[{"arxiv":1,"date_updated":"2023-08-14T11:43:55Z","volume":245,"oa":1,"article_processing_charge":"No","_id":"10220","publication_identifier":{"eissn":["1565-8511"],"issn":["0021-2172"]},"acknowledgement":"Research supported by the Swiss National Science Foundation (Project SNSF-PP00P2-138948), by the Austrian Science Fund (FWF Project P31312-N35), by the Russian Foundation for Basic Research (Grants No. 15-01-06302 and 19-01-00169), by a Simons-IUM Fellowship, and by the D. Zimin Dynasty Foundation Grant. We would like to thank E. Alkin, A. Klyachko, V. Krushkal, S. Melikhov, M. Tancer, P. Teichner and anonymous referees for helpful comments and discussions.","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","project":[{"grant_number":"P31312","_id":"26611F5C-B435-11E9-9278-68D0E5697425","name":"Algorithms for Embeddings and Homotopy Theory","call_identifier":"FWF"}],"quality_controlled":"1","oa_version":"Preprint","publication_status":"published","citation":{"chicago":"Avvakumov, Sergey, Isaac Mabillard, Arkadiy B. Skopenkov, and Uli Wagner. “Eliminating Higher-Multiplicity Intersections. III. Codimension 2.” <i>Israel Journal of Mathematics</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1007/s11856-021-2216-z\">https://doi.org/10.1007/s11856-021-2216-z</a>.","apa":"Avvakumov, S., Mabillard, I., Skopenkov, A. B., &#38; Wagner, U. (2021). Eliminating higher-multiplicity intersections. III. Codimension 2. <i>Israel Journal of Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s11856-021-2216-z\">https://doi.org/10.1007/s11856-021-2216-z</a>","ieee":"S. Avvakumov, I. Mabillard, A. B. Skopenkov, and U. Wagner, “Eliminating higher-multiplicity intersections. III. Codimension 2,” <i>Israel Journal of Mathematics</i>, vol. 245. Springer Nature, pp. 501–534, 2021.","short":"S. Avvakumov, I. Mabillard, A.B. Skopenkov, U. Wagner, Israel Journal of Mathematics 245 (2021) 501–534.","ista":"Avvakumov S, Mabillard I, Skopenkov AB, Wagner U. 2021. Eliminating higher-multiplicity intersections. III. Codimension 2. Israel Journal of Mathematics. 245, 501–534.","ama":"Avvakumov S, Mabillard I, Skopenkov AB, Wagner U. Eliminating higher-multiplicity intersections. III. Codimension 2. <i>Israel Journal of Mathematics</i>. 2021;245:501–534. doi:<a href=\"https://doi.org/10.1007/s11856-021-2216-z\">10.1007/s11856-021-2216-z</a>","mla":"Avvakumov, Sergey, et al. “Eliminating Higher-Multiplicity Intersections. III. Codimension 2.” <i>Israel Journal of Mathematics</i>, vol. 245, Springer Nature, 2021, pp. 501–534, doi:<a href=\"https://doi.org/10.1007/s11856-021-2216-z\">10.1007/s11856-021-2216-z</a>."},"abstract":[{"lang":"eng","text":"We study conditions under which a finite simplicial complex K can be mapped to ℝd without higher-multiplicity intersections. An almost r-embedding is a map f: K → ℝd such that the images of any r pairwise disjoint simplices of K do not have a common point. We show that if r is not a prime power and d ≥ 2r + 1, then there is a counterexample to the topological Tverberg conjecture, i.e., there is an almost r-embedding of the (d +1)(r − 1)-simplex in ℝd. This improves on previous constructions of counterexamples (for d ≥ 3r) based on a series of papers by M. Özaydin, M. Gromov, P. Blagojević, F. Frick, G. Ziegler, and the second and fourth present authors.\r\n\r\nThe counterexamples are obtained by proving the following algebraic criterion in codimension 2: If r ≥ 3 and if K is a finite 2(r − 1)-complex, then there exists an almost r-embedding K → ℝ2r if and only if there exists a general position PL map f: K → ℝ2r such that the algebraic intersection number of the f-images of any r pairwise disjoint simplices of K is zero. This result can be restated in terms of a cohomological obstruction and extends an analogous codimension 3 criterion by the second and fourth authors. As another application, we classify ornaments f: S3 ⊔ S3 ⊔ S3 → ℝ5 up to ornament concordance.