math_groups_discgeom

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
Datum2005
Quelle/n
Erschienen inJ. Combinatorial Theory, Ser.~A, volume 112, pages 82-104
ArtText