2004/09/06 by Torsten Schöneborn, Günter M. Ziegler, Schöneborn, Torsten +1
Computer Science · Mathematics · #052A35 #05C62 #55M20 #5A35 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #FOS: Mathematics #Metric Geometry (math.MG) #Topological and Geometric Data Analysis #math.CO #math.MG #msc:052A35 #msc:05C62 #msc:55M20 #msc:5A35
paper · pdf · doi:10.48550/arxiv.math/0409081
19 pages. J. Combinatorial Theory, Ser. A, to appear
openalex publication_date 2004/09/06 · arxiv created 2005/01/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Topological Tverberg Theorem claims that any continuous map of a (q-1)(d+1)-simplex to \Rd 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 \Rd. ``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.