\r\n\r\nIt follows from work of M. Freedman, V. Krushkal and P. Teichner that the analogous criterion for r = 2 is false. We prove a lemma on singular higher-dimensional Borromean rings, yielding an elementary proof of the counterexample."}],"author":[{"last_name":"Avvakumov","full_name":"Avvakumov, Sergey","first_name":"Sergey","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Mabillard, Isaac","last_name":"Mabillard","first_name":"Isaac","id":"32BF9DAA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Skopenkov","full_name":"Skopenkov, Arkadiy B.","first_name":"Arkadiy B."},{"id":"36690CA2-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-1494-0568","last_name":"Wagner","full_name":"Wagner, Uli","first_name":"Uli"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1511.03501"}],"isi":1,"related_material":{"record":[{"status":"public","relation":"earlier_version","id":"8183"},{"status":"public","id":"9308","relation":"earlier_version"}]},"doi":"10.1007/s11856-021-2216-z","year":"2021","external_id":{"isi":["000712942100013"],"arxiv":["1511.03501"]},"title":"Eliminating higher-multiplicity intersections. III. Codimension 2","publication":"Israel Journal of Mathematics","page":"501–534 ","day":"30","type":"journal_article","intvolume":"       245","status":"public","department":[{"_id":"UlWa"}],"date_created":"2021-11-07T23:01:24Z","month":"10","date_published":"2021-10-30T00:00:00Z","article_type":"original","publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}]},{"date_created":"2020-05-10T22:00:48Z","conference":{"name":"SODA: Symposium on Discrete Algorithms","end_date":"2020-01-08","location":"Salt Lake City, UT, United States","start_date":"2020-01-05"},"department":[{"_id":"UlWa"}],"language":[{"iso":"eng"}],"publisher":"SIAM","scopus_import":1,"date_published":"2020-01-01T00:00:00Z","month":"01","page":"767-785","publication":"Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms","status":"public","type":"conference","day":"01","main_file_link":[{"url":"https://doi.org/10.1137/1.9781611975994.47","open_access":"1"}],"title":"Embeddability of simplicial complexes is undecidable","year":"2020","doi":"10.1137/1.9781611975994.47","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Published Version","quality_controlled":"1","project":[{"grant_number":"P31312","name":"Algorithms for Embeddings and Homotopy Theory","_id":"26611F5C-B435-11E9-9278-68D0E5697425","call_identifier":"FWF"}],"_id":"7806","publication_identifier":{"isbn":["9781611975994"]},"oa":1,"date_updated":"2021-01-12T08:15:38Z","volume":"2020-January","article_processing_charge":"No","author":[{"id":"3E8AF77E-F248-11E8-B48F-1D18A9856A87","first_name":"Marek","last_name":"Filakovský","full_name":"Filakovský, Marek"},{"id":"36690CA2-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-1494-0568","last_name":"Wagner","full_name":"Wagner, Uli","first_name":"Uli"},{"id":"3AA52972-F248-11E8-B48F-1D18A9856A87","last_name":"Zhechev","full_name":"Zhechev, Stephan Y","first_name":"Stephan Y"}],"abstract":[{"text":"We consider the following decision problem EMBEDk→d in computational topology (where k ≤ d are fixed positive integers): Given a finite simplicial complex K of dimension k, does there exist a (piecewise-linear) embedding of K into ℝd?\r\nThe special case EMBED1→2 is graph planarity, which is decidable in linear time, as shown by Hopcroft and Tarjan. In higher dimensions, EMBED2→3 and EMBED3→3 are known to be decidable (as well as NP-hard), and recent results of Čadek et al. in computational homotopy theory, in combination with the classical Haefliger–Weber theorem in geometric topology, imply that EMBEDk→d can be solved in polynomial time for any fixed pair (k, d) of dimensions in the so-called metastable range .\r\nHere, by contrast, we prove that EMBEDk→d is algorithmically undecidable for almost all pairs of dimensions outside the metastable range, namely for . This almost completely resolves the decidability vs. undecidability of EMBEDk→d in higher dimensions and establishes a sharp dichotomy between polynomial-time solvability and undecidability.