Topological Tverberg theorems for products of polytopes

[2608.02979] Topological Tverberg theorems for products of polytopes Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2608.02979 (math) [Submitted on 4 Aug 2026 ( v1 ), last revised 14 Aug 2026 (this version, v2)] Title: Topological Tverberg theorems for products of polytopes Authors: Steven Simon View a PDF of the paper titled Topological Tverberg theorems for products of polytopes, by Steven Simon View PDF HTML (experimental) Abstract: The topological Tverberg theorem asserts that if $r$ is a prime power then for any continuous map $f\colon \Delta_{(r-1)(d+1)}\rightarrow \mathbb{R}^d$ from the $(r-1)(d+1)$-dimensional simplex $\Delta_{(r-1)(d+1)}$ to $\mathbb{R}^d$ there exist $r$ pairwise disjoint faces of the simplex whose images have non-empty $r$-fold intersection. By refinement, the same conclusion holds if the simplex is replaced by any polytope of the same dimension. While this dimension is tight for simplices, recent work of Soberón and Zerbib shows that this need not be true for polytopes in general. Here we give topological Tverberg theorems for products of simplices. Each of these improves upon the $(r-1)(d+1)$-dimensional threshold, even while imposing the structural condition that the ``Tverberg faces'' of the product are themselves the products of pairwise disjoint faces from each simplex factor. As before, refinement extends these results, and in particular their dimensional improvements, to products of arbitrary polytopes. As an example, if $d+1$ is a power of two then whenever $m\geq n\geq d+1$ and $m+n=3d+2$ we show that any continuous map $f\colon \Delta_m\times \Delta_n\rightarrow \mathbb{R}^d$ admits disjoint faces $\sigma_1,\sigma_2$ of $\Delta_m$ and $\tau_1,\tau_2$ of $\Delta_n$ such that $\cap_{i,j\in[2]} f(\sigma_i\times\tau_j)\neq \emptyset$. In the case of multilinear maps, our results imply partitions of grid-indexed point sets in $\mathbb{R}^d$ by specialized subsets with stronger intersection conclusions than given by Tverberg's original theorem. Lastly, we extend our results to van Kampen--Flores type theorems which impose dimensional restrictions on the faces of each product factor. Comments: 20 pages Subjects: Combinatorics (math.CO) ; Algebraic Topology (math.AT) MSC classes: 52a37, 52B12, 52a30, 55S91 Cite as: arXiv:2608.02979 [math.CO] (or arXiv:2608.02979v2 [math.CO] for this version) https://doi.org/10.48550/arXiv.2608.02979 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Steven Simon [ view email ] [v1] Tue, 4 Aug 2026 00:30:59 UTC (24 KB) [v2] Fri, 14 Aug 2026 21:25:22 UTC (24 KB) Full-text links: Access Paper: View a PDF of the paper titled Topological Tverberg theorems for products of polytopes, by Steven Simon View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2026-08 Change to browse by: math math.AT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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