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