Tverberg's theorem for cell complexes

The topological Tverberg theorem states that any continuous map of a $(d+1)(r-1)$-simplex into the Euclidean $d$-space maps some points from $r$ pairwise disjoint faces of the simplex to the same point whenever $r$ is a prime power. We substantially generalize this theorem to continuous maps of certain CW complexes, including simplicial $((d+1)(r-1)-1)$-spheres, into the Euclidean $d$-space. We also discuss the atomicity of the Tverberg property.