The topological Tverberg problem beyond prime powers

Tverberg-type theory aims to establish sufficient conditions for a simplicial complex $\mathrm{\Sigma }$ such that every continuous map $f\colon \Sigma \to $\mathbb{R}$^d$maps$q$pointsfrompairwisedisjointfacestothesamepointin$$\mathbb{R}$^d$\mathrm{.}Suchresultsareplentifulfor$q$ a power of a prime. However, for $q$ with at least two distinct prime divisors, results that guarantee the existence of $q$-fold points of coincidence are non-existent – aside from immediate corollaries of the prime power case. Here we present a general method that yields such results beyond the case of prime powers. In particular, we prove previously conjectured upper bounds for the topological Tverberg problem for all $q$.