\r\nOur result complements (and in a wide range of dimensions strengthens) earlier results of Matoušek, Tancer, and the second author, who showed that EMBEDk→d is undecidable for 4 ≤ k ϵ {d – 1, d}, and NP-hard for all remaining pairs (k, d) outside the metastable range and satisfying d ≥ 4.","lang":"eng"}],"publication_status":"published","citation":{"ama":"Filakovský M, Wagner U, Zhechev SY. Embeddability of simplicial complexes is undecidable. In: <i>Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms</i>. Vol 2020-January. SIAM; 2020:767-785. doi:<a href=\"https://doi.org/10.1137/1.9781611975994.47\">10.1137/1.9781611975994.47</a>","mla":"Filakovský, Marek, et al. “Embeddability of Simplicial Complexes Is Undecidable.” <i>Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms</i>, vol. 2020–January, SIAM, 2020, pp. 767–85, doi:<a href=\"https://doi.org/10.1137/1.9781611975994.47\">10.1137/1.9781611975994.47</a>.","ista":"Filakovský M, Wagner U, Zhechev SY. 2020. Embeddability of simplicial complexes is undecidable. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms. SODA: Symposium on Discrete Algorithms vol. 2020–January, 767–785.","short":"M. Filakovský, U. Wagner, S.Y. Zhechev, in:, Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2020, pp. 767–785.","apa":"Filakovský, M., Wagner, U., &#38; Zhechev, S. Y. (2020). Embeddability of simplicial complexes is undecidable. In <i>Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms</i> (Vol. 2020–January, pp. 767–785). Salt Lake City, UT, United States: SIAM. <a href=\"https://doi.org/10.1137/1.9781611975994.47\">https://doi.org/10.1137/1.9781611975994.47</a>","ieee":"M. Filakovský, U. Wagner, and S. Y. Zhechev, “Embeddability of simplicial complexes is undecidable,” in <i>Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms</i>, Salt Lake City, UT, United States, 2020, vol. 2020–January, pp. 767–785.","chicago":"Filakovský, Marek, Uli Wagner, and Stephan Y Zhechev. “Embeddability of Simplicial Complexes Is Undecidable.” In <i>Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms</i>, 2020–January:767–85. SIAM, 2020. <a href=\"https://doi.org/10.1137/1.9781611975994.47\">https://doi.org/10.1137/1.9781611975994.47</a>."}},{"date_published":"2020-06-01T00:00:00Z","month":"06","language":[{"iso":"eng"}],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","scopus_import":"1","department":[{"_id":"UlWa"}],"license":"https://creativecommons.org/licenses/by/3.0/","has_accepted_license":"1","file":[{"relation":"main_file","content_type":"application/pdf","file_id":"8007","creator":"dernst","access_level":"open_access","date_updated":"2020-07-14T12:48:06Z","file_size":575896,"file_name":"2020_LIPIcsSoCG_Avvakumov.pdf","checksum":"6872df6549142f709fb6354a1b2f2c06","date_created":"2020-06-23T11:13:49Z"}],"date_created":"2020-06-22T09:14:19Z","conference":{"start_date":"2020-06-22","location":"Zürich, Switzerland","name":"SoCG: Symposium on Computational Geometry","end_date":"2020-06-26"},"type":"conference","day":"01","status":"public","intvolume":"       164","file_date_updated":"2020-07-14T12:48:06Z","publication":"36th International Symposium on Computational Geometry","doi":"10.4230/LIPIcs.SoCG.2020.12","year":"2020","external_id":{"arxiv":["1909.00263"]},"title":"Homotopic curve shortening and the affine curve-shortening flow","article_number":"12:1 - 12:15","alternative_title":["LIPIcs"],"tmp":{"name":"Creative Commons Attribution 3.0 Unported (CC BY 3.0)","short":"CC BY (3.0)","image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/3.0/legalcode"},"ddc":["510"],"publication_status":"published","citation":{"ieee":"S. Avvakumov and G. Nivasch, “Homotopic curve shortening and the affine curve-shortening flow,” in <i>36th International Symposium on Computational Geometry</i>, Zürich, Switzerland, 2020, vol. 164.","apa":"Avvakumov, S., &#38; Nivasch, G. (2020). Homotopic curve shortening and the affine curve-shortening flow. In <i>36th International Symposium on Computational Geometry</i> (Vol. 164). Zürich, Switzerland: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2020.12\">https://doi.org/10.4230/LIPIcs.SoCG.2020.12</a>","chicago":"Avvakumov, Sergey, and Gabriel Nivasch. “Homotopic Curve Shortening and the Affine Curve-Shortening Flow.” In <i>36th International Symposium on Computational Geometry</i>, Vol. 