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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Bockstein homomorphism canonical | 2 |
| integral Bockstein homomorphism | 1 |
| mod p Bockstein homomorphism | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11219616 — 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: Bockstein homomorphism Context triple: [Characteristic Classes, hasSubject, Bockstein homomorphism]
-
A.
Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
-
B.
Hopf invariant
The Hopf invariant is a topological integer-valued invariant that classifies certain continuous maps between spheres, playing a central role in homotopy theory and the study of higher-dimensional linking.
-
C.
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.
-
D.
Serre spectral sequence
The Serre spectral sequence is a fundamental tool in algebraic topology that relates the homology or cohomology of a fibration to that of its base and fiber, enabling complex computations in a systematic way.
-
E.
Tate cohomology
Tate cohomology is a refinement of group cohomology that extends cohomology and homology to negative degrees, playing a key role in algebraic number theory and Galois cohomology.
- 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: Bockstein homomorphism Target entity description: 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.
-
A.
Grothendieck spectral sequence
The Grothendieck spectral sequence is a fundamental tool in homological algebra that relates the derived functors of a composite functor to the derived functors of its components, enabling efficient computation of cohomology.
-
B.
Hopf invariant
The Hopf invariant is a topological integer-valued invariant that classifies certain continuous maps between spheres, playing a central role in homotopy theory and the study of higher-dimensional linking.
-
C.
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.
-
D.
Serre spectral sequence
The Serre spectral sequence is a fundamental tool in algebraic topology that relates the homology or cohomology of a fibration to that of its base and fiber, enabling complex computations in a systematic way.
-
E.
Tate cohomology
Tate cohomology is a refinement of group cohomology that extends cohomology and homology to negative degrees, playing a key role in algebraic number theory and Galois cohomology.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
cohomology operation
ⓘ
connecting homomorphism ⓘ homomorphism ⓘ natural transformation ⓘ |
| appearsIn | Bockstein spectral sequence ⓘ |
| 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
ⓘ
linked to:
Bockstein homomorphism
mod p Bockstein homomorphism ⓘ
linked to:
Bockstein homomorphism
|
| 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 ⓘ |
| 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 ⓘ |
| specialCaseOf | connecting homomorphism in derived functors ⓘ |
| usedIn | cohomology theory ⓘ |
| usedToConstruct | Bockstein spectral sequence ⓘ |
| 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 ⓘ characteristic classes of vector bundles ⓘ cohomology operations ⓘ |
| usedWithCoefficients |
Z/nZ
ⓘ
Z/pZ ⓘ local coefficient systems ⓘ |
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: Bockstein homomorphism Description of subject: 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.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Bockstein homomorphism
linked to: Bockstein homomorphism