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A generalized van Kampen-Flores theorem


Author: K. S. Sarkaria
Journal: Proc. Amer. Math. Soc. 111 (1991), 559-565
MSC: Primary 57N10; Secondary 57M05
DOI: https://doi.org/10.1090/S0002-9939-1991-1004423-6
MathSciNet review: 1004423
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Abstract | References | Similar Articles | Additional Information

Abstract: The $ n$-skeleton of a $ (2n + 2)$-simplex does not embed in $ {{\mathbf{R}}^{2n}}$. This well-known result is due (independently) to van Kampen, 1932, and Flores, 1933, who proved the case $ p = 2$ of the following:

Theorem. Let $ p$ be a prime, and let $ s$ and $ l$ be positive integers such that $ l(p - 1) \leq p(s - 1)$. Then, for any continuous map $ f$ from a $ (ps + p - 2)$-dimensional simplex into $ {{\mathbf{R}}^l}$, there must exist $ p$ points $ \{ {x_1}, \ldots ,{x_p}\} $, lying in pairwise disjoint faces of dimensions $ \leq s - 1$ of this simplex, such that $ f({x_1}) = \cdots = f({x_p})$.


References [Enhancements On Off] (What's this?)

  • [1] E. G. Bajmóczy and I. Bárány, On a common generalization of Borsuk's and Radon's theorem, Acta Math. Acad. Sci. Hungar. 34 (1979), 347-350. MR 565677 (81f:52004)
  • [2] I. Bárány, S. B. Shlosman and A. Szücs, On a topological generalization of a theorem of Tverberg, J. London Math. Soc. (2) 23 (1981), 158-164. MR 602247 (82d:55001)
  • [3] K. Borsuk, Drei Sätze über die $ n$-dimensionale euklidische Sphäre, Fund. Math. 20 (1933), 177-190.
  • [4] A. Dold, Simple proofs of some Borsuk-Ulam results, Contemp. Math., vol. 19, Amer. Math. Soc., 1983, pp. 65-69. MR 711043 (85e:55003)
  • [5] A. Flores, Über $ n$-dimensionale Komplexe die im $ {R_{2n + 1}}$ absolut selbstverschlungen sind, Ergeb. Math. Kolloq. 6 (1933/34), 4-7.
  • [6] J. Radon, Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten, Math. Ann. 83 (1921), 113-115. MR 1512002
  • [7] K. S. Sarkaria, Kneser colorings of polyhedra, Illinois J. Math. (to appear). MR 1007897 (90h:57002)
  • [8] -, A generalized Kneser conjecture, J. Combin. Theory Ser B (to appear). MR 1064678 (91g:05003)
  • [9] -, Van Kampen obstructions, in preparation.
  • [10] H. Tverberg, A generalization of Radon's theorem, J. London Math. Soc. 41 (1966), 123-128. MR 0187147 (32:4601)
  • [11] E. R. van Kampen, Komplexe in euklidischen Räumen, Abh. Math. Sem. Hamburg 9 (1932), 72-78; Berichtigung dazu, ibid (1932), 152-153.
  • [12] C. Weber, Plongements de polyèdres dans le domaine métastable, Comment. Math. Helv. 42 (1967), 1-27. MR 0238330 (38:6606)
  • [13] W.-T. Wu, A theory of imbedding, immersion, and isotopy of polytopes in a Euclidean space, Science Press, Peking, 1965. MR 0215305 (35:6146)

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DOI: https://doi.org/10.1090/S0002-9939-1991-1004423-6
Article copyright: © Copyright 1991 American Mathematical Society

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