A topological product Tverberg Theorem

We prove a generalization of the topological Tverberg theorem. One special instance of our general theorem is the following: Let $\mathrm{\Delta }$ denote the 8-dimensional simplex viewed as an abstract simplicial complex, and suppose that its vertices are arranged in a $3\times 3$ array. Then for any continuous map $f:\mathrm{\Delta }\to {\mathbb{R}}^3$ it is possible to partition the rows or the columns of the vertex array into two parts, such that the disjoint faces $\sigma$ and $\tau$ induced by the two parts satisfy $f(\sigma )\cap f(\tau )\mathrm{\ne }\mathrm{\varnothing }$. Our result also has consequences for geometric transversals and topological Helly.