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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

A generalized van Kampen-Flores theorem


Author: K. S. Sarkaria
Journal: Proc. Amer. Math. Soc. 111 (1991), 559-565
MSC: Primary 57N10; Secondary 57M05
MathSciNet review: 1004423
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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})$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1004423-6
PII: S 0002-9939(1991)1004423-6
Article copyright: © Copyright 1991 American Mathematical Society