zbMATH — the first resource for mathematics

A combinatorial analog of a theorem of F.J. Dyson. (English) Zbl 1196.54073
F. J. Dyson’s theorem [Ann. Math. (2) 54, 534–536 (1951; Zbl 0045.02901)] states that for any real-valued function on \(S^2\) there are two orthogonal diameters whose endpoints are mapped to the same point. The authors give a combinatorial proof of this theorem exploiting Tucker labellings: Let \(T\) be a triangulation of \(S^2\) which is invariant under the antipodal map \(A\). A Tucker labelling \(\ell\) assigns the values \(\pm1\) to the vertices of \(T\) such that \(\ell(-v)=\ell(v)\). The authors show that for the linear extension \(L\) of \(\ell\) there exists a polygonal simple closed path that is invariant under \(A\) and is mapped to zero under \(L\). They then show that this result is “equivalent” to Dyson’s theorem.

54H25 Fixed-point and coincidence theorems (topological aspects)
Full Text: DOI arXiv
[1] Dyson, F., Continuous functions defined on spheres, Ann. of math., 54, 534-536, (1951) · Zbl 0045.02901
[2] Fan, K., A generalization of Tucker’s combinatorial lemma with topological applications, Ann. of math., 56, 2, 431-437, (1952) · Zbl 0047.42004
[3] Freund, R.; Todd, M., A constructive proof of Tucker’s combinatorial lemma, J. combin. theory ser. A, 30, 3, 321-325, (1981) · Zbl 0462.05026
[4] Hirsch, M., Differential topology, (1976), Springer-Verlag · Zbl 0356.57001
[5] Kulpa, W.; Turzański, M., A combinatorial lemma for a symmetric triangulation of the sphere \(\mathbb{S}^2\), Acta univ. carolin. math. phys., 42, 2, 69-74, (2001) · Zbl 1003.05033
[6] Kuratowski, K., Topology, vol. II, (), 170-171
[7] Lefschetz, S., Introduction to topology, (), 134-141
[8] Livesay, G., On a theorem of F.J. Dyson, Ann. of math., 59, 227-229, (1954)
[9] de Longueville, M.; Živaljević, R., The borsuk – ulam-property, Tucker-property and constructive proofs in combinatorics, J. combin. theory ser. A, 113, 839-850, (2006) · Zbl 1093.05006
[10] Matoušek, J., Using the borsuk – ulam theorem, (), written in cooperation with A. Björner and G. Ziegler
[11] Matoušek, J., A combinatorial proof of Kneser’s conjecture, Combinatorica, 24, 163-170, (2004) · Zbl 1047.05018
[12] F. Meunier, doctoral thesis, available at http://www.enpc.fr/lvmt/frederic.meunier/These.pdf
[13] Prescott, T.; Su, F., A constructive proof of Ky Fan’s generalization of Tucker’s lemma, J. combin. theory ser. A, 111, 257-265, (2005) · Zbl 1080.55005
[14] Simmons, F.; Su, F., Consensus-halving via theorems of borsuk – ulam and Tucker, Math. social sci., 45, 1, 15-25, (2003) · Zbl 1027.91047
[15] Sperner, E., Ein satz über untermengen einer endlichen menge, Math. Z., 27, 1, 544-548, (1928) · JFM 54.0090.06
[16] Tucker, A., Some topological properties of disk and sphere, (), 285-309
[17] Yang, C., On theorems of borsuk – ulam, kakutani – yamabe – yujobo and Dyson, II, Ann. of math., 62, 271-283, (1955) · Zbl 0067.15202
[18] Zarankiewicz, K., Un théorème sur l’uniformisation des fonctions continues et son application à la démonstration du théorème de F.J. Dyson sur LES transformations de la surface sphérique, Bull. acad. Pol. sci., 3, 117-120, (1954) · Zbl 0056.41902
[19] Ziegler, G., Generalized Kneser coloring theorems with combinatorial proofs, Invent. math., 147, 671-691, (2002) · Zbl 1029.05058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.