The Topological Tverberg Problem and winding numbers

Details The Topological Tverberg Problem and winding numbers The Topological Tverberg Problem and winding numbers

2004-09-06

arxiv.org/abs/math/0409081v2

Mathematics

Combinatorics

19 pages. J. Combinatorial Theory, Ser. A, to appear

Scientific paper

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.

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