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. <a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2020.12\">https://doi.org/10.4230/LIPIcs.SoCG.2020.12</a>.","mla":"Avvakumov, Sergey, and Gabriel Nivasch. “Homotopic Curve Shortening and the Affine Curve-Shortening Flow.” <i>36th International Symposium on Computational Geometry</i>, vol. 164, 12:1-12:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2020.12\">10.4230/LIPIcs.SoCG.2020.12</a>.","ama":"Avvakumov S, Nivasch G. Homotopic curve shortening and the affine curve-shortening flow. In: <i>36th International Symposium on Computational Geometry</i>. Vol 164. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2020. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SoCG.2020.12\">10.4230/LIPIcs.SoCG.2020.12</a>","ista":"Avvakumov S, Nivasch G. 2020. Homotopic curve shortening and the affine curve-shortening flow. 36th International Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 164, 12:1-12:15.","short":"S. Avvakumov, G. Nivasch, in:, 36th International Symposium on Computational Geometry, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020."},"author":[{"id":"3827DAC8-F248-11E8-B48F-1D18A9856A87","full_name":"Avvakumov, Sergey","last_name":"Avvakumov","first_name":"Sergey"},{"first_name":"Gabriel","full_name":"Nivasch, Gabriel","last_name":"Nivasch"}],"abstract":[{"text":"We define and study a discrete process that generalizes the convex-layer decomposition of a planar point set. Our process, which we call homotopic curve shortening (HCS), starts with a closed curve (which might self-intersect) in the presence of a set P⊂ ℝ² of point obstacles, and evolves in discrete steps, where each step consists of (1) taking shortcuts around the obstacles, and (2) reducing the curve to its shortest homotopic equivalent. We find experimentally that, if the initial curve is held fixed and P is chosen to be either a very fine regular grid or a uniformly random point set, then HCS behaves at the limit like the affine curve-shortening flow (ACSF). This connection between HCS and ACSF generalizes the link between \"grid peeling\" and the ACSF observed by Eppstein et al. (2017), which applied only to convex curves, and which was studied only for regular grids. We prove that HCS satisfies some properties analogous to those of ACSF: HCS is invariant under affine transformations, preserves convexity, and does not increase the total absolute curvature. Furthermore, the number of self-intersections of a curve, or intersections between two curves (appropriately defined), does not increase. Finally, if the initial curve is simple, then the number of inflection points (appropriately defined) does not increase.","lang":"eng"}],"volume":164,"date_updated":"2021-01-12T08:16:23Z","oa":1,"article_processing_charge":"No","arxiv":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Published Version","project":[{"grant_number":"P31312","_id":"26611F5C-B435-11E9-9278-68D0E5697425","name":"Algorithms for Embeddings and Homotopy Theory","call_identifier":"FWF"}],"quality_controlled":"1","_id":"7991","publication_identifier":{"issn":["18688969"],"isbn":["9783959771436"]}},{"date_updated":"2023-08-17T13:50:44Z","oa":1,"volume":20,"article_processing_charge":"No","arxiv":1,"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","quality_controlled":"1","oa_version":"Preprint","project":[{"call_identifier":"FWF","_id":"26611F5C-B435-11E9-9278-68D0E5697425","name":"Algorithms for Embeddings and Homotopy