  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 $r=6$.
