Stronger counterexamples to the topological Tverberg conjecture

S. Avvakumov, R. Karasev, A. Skopenkov, ArXiv:1908.08731 (n.d.).

Preprint | Submitted | English
Author
Avvakumov, SerhiiIST Austria; Karasev, R.; Skopenkov, A.
Department
Abstract
Denote by ∆N the N-dimensional simplex. A map f : ∆N → Rd is an almost r-embedding if fσ1∩. . .∩fσr = ∅ 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 ≥ 2r + 1, then there is an almost r-embedding ∆(d+1)(r−1) → Rd. This was improved by Blagojevi´c–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 l d + 2 r + 1 m−2, then there is an almost r-embedding ∆N → Rd. For the r-fold van Kampen–Flores conjecture we also produce counterexamples which are stronger than previously known. Our proof is based on generalizations of the Mabillard–Wagner theorem on construction of almost r-embeddings from equivariant maps, and of the Ozaydin theorem on existence of equivariant maps.
Publishing Year
Date Published
2019-08-23
Journal Title
arXiv:1908.08731
Acknowledgement
We would like to thank F. Frick for helpful discussions
Article Number
1908.08731
Page
7
IST-REx-ID

Cite this

Avvakumov S, Karasev R, Skopenkov A. Stronger counterexamples to the topological Tverberg conjecture. arXiv:190808731.
Avvakumov, S., Karasev, R., & Skopenkov, A. (n.d.). Stronger counterexamples to the topological Tverberg conjecture. ArXiv:1908.08731. arXiv.
Avvakumov, Sergey, R. Karasev, and A. Skopenkov. “Stronger Counterexamples to the Topological Tverberg Conjecture.” ArXiv:1908.08731. arXiv, n.d.
S. Avvakumov, R. Karasev, and A. Skopenkov, “Stronger counterexamples to the topological Tverberg conjecture,” arXiv:1908.08731. arXiv.
Avvakumov S, Karasev R, Skopenkov A. Stronger counterexamples to the topological Tverberg conjecture. arXiv:1908.08731.
Avvakumov, Sergey, et al. “Stronger Counterexamples to the Topological Tverberg Conjecture.” ArXiv:1908.08731, 1908.08731, arXiv.
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