Whitehead groups
E924762
algebraic K-theory invariant
functor on groups
invariant in geometric group theory
invariant in geometric topology
Whitehead groups are algebraic K-theory invariants associated with groups that measure the failure of certain projective modules or h-cobordisms to be trivial, playing a central role in high-dimensional topology and geometric group theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Whitehead groups canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11425542 — 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 groups Context triple: [Introduction to Algebraic K-Theory, topic, Whitehead groups]
-
A.
Whitehead product 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.
-
B.
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.
-
C.
Whitehead product
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.
-
D.
Picard group
The Picard group is an algebraic invariant of a variety or scheme that classifies line bundles (or divisor classes) up to isomorphism, playing a central role in algebraic geometry.
-
E.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
- 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 groups Target entity description: Whitehead groups are algebraic K-theory invariants associated with groups that measure the failure of certain projective modules or h-cobordisms to be trivial, playing a central role in high-dimensional topology and geometric group theory.
-
A.
Whitehead product 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.
-
B.
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.
-
C.
Whitehead product
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.
-
D.
Picard group
The Picard group is an algebraic invariant of a variety or scheme that classifies line bundles (or divisor classes) up to isomorphism, playing a central role in algebraic geometry.
-
E.
Weil group
The Weil group is an extension of the absolute Galois group introduced by André Weil to refine class field theory and play a central role in the formulation of the local and global Langlands correspondences.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic K-theory invariant
ⓘ
functor on groups ⓘ invariant in geometric group theory ⓘ invariant in geometric topology ⓘ |
| alsoKnownAs | Whitehead groups ⓘ |
| appearsIn |
structure sets of manifolds
ⓘ
surgery theory ⓘ |
| codomain | abelian group ⓘ |
| constructedAsQuotientOf | K1(Z[G]) ⓘ |
| constructedFrom | K1(Z[G]) ⓘ |
| context | topological applications of algebraic K-theory ⓘ |
| definedFor |
discrete groups
ⓘ
fundamental groups of CW-complexes ⓘ |
| dependsOn | integral group ring Z[G] ⓘ |
| domain | group ⓘ |
| generalizedBy |
higher Whitehead groups
ⓘ
higher algebraic K-groups ⓘ |
| isAbelianGroup | true ⓘ |
| isConjecturedZeroFor |
many classes of aspherical manifold groups
ⓘ
torsion-free hyperbolic groups ⓘ |
| isFunctorialIn | group homomorphisms ⓘ |
| isZeroFor |
finite cyclic groups
ⓘ
free groups ⓘ fundamental groups of compact surfaces ⓘ trivial group ⓘ |
| measures |
failure of certain projective modules to be free
ⓘ
obstruction to triviality of h-cobordisms ⓘ |
| namedAfter | J. H. C. Whitehead NERFINISHED ⓘ |
| quotientsOut |
image of trivial units in Z[G]
ⓘ
±G ⓘ |
| relatedConjecture |
Borel conjecture
NERFINISHED
ⓘ
Farrell–Jones conjecture NERFINISHED ⓘ |
| relatedTo |
K1 of a group ring
ⓘ
Whitehead torsion NERFINISHED ⓘ algebraic K-theory ⓘ geometric group theory ⓘ h-cobordism theorem NERFINISHED ⓘ high-dimensional manifold topology ⓘ projective modules over group rings ⓘ s-cobordism theorem NERFINISHED ⓘ simple homotopy theory ⓘ |
| symbol | Wh(G) ⓘ |
| usedIn |
classification of h-cobordisms
ⓘ
classification of high-dimensional manifolds up to homeomorphism ⓘ study of simple homotopy equivalences ⓘ |
| vanishingImplies |
every h-cobordism is simple
ⓘ
every homotopy equivalence is simple up to stabilization ⓘ |
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 groups Description of subject: Whitehead groups are algebraic K-theory invariants associated with groups that measure the failure of certain projective modules or h-cobordisms to be trivial, playing a central role in high-dimensional topology and geometric group theory.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.