[{"date_created":"2020-05-10T22:00:48Z","volume":"2020-January","title":"Embeddability of simplicial complexes is undecidable","oa_version":"Published Version","day":"01","scopus_import":1,"author":[{"first_name":"Marek","full_name":"Filakovský, Marek","id":"3E8AF77E-F248-11E8-B48F-1D18A9856A87","last_name":"Filakovský"},{"last_name":"Wagner","full_name":"Wagner, Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-1494-0568","first_name":"Uli"},{"full_name":"Zhechev, Stephan Y","id":"3AA52972-F248-11E8-B48F-1D18A9856A87","last_name":"Zhechev","first_name":"Stephan Y"}],"publication_status":"published","publication_identifier":{"isbn":["9781611975994"]},"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"}],"department":[{"_id":"UlWa"}],"month":"01","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"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>.","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.","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>.","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>","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>","short":"M. Filakovský, U. Wagner, S.Y. Zhechev, in:, Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2020, pp. 767–785.","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."},"language":[{"iso":"eng"}],"oa":1,"type":"conference","_id":"7806","date_updated":"2021-01-12T08:15:38Z","publisher":"SIAM","article_processing_charge":"No","doi":"10.1137/1.9781611975994.47","quality_controlled":"1","main_file_link":[{"open_access":"1","url":"https://doi.org/10.1137/1.9781611975994.47"}],"page":"767-785","year":"2020","conference":{"location":"Salt Lake City, UT, United States","start_date":"2020-01-05","end_date":"2020-01-08","name":"SODA: Symposium on Discrete Algorithms"},"date_published":"2020-01-01T00:00:00Z","status":"public","publication":"Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms","project":[{"call_identifier":"FWF","grant_number":"P31312","name":"Algorithms for Embeddings and Homotopy Theory","_id":"26611F5C-B435-11E9-9278-68D0E5697425"}]},{"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","citation":{"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>.","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>","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>.","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.","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.","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>"},"language":[{"iso":"eng"}],"oa":1,"department":[{"_id":"UlWa"}],"month":"04","arxiv":1,"publication_identifier":{"issn":["16153375"],"eissn":["16153383"]},"publication_status":"published","abstract":[{"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.","lang":"eng"}],"intvolume":"        20","article_type":"original","date_created":"2019-06-16T21:59:14Z","volume":20,"title":"Are two given maps homotopic? An algorithmic viewpoint","oa_version":"Preprint","author":[{"full_name":"Filakovský, Marek","id":"3E8AF77E-F248-11E8-B48F-1D18A9856A87","last_name":"Filakovský","first_name":"Marek"},{"last_name":"Vokřínek","full_name":"Vokřínek, Lukas","first_name":"Lukas"}],"scopus_import":"1","day":"01","date_published":"2020-04-01T00:00:00Z","project":[{"grant_number":"P31312","name":"Algorithms for Embeddings and Homotopy Theory","call_identifier":"FWF","_id":"26611F5C-B435-11E9-9278-68D0E5697425"}],"status":"public","publication":"Foundations of Computational Mathematics","external_id":{"arxiv":["1312.2337"],"isi":["000522437400004"]},"isi":1,"year":"2020","quality_controlled":"1","main_file_link":[{"url":"https://arxiv.org/abs/1312.2337","open_access":"1"}],"page":"311-330","type":"journal_article","date_updated":"2023-08-17T13:50:44Z","_id":"6563","publisher":"Springer Nature","doi":"10.1007/s10208-019-09419-x","article_processing_charge":"No"},{"project":[{"_id":"25F8B9BC-B435-11E9-9278-68D0E5697425","call_identifier":"FWF","name":"Robust invariants of Nonlinear Systems","grant_number":"M01980"},{"name":"FWF Open Access Fund","call_identifier":"FWF","_id":"3AC91DDA-15DF-11EA-824D-93A3E7B544D1"}],"publication":"Journal of Applied and Computational Topology","status":"public","date_published":"2018-12-01T00:00:00Z","year":"2018","related_material":{"record":[{"id":"6681","relation":"dissertation_contains","status":"public"}]},"page":"177-231","ddc":["514"],"quality_controlled":"1","doi":"10.1007/s41468-018-0021-5","publisher":"Springer","date_updated":"2023-09-07T13:10:36Z","_id":"6774","type":"journal_article","oa":1,"language":[{"iso":"eng"}],"issue":"3-4","citation":{"ama":"Filakovský M, Franek P, Wagner U, Zhechev SY. Computing simplicial representatives of homotopy group elements. <i>Journal of Applied and Computational Topology</i>. 