Whitehead product in homotopy theory

E886920

algebraic operation bilinear operation on homotopy groups construction 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.

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Observed surface forms (1)

Surface form	Occurrences
Whitehead product	0

Statements (48)

Predicate	Object
instanceOf	algebraic operation ⓘ bilinear operation on homotopy groups ⓘ construction in homotopy theory ⓘ
actsOn	homotopy groups ⓘ π_m(X) ⓘ π_n(X) ⓘ
appearsIn	homotopy spectral sequences ⓘ long exact sequences of homotopy groups for pairs ⓘ
arity	binary operation ⓘ
captures	interaction of homotopy classes of maps from spheres ⓘ linking behavior of spheres in a space ⓘ nontrivial compositions of sphere maps ⓘ
codomain	π_{n+m-1}(X) ⓘ
constructionUses	attaching map from S^{m+n-1} to S^m ∨ S^n ⓘ commutator-like map on spheres ⓘ wedge sum S^m ∨ S^n ⓘ
context	graded homotopy groups π_*(X) ⓘ
definedIn	homotopy category of pointed spaces ⓘ
domainCondition	α ∈ π_m(X), β ∈ π_n(X) ⓘ
field	algebraic topology ⓘ
firstIntroducedBy	J. H. C. Whitehead NERFINISHED ⓘ
generalizes	commutator in fundamental groups ⓘ
gives	higher order operations in homotopy ⓘ
isDefinedFor	pointed topological spaces ⓘ
isToolFor	detecting nontrivial elements in homotopy groups ⓘ studying attaching maps of cells in CW-complexes ⓘ
isZeroFor	simply connected suspensions under suitable conditions ⓘ
namedAfter	J. H. C. Whitehead NERFINISHED ⓘ
nontrivialExample	Hopf invariant phenomena in homotopy groups of spheres ⓘ [ι_n,ι_n] in π_{2n-1}(S^n) for n>1 ⓘ
outputCondition	[α,β] ∈ π_{m+n-1}(X) ⓘ
property	bilinear ⓘ functorial ⓘ graded skew-commutative ⓘ natural with respect to continuous maps ⓘ
relatedTo	Lie algebra structure on homotopy groups of an H-space ⓘ Samelson product NERFINISHED ⓘ
requires	chosen basepoint ⓘ
satisfies	[α,β] = −(−1)^{mn}[β,α] for α∈π_m, β∈π_n ⓘ graded Jacobi identity up to sign ⓘ
structureType	graded Lie algebra up to homotopy ⓘ
usedIn	Postnikov tower computations ⓘ description of k-invariants ⓘ homotopy Lie algebra of a space ⓘ obstruction theory ⓘ study of homotopy groups of spheres ⓘ
vanishesOn	H-spaces under suitable conditions ⓘ
yearIntroducedApprox	mid 20th century ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

J. H. C. Whitehead → knownFor → Whitehead product in homotopy theory ⓘ