The Topological Tverberg Problem and winding numbers

Torsten Schöneborn and Günter M. Ziegler— 2005

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 K_{3q-2} in the plane. We investigate graphs that are minimal with respect to the winding number condition.

TitelThe Topological Tverberg Problem and winding numbers
VerfasserTorsten Schöneborn and Günter M. Ziegler
Erschienen inJ. Combinatorial Theory, Ser.~A, volume 112, pages 82-104