In mathematics, Tucker's lemma is a combinatorial

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

 analog of the Borsuk–Ulam theorem.

Let T be a triangulation of the closed n-dimensional ball . Assume T is antipodally symmetric on the boundary . That means that the subset of simplices of T which are in provides a triangulation of where if σ is a simplex then so is −σ. Let
be a labelling of the vertices of T which satisfies L(−v)=−L(v) for all vertices v in . Then Tucker's Lemma states that there exists a 1-simplex in T whose vertices are labelled by the same number but with opposite signs.
In the above example, where n=2, the red 1-simplex has vertices which are labelled by the same number with opposite signs. Tucker's lemma states that for such a triangulation at least one such 1-simplex must exist.

See also

• Brouwer fixed point theorem
  Brouwer fixed point theorem
  Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...
  
  
• Borsuk–Ulam theorem
• Topological combinatorics
  Topological combinatorics
  The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this gradually turned into the field of algebraic topology....