The Topological Tverberg Theorem and winding numbers

Abstract

The Topological Tverberg Theorem claims that any continuous map of a $(q-1)(d+1)$-simplex to ${\mathbb{R}}^d$ identifies points from q disjoint faces. (This has been proved for affine maps, for $d⩽1$, and if q is a prime power, but not yet in general.)

The Topological Tverberg Theorem can be restricted to maps of the d-skeleton of the simplex. We further show that it is equivalent to a “Winding Number Conjecture” that concerns only maps of the $(d-1)$-skeleton of a $(q-1)(d+1)$-simplex to ${\mathbb{R}}^d$. “Many Tverberg partitions” arise if and only if there are “many q-winding partitions.”

The $d=2$ case of the Winding Number Conjecture is a problem about drawings of the complete graphs $K_{3q-2}$ in the plane. We investigate graphs that are minimal with respect to the winding number condition.