A topological product Tverberg Theorem

TL;DR

This paper generalizes the topological Tverberg theorem, demonstrating that for certain high-dimensional simplices mapped into three-dimensional space, specific partitions of vertices guarantee intersecting images, impacting geometric transversals and Helly-type results.

Contribution

It introduces a new topological Tverberg theorem for an 8-dimensional simplex with vertex arrangements, extending previous results and providing applications to geometric transversals and Helly theorems.

Findings

Partitioning vertices in a 3x3 array yields intersecting images under continuous maps.

Generalization of the topological Tverberg theorem to higher dimensions.

Implications for geometric transversals and topological Helly theorems.

Abstract

We prove a generalization of the topological Tverberg theorem. One special instance of our general theorem is the following: Let $\Delta$ 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 $f:\Delta \to \mathbb{R}^3$ 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 $f(\sigma)\cap f(\tau) \neq \emptyset$. Our result also has consequences for geometric transversals and topological Helly.