Whitehead product in homotopy theory
E886920
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Whitehead product in homotopy theory canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10829222 — 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: Whitehead product in homotopy theory Context triple: [J. H. C. Whitehead, knownFor, Whitehead product in homotopy theory]
-
A.
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.
-
B.
Classifying Spaces and Fibrations
"Classifying Spaces and Fibrations" is a mathematical work that develops the theory of classifying spaces in algebraic topology and their relationship to fiber bundles and fibrations.
-
C.
Sullivan minimal model in rational homotopy theory
The Sullivan minimal model in rational homotopy theory is a canonical commutative differential graded algebra that encodes the rational homotopy type of a topological space in an algebraic form.
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D.
Whitney approximation theorem
The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
-
E.
"Algebraic Topology"
"Algebraic Topology" is a foundational mathematical text that develops topological concepts using algebraic methods such as homology and cohomology theories.
- 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: Whitehead product in homotopy theory Target entity description: 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.
-
A.
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.
-
B.
Classifying Spaces and Fibrations
"Classifying Spaces and Fibrations" is a mathematical work that develops the theory of classifying spaces in algebraic topology and their relationship to fiber bundles and fibrations.
-
C.
Sullivan minimal model in rational homotopy theory
The Sullivan minimal model in rational homotopy theory is a canonical commutative differential graded algebra that encodes the rational homotopy type of a topological space in an algebraic form.
-
D.
Whitney approximation theorem
The Whitney approximation theorem is a fundamental result in differential topology stating that any continuous function between smooth manifolds can be uniformly approximated by smooth functions.
-
E.
"Algebraic Topology"
"Algebraic Topology" is a foundational mathematical text that develops topological concepts using algebraic methods such as homology and cohomology theories.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Whitehead product in homotopy theory Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.