On the Topological Tverberg Theorem

2004/05/20 by Torsten Schöneborn, Schöneborn, Torsten · 2 citations
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #Digital Image Processing Techniques #Topological and Geometric Data Analysis #math.AT #math.CO #msc:05A18 #msc:05C10 #msc:05C38 #msc:05C83 #msc:52A35 #msc:57Q91

paper · pdf · doi:10.48550/arxiv.math/0405393

45 pages, 13 figures, 3 tables

arxiv created 2004/05/20 · arxiv updated 2009/12/01

Abstract

We introduce a new ``Winding Number Conjecture'' about maps from the (d-1)-skeleton of the ((d+1)(q-1))-simplex into \real^d. This conjecture is equivalent to the Topological Tverberg Theorem. Furthermore, many statements about the Topological Tverberg Theorem transfer to the Winding Number Conjecture, for example all currently proven cases of the Topological Tverberg Theorem as well as Sierksma's conjecture about the number of Tverberg partitions. In the case d=2, the Winding Number Conjecture is a statement about complete graphs: It claims that in every image of K_3(q-1)+1 in the plane either q-1 triangles wind around one vertex or q-2 triangles wind around the intersection of two edges, where the triangles, edges and vertices are disjoint. We examine which other graphs have this property and find the minimal subgraph of K₇ having this property.

Cited by

• The Topological Tverberg Problem and winding numbers
• Using the Borsuk–Ulam Theorem

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