Tucker-Ky Fan colorings
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- by K. S. Sarkaria PDF
- Proc. Amer. Math. Soc. 110 (1990), 1075-1081 Request permission
Abstract:
The existence of a continuous ${{\mathbf {Z}}_2}$-map from a free $m$-dimensional ${{\mathbf {Z}}_2}$-simplicial complex $E$ to the $(m - 1)$-dimensional antipodal sphere ${S^{m - 1}}$ is characterized by means of an enumerative combinatorial criterion involving a coloring of the vertices of $E$. The Borsuk-Ulam theorem, 1933, and the combinatorial lemmas of Tucker, 1945, and Ky Fan, 1952, are easy consequences of this result for the case $|E| = {S^m}$.References
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K. Borsuk, Drei Sätze über die $n$-dimensionale euklidische Sphäre, Fund. Math. 20 (1933), 177-190.
- Albrecht Dold, Simple proofs of some Borsuk-Ulam results, Proceedings of the Northwestern Homotopy Theory Conference (Evanston, Ill., 1982) Contemp. Math., vol. 19, Amer. Math. Soc., Providence, RI, 1983, pp. 65–69. MR 711043, DOI 10.1090/conm/019/711043 A. Flores, Über $n$-dimensionale Komplexe, die im ${R_{2n + 1}}$ absolut selbstverschlungen sind, Ergeb. math. Kolloq. 6 (1933/34), 4-7.
- Ky Fan, A generalization of Tucker’s combinatorial lemma with topological applications, Ann. of Math. (2) 56 (1952), 431–437. MR 51506, DOI 10.2307/1969651
- John Milnor, Construction of universal bundles. I, Ann. of Math. (2) 63 (1956), 272–284. MR 77122, DOI 10.2307/1969609
- M. Richardson and P. A. Smith, Periodic transformations of complexes, Ann. of Math. (2) 39 (1938), no. 3, 611–633. MR 1503428, DOI 10.2307/1968638
- K. S. Sarkaria, A generalized Kneser conjecture, J. Combin. Theory Ser. B 49 (1990), no. 2, 236–240. MR 1064678, DOI 10.1016/0095-8956(90)90029-Y
- K. S. Sarkaria, A generalized van Kampen-Flores theorem, Proc. Amer. Math. Soc. 111 (1991), no. 2, 559–565. MR 1004423, DOI 10.1090/S0002-9939-1991-1004423-6
- K. S. Sarkaria, Kuratowski complexes, Topology 30 (1991), no. 1, 67–76. MR 1081934, DOI 10.1016/0040-9383(91)90034-2
- Arnold Shapiro, Obstructions to the imbedding of a complex in a euclidean space. I. The first obstruction, Ann. of Math. (2) 66 (1957), 256–269. MR 89410, DOI 10.2307/1969998
- Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258
- A. W. Tucker, Some topological properties of disk and sphere, Proc. First Canadian Math. Congress, Montreal, 1945, University of Toronto Press, Toronto, 1946, pp. 285–309. MR 0020254
- S. M. Ulam, A collection of mathematical problems, Interscience Tracts in Pure and Applied Mathematics, no. 8, Interscience Publishers, New York-London, 1960. MR 0120127 E. R. van Kampen, Komplexe in euklidischen Räumen and Berichtigung, Abh. Math. Sem. Univ. Hamburg 9 (1932), 72-78 and 152-153.
- James W. Walker, A homology version of the Borsuk-Ulam theorem, Amer. Math. Monthly 90 (1983), no. 7, 466–468. MR 711647, DOI 10.2307/2975728
- Wen Tsün Wu, On the realization of complexes in euclidean spaces. I, Sci. Sinica 7 (1958), 251–297. MR 99026
- Wu Wen-tsün, A theory of imbedding, immersion, and isotopy of polytopes in a euclidean space, Science Press, Peking, 1965. MR 0215305
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 110 (1990), 1075-1081
- MSC: Primary 57N10; Secondary 05C15, 57M05
- DOI: https://doi.org/10.1090/S0002-9939-1990-1004424-7
- MathSciNet review: 1004424