Theory","grant_number":"P31312"}],"_id":"6563","publication_identifier":{"issn":["16153375"],"eissn":["16153383"]},"publication_status":"published","citation":{"apa":"Filakovský, M., &#38; Vokřínek, L. (2020). Are two given maps homotopic? An algorithmic viewpoint. <i>Foundations of Computational Mathematics</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s10208-019-09419-x\">https://doi.org/10.1007/s10208-019-09419-x</a>","ieee":"M. Filakovský and L. Vokřínek, “Are two given maps homotopic? An algorithmic viewpoint,” <i>Foundations of Computational Mathematics</i>, vol. 20. Springer Nature, pp. 311–330, 2020.","chicago":"Filakovský, Marek, and Lukas Vokřínek. “Are Two given Maps Homotopic? An Algorithmic Viewpoint.” <i>Foundations of Computational Mathematics</i>. Springer Nature, 2020. <a href=\"https://doi.org/10.1007/s10208-019-09419-x\">https://doi.org/10.1007/s10208-019-09419-x</a>.","ama":"Filakovský M, Vokřínek L. Are two given maps homotopic? An algorithmic viewpoint. <i>Foundations of Computational Mathematics</i>. 2020;20:311-330. doi:<a href=\"https://doi.org/10.1007/s10208-019-09419-x\">10.1007/s10208-019-09419-x</a>","mla":"Filakovský, Marek, and Lukas Vokřínek. “Are Two given Maps Homotopic? An Algorithmic Viewpoint.” <i>Foundations of Computational Mathematics</i>, vol. 20, Springer Nature, 2020, pp. 311–30, doi:<a href=\"https://doi.org/10.1007/s10208-019-09419-x\">10.1007/s10208-019-09419-x</a>.","ista":"Filakovský M, Vokřínek L. 2020. Are two given maps homotopic? An algorithmic viewpoint. Foundations of Computational Mathematics. 20, 311–330.","short":"M. Filakovský, L. Vokřínek, Foundations of Computational Mathematics 20 (2020) 311–330."},"author":[{"id":"3E8AF77E-F248-11E8-B48F-1D18A9856A87","full_name":"Filakovský, Marek","last_name":"Filakovský","first_name":"Marek"},{"first_name":"Lukas","full_name":"Vokřínek, Lukas","last_name":"Vokřínek"}],"abstract":[{"lang":"eng","text":"This paper presents two algorithms. The first decides the existence of a pointed homotopy between given simplicial maps 𝑓,𝑔:𝑋→𝑌, and the second computes the group [𝛴𝑋,𝑌]∗ of pointed homotopy classes of maps from a suspension; in both cases, the target Y is assumed simply connected. More generally, these algorithms work relative to 𝐴⊆𝑋."}],"main_file_link":[{"url":"https://arxiv.org/abs/1312.2337","open_access":"1"}],"isi":1,"doi":"10.1007/s10208-019-09419-x","year":"2020","external_id":{"isi":["000522437400004"],"arxiv":["1312.2337"]},"title":"Are two given maps homotopic? An algorithmic viewpoint","page":"311-330","publication":"Foundations of Computational Mathematics","type":"journal_article","day":"01","status":"public","intvolume":"        20","department":[{"_id":"UlWa"}],"date_created":"2019-06-16T21:59:14Z","date_published":"2020-04-01T00:00:00Z","article_type":"original","month":"04","language":[{"iso":"eng"}],"publisher":"Springer Nature","scopus_import":"1"},{"citation":{"chicago":"Avvakumov, Sergey, and Sergey Kudrya. “Vanishing of All Equivariant Obstructions and the Mapping Degree.” <i>ArXiv</i>. arXiv, n.d.","apa":"Avvakumov, S., &#38; Kudrya, S. (n.d.). Vanishing of all equivariant obstructions and the mapping degree. <i>arXiv</i>. arXiv.","ieee":"S. Avvakumov and S. Kudrya, “Vanishing of all equivariant obstructions and the mapping degree,” <i>arXiv</i>. arXiv.","ista":"Avvakumov S, Kudrya S. Vanishing of all equivariant obstructions and the mapping degree. arXiv, 1910.12628.","short":"S. Avvakumov, S. Kudrya, ArXiv (n.d.).","ama":"Avvakumov S, Kudrya S. Vanishing of all equivariant obstructions and the mapping degree. <i>arXiv</i>.","mla":"Avvakumov, Sergey, and Sergey Kudrya. “Vanishing of All Equivariant Obstructions and the Mapping Degree.” <i>ArXiv</i>, 1910.12628, arXiv."},"day":"28","publication_status":"submitted","type":"preprint","abstract":[{"lang":"eng","text":"Suppose that $n\\neq p^k$ and $n\\neq 2p^k$ for all $k$ and all primes $p$. We prove that for any Hausdorff compactum $X$ with a free action of the symmetric group $\\mathfrak S_n$ there exists an $\\mathfrak S_n$-equivariant map $X \\to\r\n{\\mathbb R}^n$ whose image avoids the diagonal $\\{(x,x\\dots,x)\\in {\\mathbb R}^n|x\\in {\\mathbb R}\\}$.