Tverbergs theorem for cell complexes

The topological Tverberg theorem states that given any continuous map $fcolonDelta^{(d+1)(r-1)}tomathbb{R}^d$, there are pairwise disjoint faces $sigma_1,ldots,sigma_r$ of $Delta^{(d+1)(r-1)}$ such that $f(sigma_1)capcdotscap f(sigma_r) eemptyset$ whenever $r$ is a prime power. We generalize this theorem to a continuous map from a certain CW complex into a Euclidean space.