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arXiv · 2412.16055

A topological product Tverberg Theorem

Andreas F. Holmsen·Grace McCourt·Daniel McGinnis·Shira Zerbib
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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.

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Andreas F. Holmsen, Grace McCourt, Daniel McGinnis, Shira Zerbib. 2024-12-20. A topological product Tverberg Theorem. https://arxiv.org/abs/2412.16055

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