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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Whitehead group | 0 |
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.