preprint
Stronger counterexamples to the topological Tverberg conjecture
submitted
Sergey
Avvakumov
author 3827DAC8-F248-11E8-B48F-1D18A9856A87
R.
Karasev
author
A.
Skopenkov
author
UlWa
department
Algorithms for Embeddings and Homotopy Theory
project
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.
arXiv2019
eng
arXiv
1908.08731
000986519600004
https://research-explorer.ista.ac.at/record/8156
Avvakumov, Sergey, R. Karasev, and A. Skopenkov. “Stronger Counterexamples to the Topological Tverberg Conjecture.” <i>ArXiv</i>. arXiv, n.d.
Avvakumov S, Karasev R, Skopenkov A. Stronger counterexamples to the topological Tverberg conjecture. arXiv, 1908.08731.
Avvakumov, S., Karasev, R., & Skopenkov, A. (n.d.). Stronger counterexamples to the topological Tverberg conjecture. <i>arXiv</i>. arXiv.
S. Avvakumov, R. Karasev, A. Skopenkov, ArXiv (n.d.).
S. Avvakumov, R. Karasev, and A. Skopenkov, “Stronger counterexamples to the topological Tverberg conjecture,” <i>arXiv</i>. arXiv.
Avvakumov, Sergey, et al. “Stronger Counterexamples to the Topological Tverberg Conjecture.” <i>ArXiv</i>, 1908.08731, arXiv.
Avvakumov S, Karasev R, Skopenkov A. Stronger counterexamples to the topological Tverberg conjecture. <i>arXiv</i>.
81842020-07-30T10:45:34Z2023-09-08T11:20:02Z