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

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