\r\n  Previously, the special cases of this statement for certain $X$ were usually proved using the equivartiant obstruction theory. Such calculations are difficult and may become infeasible past the first (primary) obstruction. We\r\ntake a different approach which allows us to prove the vanishing of all obstructions simultaneously. The essential step in the proof is classifying the possible degrees of $\\mathfrak S_n$-equivariant maps from the boundary\r\n$\\partial\\Delta^{n-1}$ of $(n-1)$-simplex to itself.  Existence of equivariant maps between spaces is important for many questions arising from discrete mathematics and geometry, such as Kneser's conjecture, the Square Peg conjecture, the Splitting Necklace problem, and the Topological Tverberg conjecture, etc. We demonstrate the utility of our result  applying it to one such question, a specific instance of envy-free division problem."}],"status":"public","author":[{"full_name":"Avvakumov, Sergey","last_name":"Avvakumov","first_name":"Sergey","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87"},{"id":"ecf01965-d252-11ea-95a5-8ada5f6c6a67","first_name":"Sergey","full_name":"Kudrya, Sergey","last_name":"Kudrya"}],"publication":"arXiv","arxiv":1,"article_processing_charge":"No","date_updated":"2023-09-07T13:12:17Z","oa":1,"_id":"8182","project":[{"grant_number":"P31312","call_identifier":"FWF","_id":"26611F5C-B435-11E9-9278-68D0E5697425","name":"Algorithms for Embeddings and Homotopy Theory"}],"oa_version":"Preprint","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"10","year":"2019","date_published":"2019-10-28T00:00:00Z","external_id":{"arxiv":["1910.12628"]},"title":"Vanishing of all equivariant obstructions and the mapping degree","publisher":"arXiv","language":[{"iso":"eng"}],"article_number":"1910.12628","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1910.12628"}],"department":[{"_id":"UlWa"}],"related_material":{"record":[{"relation":"later_version","id":"11446","status":"public"},{"status":"public","id":"8156","relation":"dissertation_contains"}]},"date_created":"2020-07-30T10:45:08Z"},{"citation":{"mla":"Avvakumov, Sergey, et al. “Stronger Counterexamples to the Topological Tverberg Conjecture.” <i>ArXiv</i>, 1908.08731, arXiv.","ama":"Avvakumov S, Karasev R, Skopenkov A. Stronger counterexamples to the topological Tverberg conjecture. <i>arXiv</i>.","ista":"Avvakumov S, Karasev R, Skopenkov A. Stronger counterexamples to the topological Tverberg conjecture. arXiv, 1908.08731.","short":"S. Avvakumov, R. Karasev, A. Skopenkov, ArXiv (n.d.).","ieee":"S. Avvakumov, R. Karasev, and A. Skopenkov, “Stronger counterexamples to the topological Tverberg conjecture,” <i>arXiv</i>. arXiv.","apa":"Avvakumov, S., Karasev, R., &#38; Skopenkov, A. (n.d.). Stronger counterexamples to the topological Tverberg conjecture. <i>arXiv</i>. arXiv.","chicago":"Avvakumov, Sergey, R. Karasev, and A. Skopenkov. “Stronger Counterexamples to the Topological Tverberg Conjecture.” <i>ArXiv</i>. arXiv, n.d."},"day":"23","publication_status":"submitted","type":"preprint","abstract":[{"lang":"eng","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. "}],"author":[{"id":"3827DAC8-F248-11E8-B48F-1D18A9856A87","first_name":"Sergey","full_name":"Avvakumov, Sergey","last_name":"Avvakumov"},{"first_name":"R.","full_name":"Karasev, R.","last_name":"Karasev"},{"full_name":"Skopenkov, A.","last_name":"Skopenkov","first_name":"A."