We prove a generalization of the topological Tverberg theorem. One special instance of our general theorem is the following: Let 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: R³ it is possible to partition the rows or the columns of the vertex array into two parts, such that the disjoint faces σ and τ induced by the two parts satisfy f (σ) ∩ f (τ) ≠ ∅ f () f (). Our result also has consequences for geometric transversals and topological Helly theorems.
Holmsen et al. (Fri,) studied this question.