A topological product Tverberg Theorem
Andreas F. Holmsen, Grace McCourt, Daniel McGinnis, and Shira Zerbib

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 denote the 8-dimensional simplex viewed as an abstract simplicial complex, and suppose that its vertices are arranged in a array. Then for any continuous map it is possible to partition the rows or the columns of the vertex array into two parts, such that the disjoint faces and induced by the two parts satisfy . Our result also has consequences for geometric transversals and topological Helly.
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Taxonomy
TopicsDigital Image Processing Techniques
