Tucker's lemma

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In this 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.

In mathematics, Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem, named after Albert W. Tucker.

Let T be a triangulation of the closed n-dimensional ball Bn. Assume T is antipodally symmetric on the boundary sphere Sn-1. That means that the subset of simplices of T which are in Sn-1 provides a triangulation of Sn-1 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 Sn-1. 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.

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