A generalized van Kampen-Flores theorem

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 $f({x_1}) = \cdots = f({x_p})$.