Steenrod operations
E911362
Steenrod operations are cohomology operations in algebraic topology that act on cohomology groups, providing powerful tools for distinguishing topological spaces and defining and studying characteristic classes.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Steenrod operations canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11219603 — 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: Steenrod operations Context triple: [Characteristic Classes, hasSubject, Steenrod operations]
-
A.
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.
-
B.
Whitehead product in homotopy theory
The Whitehead product in homotopy theory is a bilinear operation on homotopy groups that captures how spheres can be nontrivially linked or composed within a topological space.
-
C.
Stiefel–Whitney classes
Stiefel–Whitney classes are characteristic classes in algebraic topology that assign cohomology invariants to real vector bundles, capturing their topological and orientability properties.
-
D.
Alexander–Spanier cohomology
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.
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E.
Eilenberg–MacLane spaces
Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of 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: Steenrod operations Target entity description: Steenrod operations are cohomology operations in algebraic topology that act on cohomology groups, providing powerful tools for distinguishing topological spaces and defining and studying characteristic classes.
-
A.
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.
-
B.
Whitehead product in homotopy theory
The Whitehead product in homotopy theory is a bilinear operation on homotopy groups that captures how spheres can be nontrivially linked or composed within a topological space.
-
C.
Stiefel–Whitney classes
Stiefel–Whitney classes are characteristic classes in algebraic topology that assign cohomology invariants to real vector bundles, capturing their topological and orientability properties.
-
D.
Alexander–Spanier cohomology
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.
-
E.
Eilenberg–MacLane spaces
Eilenberg–MacLane spaces are topological spaces characterized by having a single nontrivial homotopy group, serving as fundamental building blocks in homotopy theory and the definition of cohomology.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
cohomology operation
ⓘ
concept in algebraic topology ⓘ |
| actsOn | cohomology groups ⓘ |
| appliesTo |
cohomology of CW complexes
ⓘ
cohomology of topological spaces ⓘ |
| are | natural transformations of cohomology functors ⓘ |
| basisFor | structure of the mod p cohomology of spaces ⓘ |
| coefficientField |
F_2
ⓘ
F_p ⓘ |
| component |
Bockstein homomorphism
NERFINISHED
ⓘ
Steenrod squares NERFINISHED ⓘ reduced pth power operations ⓘ |
| definedOn |
cohomology with coefficients in a finite field
ⓘ
mod p cohomology ⓘ singular cohomology ⓘ |
| degreeShift | nonnegative ⓘ |
| field | algebraic topology ⓘ |
| generalizationOf | cup product powers ⓘ |
| introducedBy | Norman Steenrod NERFINISHED ⓘ |
| invariantUnder | homotopy equivalence ⓘ |
| namedAfter | Norman Steenrod NERFINISHED ⓘ |
| property |
determine the structure of the Steenrod algebra
ⓘ
functorial ⓘ graded ⓘ satisfy Adem relations ⓘ satisfy Cartan formula ⓘ stable under suspension ⓘ |
| relatedTo |
Eilenberg–MacLane spaces
NERFINISHED
ⓘ
Steenrod algebra NERFINISHED ⓘ cup product in cohomology ⓘ |
| satisfies |
Adem relations among operations
ⓘ
Cartan formula for products NERFINISHED ⓘ naturality with respect to continuous maps ⓘ stability with respect to suspension ⓘ |
| timePeriod | mid 20th century ⓘ |
| usedFor |
defining characteristic classes
ⓘ
distinguishing topological spaces ⓘ studying characteristic classes ⓘ |
| usedIn |
calculation of cohomology rings
ⓘ
classification of manifolds ⓘ homotopy theory ⓘ obstruction theory ⓘ stable homotopy theory ⓘ study of Postnikov towers ⓘ study of characteristic classes of vector bundles ⓘ theory of fiber bundles ⓘ |
| usedToDetect |
nontrivial characteristic classes
ⓘ
nontrivial homotopy classes ⓘ |
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: Steenrod operations Description of subject: Steenrod operations are cohomology operations in algebraic topology that act on cohomology groups, providing powerful tools for distinguishing topological spaces and defining and studying characteristic classes.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.