Tucker’s lemma

E83404

combinatorial lemma topological combinatorics result

Tucker’s lemma is a combinatorial analog of the Borsuk–Ulam theorem that provides conditions guaranteeing the existence of certain complementary edge labels in triangulated spheres.

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Observed surface forms (3)

Surface form	Occurrences
Borsuk–Ulam theorem	1
Tucker's lemma	1
octahedral Tucker lemma	1

Statements (47)

Predicate	Object
instanceOf	combinatorial lemma ⓘ topological combinatorics result ⓘ
appliesTo	triangulated spheres ⓘ
assumes	antipodal symmetry of the triangulation ⓘ
concerns	Z2-equivariant combinatorial structures ⓘ antipodal labeling of triangulations ⓘ
field	combinatorics ⓘ topology ⓘ
generalizedBy	Ky Fan’s lemma ⓘ
guaranteesExistenceOf	complementary edge labels ⓘ
hasApplication	discrete versions of fair division theorems ⓘ
hasConclusion	existence of an edge whose endpoints have opposite labels ⓘ
hasConsequence	existence of complementary labeled simplex edges ⓘ
hasConstraint	labeling must be antipodal ⓘ labels exclude zero in the classical formulation ⓘ
hasDimensionParameter	n ⓘ
hasDomain	combinatorial topology ⓘ
hasInput	antipodally symmetric triangulation of a sphere ⓘ labeling of vertices by integers with opposite signs on antipodal points ⓘ
hasLabelSet	{±1,±2,…,±n} in the classical n-dimensional version ⓘ
hasNature	non-constructive existence result ⓘ
hasProofMethod	combinatorial methods ⓘ topological methods ⓘ
hasSpecialCase	discrete ham sandwich–type results ⓘ
hasVariant	Tucker–Fan type lemmas ⓘ cubical Tucker lemma ⓘ Tucker’s lemma self-linksurface differs ⓘ surface form: octahedral Tucker lemma
holdsOn	triangulations of the n-dimensional sphere ⓘ
implies	discrete Borsuk–Ulam type results ⓘ
isAnalogOf	Tucker’s lemma self-linksurface differs ⓘ surface form: Borsuk–Ulam theorem
isEquivalentTo	Borsuk–Ulam theorem over Z2 in appropriate formulations ⓘ
isToolFor	combinatorial fixed-point theory ⓘ discrete geometry ⓘ topological combinatorics ⓘ
namedAfter	Albert W. Tucker ⓘ
relatedTo	Ky Fan’s lemma ⓘ Sperner's lemma ⓘ surface form: Sperner’s lemma
usedFor	combinatorial proofs in consensus division ⓘ combinatorial proofs in fair division problems ⓘ combinatorial proofs in game theory ⓘ
usedIn	combinatorial proofs of fixed-point theorems ⓘ discrete versions of topological results ⓘ equivariant topology ⓘ proofs of the Borsuk–Ulam theorem ⓘ
usedToProve	consensus halving theorems ⓘ necklace splitting theorems ⓘ
yearIntroducedApprox	1940s ⓘ

Referenced by (5)

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

Tucker’s lemma → hasVariant → Tucker’s lemma self-linksurface differs ⓘ

this entity surface form: octahedral Tucker lemma

Tucker’s lemma → isAnalogOf → Tucker’s lemma self-linksurface differs ⓘ

this entity surface form: Borsuk–Ulam theorem

Albert W. Tucker → knownFor → Tucker’s lemma ⓘ

Tucker → knownFor → Tucker’s lemma ⓘ

subject surface form: Albert W. Tucker

this entity surface form: Tucker's lemma

Albert W. Tucker → notableWork → Tucker’s lemma ⓘ