Stronger counterexamples to the topological Tverberg conjecture
S. Avvakumov, R. Karasev, A. Skopenkov

TL;DR
This paper advances the construction of counterexamples to the topological Tverberg conjecture, especially for non-prime power r, by developing stronger counterexamples using generalized topological and combinatorial methods.
Contribution
It provides improved bounds and stronger counterexamples for the topological Tverberg conjecture and related van Kampen-Flores conjecture, extending previous constructions with new topological tools.
Findings
Constructed larger counterexamples for non-prime power r
Developed stronger counterexamples for the van Kampen-Flores conjecture
Extended the use of equivariant topology methods in combinatorial topology
Abstract
Denote by the -dimensional simplex. A map is an almost -embedding if whenever are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if is not a prime power and , then there is an almost -embedding . This was improved by Blagojevi\'c-Frick-Ziegler using a simple construction of higher-dimensional counterexamples by taking -fold join power of lower-dimensional ones. We improve this further (for large compared to ): If is not a prime power and , then there is an almost -embedding . For the -fold van Kampen-Flores conjecture we also produce counterexamples which are stronger…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · graph theory and CDMA systems · Computational Geometry and Mesh Generation
