A topological product Tverberg Theorem

Abstract

We prove a generalization of the topological Tverberg theorem. One special instance of our general theorem is the following: Let $\hat{I}”$ 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 $f:\hat{I}”\to {\mathbb{R}}^3$ it is possible to partition the rows or the columns of the vertex array into two parts, such that the disjoint faces $\ddot{I}ƒ$ and $\ddot{I}„$ induced by the two parts satisfy $f(\ddot{I}ƒ)\cap f(\ddot{I}„)\ne \varnothing$. Our result also has consequences for geometric transversals and topological Helly.

10 pages