOn |
cellular cohomology
ⓘ
simplicial cohomology ⓘ singular cohomology ⓘ |
| dependsOn | choice of short exact sequence of coefficients ⓘ |
| domain | cohomology group ⓘ |
| field |
algebraic topology
ⓘ
homological algebra ⓘ |
| generalizesTo | cohomology theories represented by spectra ⓘ |
| hasDegree | +1 ⓘ |
| hasVariant |
integral Bockstein homomorphism
NERFINISHED
ⓘ
mod p Bockstein homomorphism NERFINISHED ⓘ |
| isConnectingHomomorphismOf | long exact sequence in cohomology ⓘ |
| isNaturalIn |
continuous map of spaces
ⓘ
topological space X ⓘ |
| maps | H^n(X;C) to H^{n+1}(X;A) ⓘ |
| namedAfter | M. Bockstein NERFINISHED ⓘ |
| oftenDenoted |
β
ⓘ
δ ⓘ |
| property |
compatible with cup products up to sign in many contexts
ⓘ
functorial with respect to maps of spaces ⓘ |
| relatedTo |
Ext functor
ⓘ
long exact sequence of Ext groups ⓘ short exact sequence 0 → Z → Z → Z/nZ → 0 ⓘ universal coefficient theorem NERFINISHED ⓘ |
| specialCaseOf | connecting homomorphism in derived functors ⓘ |
| usedIn | cohomology theory ⓘ |
| usedToConstruct | Bockstein spectral sequence NERFINISHED ⓘ |
| usedToDetect |
torsion in cohomology
ⓘ
torsion phenomena in topology ⓘ |
| usedToRelate |
characteristic classes
ⓘ
cohomology with different coefficients ⓘ integral and mod p cohomology ⓘ |
| usedToStudy |
Chern classes
ⓘ
Stiefel–Whitney classes NERFINISHED ⓘ characteristic classes of vector bundles ⓘ cohomology operations ⓘ |
| usedWithCoefficients |
Z/nZ
NERFINISHED
ⓘ
Z/pZ ⓘ local coefficient systems ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.