2018;2(3-4):177-231. doi:<a href=\"https://doi.org/10.1007/s41468-018-0021-5\">10.1007/s41468-018-0021-5</a>","short":"M. Filakovský, P. Franek, U. Wagner, S.Y. Zhechev, Journal of Applied and Computational Topology 2 (2018) 177–231.","ieee":"M. Filakovský, P. Franek, U. Wagner, and S. Y. Zhechev, “Computing simplicial representatives of homotopy group elements,” <i>Journal of Applied and Computational Topology</i>, vol. 2, no. 3–4. Springer, pp. 177–231, 2018.","chicago":"Filakovský, Marek, Peter Franek, Uli Wagner, and Stephan Y Zhechev. “Computing Simplicial Representatives of Homotopy Group Elements.” <i>Journal of Applied and Computational Topology</i>. Springer, 2018. <a href=\"https://doi.org/10.1007/s41468-018-0021-5\">https://doi.org/10.1007/s41468-018-0021-5</a>.","ista":"Filakovský M, Franek P, Wagner U, Zhechev SY. 2018. Computing simplicial representatives of homotopy group elements. Journal of Applied and Computational Topology. 2(3–4), 177–231.","mla":"Filakovský, Marek, et al. “Computing Simplicial Representatives of Homotopy Group Elements.” <i>Journal of Applied and Computational Topology</i>, vol. 2, no. 3–4, Springer, 2018, pp. 177–231, doi:<a href=\"https://doi.org/10.1007/s41468-018-0021-5\">10.1007/s41468-018-0021-5</a>.","apa":"Filakovský, M., Franek, P., Wagner, U., &#38; Zhechev, S. Y. (2018). Computing simplicial representatives of homotopy group elements. <i>Journal of Applied and Computational Topology</i>. Springer. <a href=\"https://doi.org/10.1007/s41468-018-0021-5\">https://doi.org/10.1007/s41468-018-0021-5</a>"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"12","department":[{"_id":"UlWa"}],"file":[{"relation":"main_file","checksum":"cf9e7fcd2a113dd4828774fc75cdb7e8","file_name":"2018_JourAppliedComputTopology_Filakovsky.pdf","access_level":"open_access","content_type":"application/pdf","file_id":"6775","file_size":1056278,"date_created":"2019-08-08T06:55:21Z","date_updated":"2020-07-14T12:47:40Z","creator":"dernst"}],"has_accepted_license":"1","tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","short":"CC BY (4.0)"},"intvolume":"         2","abstract":[{"text":"A central problem of algebraic topology is to understand the homotopy groups  𝜋𝑑(𝑋)  of a topological space X. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group  𝜋1(𝑋)  of a given finite simplicial complex X is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex X that is simply connected (i.e., with   𝜋1(𝑋)  trivial), compute the higher homotopy group   𝜋𝑑(𝑋)  for any given   𝑑≥2 . However, these algorithms come with a caveat: They compute the isomorphism type of   𝜋𝑑(𝑋) ,   𝑑≥2  as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of   𝜋𝑑(𝑋) . Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere   𝑆𝑑  to X has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a simply connected space X, computes   𝜋𝑑(𝑋)  and represents its elements as simplicial maps from a suitable triangulation of the d-sphere   𝑆𝑑  to X. For fixed d, the algorithm runs in time exponential in   size(𝑋) , the number of simplices of X. Moreover, we prove that this is optimal: For every fixed   𝑑≥2 , we construct a family of simply connected spaces X such that for any simplicial map representing a generator of   𝜋𝑑(𝑋) , the size of the triangulation of   𝑆𝑑  on which the map is defined, is exponential in size(𝑋) .","lang":"eng"}],"publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"publication_status":"published","file_date_updated":"2020-07-14T12:47:40Z","author":[{"first_name":"Marek","last_name":"Filakovský","id":"3E8AF77E-F248-11E8-B48F-1D18A9856A87","full_name":"Filakovský, Marek"},{"full_name":"Franek, Peter","id":"473294AE-F248-11E8-B48F-1D18A9856A87","last_name":"Franek","first_name":"Peter","orcid":"0000-0001-8878-8397"},{"last_name":"Wagner","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli","first_name":"Uli","orcid":"0000-0002-1494-0568"},{"full_name":"Zhechev, Stephan Y","id":"3AA52972-F248-11E8-B48F-1D18A9856A87","last_name":"Zhechev","first_name":"Stephan Y"}],"day":"01","title":"Computing simplicial representatives of homotopy group elements","oa_version":"Published Version","volume":2,"article_type":"original","date_created":"2019-08-08T06:47:40Z"}]