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 | 2 |
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]
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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.
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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.
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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 ⓘ |
| 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 ⓘ |
| 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 ⓘ |
| 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
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: 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 (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
John