Stronger counterexamples to the topological Tverberg conjecture

Abstract

Denote by ${\Delta }_M$ the $M$-dimensional simplex. A map $f:{\Delta }_M\to {\mathbb{R}}^d$ is an almost $r$-embedding if $f{\sigma }_1\cap \dots \cap f{\sigma }_r=\varnothing$ whenever ${\sigma }_1,\dots ,{\sigma }_r$ are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if $r$ is not a prime power and $d\ge 2r+1$, then there is an almost $r$-embedding ${\Delta }_{(d+1)(r-1)}\to {\mathbb{R}}^d$. This was improved by Blagojević-Frick-Ziegler using a simple construction of higher-dimensional counterexamples by taking $k$-fold join power of lower-dimensional ones. We improve this further (for $d$ large compared to $r$): If $r$ is not a prime power and $N:=(d+1)r-r\lceil \frac{d+2}{r+1}\rceil -2$, then there is an almost $r$-embedding ${\Delta }_N\to {\mathbb{R}}^d$. For the $r$-fold van Kampen-Flores conjecture we also produce counterexamples which are stronger than previously known. Our proof is based on generalizations of the Mabillard-Wagner theorem on construction of almost $r$-embeddings from equivariant maps, and of the Özaydin theorem on existence of equivariant maps.

9 pages, exposition improved

References in corpus (2)

• Envy-free division using mapping degree
• Topological Tverberg Theorem: the proofs and the counterexamples