Stronger counterexamples to the topological Tverberg conjecture
Abstract
Denote by $\Delta_N$ the $N$dimensional simplex. A map $f\colon \Delta_N\to\mathbb R^d$ is an almost $r$embedding if $f\sigma_1\cap\ldots\cap f\sigma_r=\emptyset$ whenever $\sigma_1,\ldots,\sigma_r$ are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if $r$ is not a prime power and $d\ge2r+1$, then there is an almost $r$embedding $\Delta_{(d+1)(r1)}\to\mathbb R^d$. This was improved by BlagojevićFrickZiegler using a simple construction of higherdimensional counterexamples by taking $k$fold join power of lowerdimensional ones. We improve this further (for $d$ large compared to $r$): If $r$ is not a prime power and $N:=(d+1)rr\Big\lceil\dfrac{d+2}{r+1}\Big\rceil2$, then there is an almost $r$embedding $\Delta_N\to\mathbb R^d$. For the $r$fold van KampenFlores conjecture we also produce counterexamples which are stronger than previously known. Our proof is based on generalizations of the MabillardWagner theorem on construction of almost $r$embeddings from equivariant maps, and of the Özaydin theorem on existence of equivariant maps.
 Publication:

arXiv eprints
 Pub Date:
 August 2019
 arXiv:
 arXiv:1908.08731
 Bibcode:
 2019arXiv190808731A
 Keywords:

 Mathematics  Geometric Topology;
 Computer Science  Computational Geometry;
 Mathematics  Combinatorics;
 52C35;
 55S91;
 57S17
 EPrint:
 7 pages