Counterexamples to the topological Tverberg conjecture
Abstract
The "topological Tverberg conjecture" by Bárány, Shlosman and Szűcs (1981) states that any continuous map of a simplex of dimension $(r1)(d+1)$ 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 Bárány et al., and for prime power $r$ by Özaydin (1987). We combine the generalized van Kampen theorem announced by Mabillard and Wagner (2014) with the constraint method of Blagojević, 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 $r=6$.
 Publication:

arXiv eprints
 Pub Date:
 February 2015
 arXiv:
 arXiv:1502.00947
 Bibcode:
 2015arXiv150200947F
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Algebraic Topology;
 Mathematics  Metric Geometry;
 52A35;
 55S35
 EPrint:
 3 pages, to appear in Oberwolfach Reports. This version differs from the Oberwolfach Reports version in two updated references. This manuscript is now part of arXiv:1510.07984