The Topological Tverberg Theorem and winding numbers

摘要

The Topological Tverberg Theorem claims that any continuous map of a (q - 1)(d + 1)-simplex to 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 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 K3q-2 in the plane. We investigate graphs that are minimal with respect to the winding number condition. (c) 2005 Elsevier Inc. All rights reserved.