Stronger Counterexamples to the Topological Tverberg Conjecture

S. Avvakumov, R. Karasev, A. Skopenkov

Research output: Contribution to journalArticlepeer-review

Abstract

Denote by Δ M the M-dimensional simplex. A map f: Δ M→ Rd is an almost r-embedding if f(σ1) ∩ … ∩ f(σr) = ∅ whenever σ1, … , σr are pairwise disjoint faces. A counterexample to the topological Tverberg conjecture asserts that if r is not a prime power and d≥ 2 r+ 1 , then there is an almost r-embedding Δ (d+1)(r-1)→ Rd . This was improved by Blagojević–Frick–Ziegler using a simple construction of higher-dimensional counterexamples by taking k-fold join power of lower-dimensional ones. We improve this further (for d large compared to r): If r is not a prime power and N=(d+1)r-r⌈d+2r+1⌉-2 , then there is an almost r-embedding Δ N→ Rd . The improvement follows from our stronger counterexamples to the r-fold van Kampen–Flores conjecture. Our proof is based on generalizations of the Mabillard–Wagner theorem on construction of almost r-embeddings from equivariant maps, and of the Özaydin theorem on existence of equivariant maps.

Original languageEnglish
Pages (from-to)717-727
Number of pages11
JournalCombinatorica
Volume43
Issue number4
DOIs
StatePublished - Aug 2023
Externally publishedYes

Keywords

  • Deleted product obstruction
  • Equivariant maps
  • Multiple points of maps
  • The topological Tverberg conjecture

All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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