A topological product Tverberg Theorem

TL;DR

This work presents a broad topological generalization of Tverberg's theorem by introducing $\Delta^{m}(n)$ as the index-structured domain and proving that for a continuous map $f: \Delta^{m}(n) \to \mathbb{R}^d$, there exists an index $i$ and a partition of $[n]$ into $p$ parts whose image faces have a nonempty intersection, provided $p$ is a prime power and $n = \lceil(\frac{d}{m}+1)(p-1)+\frac{1}{m}\rceil$. The proof relies on the configuration-space/test-map scheme, constructing a $\mathbb{Z}_p$-equivariant map from a configuration space $K(n,m,p)$ to a $p$-fold deleted join, and then building a highly connected invariant subcomplex $T$ whose join structure yields a contradiction via Dold's theorem. The result generalizes the usual topological Tverberg theorem (recovering it when $m=1$) and yields consequences for geometric transversals and topological Helly, while linking to affine versions and colorful Helly through the DHL framework. The work also discusses extensions to prime powers and highlights remaining questions about constants and optimality in related affine and transversal settings.

Abstract

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 $3\times 3$ array. Then for any continuous map $f:Δ\to \mathbb{R}^3$ it is possible to partition the rows or the columns of the vertex array into two parts, such that the disjoint faces $σ$ and $τ$ induced by the two parts satisfy $f(σ)\cap f(τ) \neq \emptyset$. Our result also has consequences for geometric transversals and topological Helly.

Theorems & Definitions (14)

• Theorem 1
• Corollary 2
• proof
• Theorem 3: Dold dold
• Lemma 4
• Lemma 5
• proof
• Lemma 6
• proof
• Lemma 7