{"date_updated":"2021-01-12T08:08:58Z","title":"Computing simplicial representatives of homotopy group elements","date_published":"2018-12-01T00:00:00Z","quality_controlled":"1","volume":2,"file":[{"access_level":"open_access","file_name":"2018_JourAppliedComputTopology_Filakovsky.pdf","creator":"dernst","date_created":"2019-08-08T06:55:21Z","date_updated":"2020-07-14T12:47:40Z","file_size":1056278,"checksum":"cf9e7fcd2a113dd4828774fc75cdb7e8","content_type":"application/pdf","file_id":"6775","relation":"main_file"}],"_id":"6774","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","issue":"3-4","publication_identifier":{"eissn":["2367-1734"],"issn":["2367-1726"]},"page":"177-231","doi":"10.1007/s41468-018-0021-5","month":"12","publication":"Journal of Applied and Computational Topology","oa":1,"date_created":"2019-08-08T06:47:40Z","ddc":["514"],"license":"https://creativecommons.org/licenses/by/4.0/","abstract":[{"lang":"eng","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) ."}],"oa_version":"Published Version","related_material":{"record":[{"id":"6681","relation":"dissertation_contains","status":"public"}]},"file_date_updated":"2020-07-14T12:47:40Z","department":[{"_id":"UlWa"}],"article_type":"original","citation":{"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>.","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.","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>","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>","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.","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>."},"publication_status":"published","intvolume":"         2","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","short":"CC BY (4.0)","image":"/images/cc_by.png"},"type":"journal_article","status":"public","language":[{"iso":"eng"}],"publisher":"Springer","author":[{"last_name":"Filakovský","full_name":"Filakovský, Marek","id":"3E8AF77E-F248-11E8-B48F-1D18A9856A87","first_name":"Marek"},{"first_name":"Peter","full_name":"Franek, Peter","last_name":"Franek","id":"473294AE-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-1494-0568","first_name":"Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","last_name":"Wagner","full_name":"Wagner, Uli"},{"last_name":"Zhechev","full_name":"Zhechev, Stephan Y","id":"3AA52972-F248-11E8-B48F-1D18A9856A87","first_name":"Stephan Y"}],"day":"01","has_accepted_license":"1","year":"2018","project":[{"call_identifier":"FWF","_id":"25F8B9BC-B435-11E9-9278-68D0E5697425","grant_number":"M01980","name":"Robust invariants of Nonlinear Systems"},{"name":"FWF Open Access Fund","call_identifier":"FWF","_id":"3AC91DDA-15DF-11EA-824D-93A3E7B544D1"}]}