Counterexamples to the topological Tverberg conjecture

TL;DR

This paper constructs counterexamples to the topological Tverberg conjecture for non-prime power r, showing that the conjecture does not hold in high-dimensional cases, thus resolving a long-standing open problem.

Contribution

It combines advanced topological theorems and methods to disprove the conjecture for all non-prime power r in sufficiently high dimensions.

Findings

Counterexamples exist for all non-prime power r

The smallest known counterexample involves a 100-dimensional simplex

Counterexamples require high-dimensional codomain maps

Abstract

The "topological Tverberg conjecture" by B\'ar\'any, Shlosman and Sz\H{u}cs (1981) states that any continuous map of a simplex of dimension $(r-1)(d+1)$ to $\mathbb{R}^d$ maps points from $r$ disjoint faces of the simplex to the same point in $\mathbb{R}^d$. This was established for affine maps by Tverberg (1966), for the case when $r$ is a prime by B\'ar\'any et al., and for prime power $r$ by \"Ozaydin (1987). We combine the generalized van Kampen theorem announced by Mabillard and Wagner (2014) with the constraint method of Blagojevi\'c, Ziegler and the author (2014), and thus prove the existence of counterexamples to the topological Tverberg conjecture for any number $r$ of faces that is not a prime power. However, these counterexamples require that the dimension $d$ of the codomain is sufficiently high: the smallest counterexample we obtain is for a map of the $100$-dimensional simplex to $\mathbb{R}^{19}$, for $r=6$.