{"acknowledgement":"We would like to thank F. Frick for helpful discussions","language":[{"iso":"eng"}],"author":[{"last_name":"Avvakumov","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87","full_name":"Avvakumov, Sergey","first_name":"Sergey"},{"full_name":"Karasev, R.","first_name":"R.","last_name":"Karasev"},{"last_name":"Skopenkov","first_name":"A.","full_name":"Skopenkov, A."}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1908.08731"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","related_material":{"record":[{"id":"8156","status":"public","relation":"dissertation_contains"}]},"department":[{"_id":"UlWa"}],"publication":"arXiv","date_updated":"2023-09-08T11:20:02Z","article_number":"1908.08731","date_created":"2020-07-30T10:45:34Z","status":"public","abstract":[{"text":"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\r\nd + 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. ","lang":"eng"}],"year":"2019","day":"23","article_processing_charge":"No","date_published":"2019-08-23T00:00:00Z","_id":"8184","external_id":{"arxiv":["1908.08731"],"isi":["000986519600004"]},"title":"Stronger counterexamples to the topological Tverberg conjecture","month":"08","oa_version":"Preprint","citation":{"apa":"Avvakumov, S., Karasev, R., &#38; Skopenkov, A. (n.d.). Stronger counterexamples to the topological Tverberg conjecture. <i>arXiv</i>. arXiv.","short":"S. Avvakumov, R. Karasev, A. Skopenkov, ArXiv (n.d.).","chicago":"Avvakumov, Sergey, R. Karasev, and A. Skopenkov. “Stronger Counterexamples to the Topological Tverberg Conjecture.” <i>ArXiv</i>. arXiv, n.d.","ieee":"S. Avvakumov, R. Karasev, and A. Skopenkov, “Stronger counterexamples to the topological Tverberg conjecture,” <i>arXiv</i>. arXiv.","mla":"Avvakumov, Sergey, et al. “Stronger Counterexamples to the Topological Tverberg Conjecture.” <i>ArXiv</i>, 1908.08731, arXiv.","ista":"Avvakumov S, Karasev R, Skopenkov A. Stronger counterexamples to the topological Tverberg conjecture. arXiv, 1908.08731.","ama":"Avvakumov S, Karasev R, Skopenkov A. Stronger counterexamples to the topological Tverberg conjecture. <i>arXiv</i>."},"type":"preprint","publisher":"arXiv","oa":1,"project":[{"call_identifier":"FWF","grant_number":"P31312","_id":"26611F5C-B435-11E9-9278-68D0E5697425","name":"Algorithms for Embeddings and Homotopy Theory"}],"isi":1,"publication_status":"submitted"}