}],"status":"public","publication":"arXiv","arxiv":1,"article_processing_charge":"No","oa":1,"date_updated":"2023-09-08T11:20:02Z","_id":"8184","project":[{"grant_number":"P31312","call_identifier":"FWF","name":"Algorithms for Embeddings and Homotopy Theory","_id":"26611F5C-B435-11E9-9278-68D0E5697425"}],"oa_version":"Preprint","acknowledgement":"We would like to thank F. Frick for helpful discussions","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","year":"2019","month":"08","date_published":"2019-08-23T00:00:00Z","title":"Stronger counterexamples to the topological Tverberg conjecture","external_id":{"arxiv":["1908.08731"],"isi":["000986519600004"]},"publisher":"arXiv","language":[{"iso":"eng"}],"main_file_link":[{"url":"https://arxiv.org/abs/1908.08731","open_access":"1"}],"article_number":"1908.08731","isi":1,"department":[{"_id":"UlWa"}],"related_material":{"record":[{"status":"public","id":"8156","relation":"dissertation_contains"}]},"date_created":"2020-07-30T10:45:34Z"},{"related_material":{"link":[{"relation":"later_version","url":"https://doi.org/10.1112/mtk.12059"}],"record":[{"id":"8156","relation":"dissertation_contains","status":"public"}]},"date_created":"2020-07-30T10:45:51Z","department":[{"_id":"UlWa"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1907.11183"}],"article_number":"1907.11183","external_id":{"arxiv":["1907.11183"]},"title":"Envy-free division using mapping degree","language":[{"iso":"eng"}],"year":"2019","doi":"10.48550/arXiv.1907.11183","month":"07","date_published":"2019-07-25T00:00:00Z","_id":"8185","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","project":[{"_id":"26611F5C-B435-11E9-9278-68D0E5697425","name":"Algorithms for Embeddings and Homotopy Theory","call_identifier":"FWF","grant_number":"P31312"}],"oa_version":"Preprint","arxiv":1,"publication":"arXiv","date_updated":"2023-09-07T13:12:17Z","oa":1,"article_processing_charge":"No","abstract":[{"text":"In this paper we study envy-free division problems. The classical approach to some of such problems, used by David Gale, reduces to considering continuous maps of a simplex to itself and finding sufficient conditions when this map hits the center of the simplex. The mere continuity is not sufficient for such a conclusion, the usual assumption (for example, in the Knaster--Kuratowski--Mazurkiewicz and the Gale theorem) is a certain boundary condition.\r\n  We follow Erel Segal-Halevi, Fr\\'ed\\'eric Meunier, and Shira Zerbib, and replace the boundary condition by another assumption, which has the economic meaning of possibility for a player to prefer an empty part in the segment\r\npartition problem. We solve the problem positively when $n$, the number of players that divide the segment, is a prime power, and we provide counterexamples for every $n$ which is not a prime power. We also provide counterexamples relevant to a wider class of fair or envy-free partition problems when $n$ is odd and not a prime power.","lang":"eng"}],"author":[{"full_name":"Avvakumov, Sergey","last_name":"Avvakumov","first_name":"Sergey","id":"3827DAC8-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Roman","last_name":"Karasev","full_name":"Karasev, Roman"}],"status":"public","publication_status":"submitted","citation":{"ista":"Avvakumov S, Karasev R. Envy-free division using mapping degree. arXiv, 1907.11183.","short":"S. Avvakumov, R. Karasev, ArXiv (n.d.).","mla":"Avvakumov, Sergey, and Roman Karasev. “Envy-Free Division Using Mapping Degree.” <i>ArXiv</i>, 1907.11183, doi:<a href=\"https://doi.org/10.48550/arXiv.1907.11183\">10.48550/arXiv.1907.11183</a>.","ama":"Avvakumov S, Karasev R. Envy-free division using mapping degree. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.1907.11183\">10.48550/arXiv.1907.11183</a>","chicago":"Avvakumov, Sergey, and Roman Karasev. “Envy-Free Division Using Mapping Degree.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.1907.11183\">https://doi.org/10.48550/arXiv.1907.11183</a>.","ieee":"S. Avvakumov and R. Karasev, “Envy-free division using mapping degree,” <i>arXiv</i>. .","apa":"Avvakumov, S., &#38; Karasev, R. (n.d.). Envy-free division using mapping degree. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.1907.11183\">https://doi.org/10.48550/arXiv.1907.11183</a>"},"day":"25","type":"preprint"}]