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 Δ Δ denote the 8-dimensional simplex viewed as an abstract simplicial complex, and suppose that its vertices are arranged in a 3 × 3 3× 3 array. Then for any continuous map f : Δ → R 3 f:Δ → R^3 it is possible to partition the rows or the columns of the vertex array into two parts, such that the disjoint faces σ σ and τ τ induced by the two parts satisfy f ( σ ) ∩ f ( τ ) ≠ ∅ f(σ )∩ f(τ ) ≠ ∅ . Our result also has consequences for geometric transversals and topological Helly theorems.