The Topological Tverberg Problem and winding numbers
Abstract
The Topological Tverberg Theorem claims that any continuous map of a (q1)(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 dskeleton of the simplex. We further show that it is equivalent to a ``Winding Number Conjecture'' that concerns only maps of the (d1)skeleton of a (q1)(d+1)simplex to \R^d. ``Many Tverberg partitions'' arise if and only if there are ``many qwinding partitions.'' The d=2 case of the Winding Number Conjecture is a problem about drawings of the complete graphs K_{3q2} in the plane. We investigate graphs that are minimal with respect to the winding number condition.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 2004
 arXiv:
 arXiv:math/0409081
 Bibcode:
 2004math......9081S
 Keywords:

 Combinatorics;
 Metric Geometry;
 052A35;
 5A35;
 05C62;
 55M20
 EPrint:
 19 pages. J. Combinatorial Theory, Ser. A, to appear