Tverberg's theorem with constraints

Abstract

The topological Tverberg theorem claims that for any continuous map of the ( q − 1 ) ( d + 1 ) -simplex σ ( d + 1 ) ( q − 1 ) to R d there are q disjoint faces of σ ( d + 1 ) ( q − 1 ) such that their images have a non-empty intersection. This has been proved for affine maps, and if q is a prime power, but not in general. We extend the topological Tverberg theorem in the following way: Pairs of vertices are forced to end up in different faces. This leads to the concept of constraint graphs. In Tverberg's theorem with constraints, we come up with a list of constraints graphs for the topological Tverberg theorem. The proof is based on connectivity results of chessboard-type complexes. Moreover, Tverberg's theorem with constraints implies new lower bounds for the number of Tverberg partitions. As a consequence, we prove Sierksma's conjecture for d = 2 and q = 3 .