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.
- Freund, Robert M.; Todd, Michael J. (1981), "A constructive proof of Tucker's combinatorial lemma", Journal of Combinatorial Theory, Series A 30 (3): 321–325, doi:10.1016/0097-3165(81)90027-3, MR 618536.
- Matoušek, Jiří (2003), Using the Borsuk–Ulam Theorem, Springer-Verlag, p. 34, ISBN 3-540-00362-2.
- Tucker, A. W. (1946), "Some topological properties of disk and sphere", Proc. First Canadian Math. Congress, Montreal, 1945, Toronto: University of Toronto Press, pp. 285–309, MR 0020254.
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