Whitehead product
E886921
The Whitehead product is a fundamental operation in algebraic topology that combines homotopy classes of maps to produce higher-order homotopy information, playing a key role in the structure of homotopy groups of spheres and related spaces.
Statements (45)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic topology concept
ⓘ
operation on homotopy groups ⓘ |
| actsOn |
homotopy classes of maps
ⓘ
homotopy groups of pointed spaces ⓘ homotopy groups of spheres ⓘ |
| appearsIn |
classical homotopy theory
ⓘ
homotopy groups of spheres computations ⓘ |
| appliesTo |
pointed CW-complexes
ⓘ
simply connected spaces ⓘ |
| arity | binary operation ⓘ |
| construction |
defined using attaching maps on S^m ∨ S^n
ⓘ
realized via the canonical map S^{m+n-1} → S^m ∨ S^n ⓘ |
| definedOn | π_*(X) for a pointed space X ⓘ |
| domain | homotopy groups ⓘ |
| field |
algebraic topology
ⓘ
homotopy theory ⓘ |
| generalizationOf | commutator in fundamental groups (up to analogy) ⓘ |
| historicalPeriod | 20th-century mathematics ⓘ |
| influenced |
development of homotopy operations
ⓘ
theory of higher order operations in topology ⓘ |
| inputType |
elements of π_m(X)
ⓘ
elements of π_n(X) ⓘ |
| keyRole |
structure of higher homotopy groups
ⓘ
structure of homotopy groups of spheres ⓘ |
| namedAfter | J. H. C. Whitehead NERFINISHED ⓘ |
| notation | [α, β] ⓘ |
| outputType | elements of π_{m+n-1}(X) ⓘ |
| property |
bilinear up to homotopy
ⓘ
depends on basepoint ⓘ graded skew-commutative ⓘ natural with respect to continuous maps ⓘ |
| relatedTo |
Eilenberg–MacLane spaces
NERFINISHED
ⓘ
Lie algebra structures on homotopy groups ⓘ Postnikov invariants NERFINISHED ⓘ Samelson product NERFINISHED ⓘ homotopy Lie algebra of a space ⓘ |
| requires | choice of basepoint in the space ⓘ |
| satisfies | graded Jacobi identity up to homotopy ⓘ |
| usedFor |
analyzing Postnikov towers
ⓘ
constructing elements in higher homotopy groups ⓘ defining Samelson product on loop spaces ⓘ describing non-abelian structure of low-dimensional homotopy groups ⓘ detecting higher-order homotopy information ⓘ studying H-spaces and loop spaces ⓘ studying homotopy groups of spheres ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.