--- _id: '6774' abstract: - lang: eng text: "A central problem of algebraic topology is to understand the homotopy groups \ \U0001D70B\U0001D451(\U0001D44B) 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 \U0001D70B1(\U0001D44B) 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 \U0001D70B1(\U0001D44B) \ trivial), compute the higher homotopy group \U0001D70B\U0001D451(\U0001D44B) \ for any given \U0001D451≥2 . However, these algorithms come with a caveat: They compute the isomorphism type of \U0001D70B\U0001D451(\U0001D44B) , \U0001D451≥2 \ as an abstract finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of \U0001D70B\U0001D451(\U0001D44B) . Converting elements of this abstract group into explicit geometric maps from the d-dimensional sphere \U0001D446\U0001D451 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 \U0001D70B\U0001D451(\U0001D44B) \ and represents its elements as simplicial maps from a suitable triangulation of the d-sphere \U0001D446\U0001D451 to X. For fixed d, the algorithm runs in time exponential in size(\U0001D44B) , the number of simplices of X. Moreover, we prove that this is optimal: For every fixed \U0001D451≥2 , we construct a family of simply connected spaces X such that for any simplicial map representing a generator of \U0001D70B\U0001D451(\U0001D44B) , the size of the triangulation of \U0001D446\U0001D451 on which the map is defined, is exponential in size(\U0001D44B) ." article_type: original author: - first_name: Marek full_name: Filakovský, Marek id: 3E8AF77E-F248-11E8-B48F-1D18A9856A87 last_name: Filakovský - first_name: Peter full_name: Franek, Peter id: 473294AE-F248-11E8-B48F-1D18A9856A87 last_name: Franek orcid: 0000-0001-8878-8397 - first_name: Uli full_name: Wagner, Uli id: 36690CA2-F248-11E8-B48F-1D18A9856A87 last_name: Wagner orcid: 0000-0002-1494-0568 - first_name: Stephan Y full_name: Zhechev, Stephan Y id: 3AA52972-F248-11E8-B48F-1D18A9856A87 last_name: Zhechev citation: ama: Filakovský M, Franek P, Wagner U, Zhechev SY. Computing simplicial representatives of homotopy group elements. Journal of Applied and Computational Topology. 2018;2(3-4):177-231. doi:10.1007/s41468-018-0021-5 apa: Filakovský, M., Franek, P., Wagner, U., & Zhechev, S. Y. (2018). Computing simplicial representatives of homotopy group elements. Journal of Applied and Computational Topology. Springer. https://doi.org/10.1007/s41468-018-0021-5 chicago: Filakovský, Marek, Peter Franek, Uli Wagner, and Stephan Y Zhechev. “Computing Simplicial Representatives of Homotopy Group Elements.” Journal of Applied and Computational Topology. Springer, 2018. https://doi.org/10.1007/s41468-018-0021-5. ieee: M. Filakovský, P. Franek, U. Wagner, and S. Y. Zhechev, “Computing simplicial representatives of homotopy group elements,” Journal of Applied and Computational Topology, vol. 2, no. 3–4. Springer, pp. 177–231, 2018. 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.” Journal of Applied and Computational Topology, vol. 2, no. 3–4, Springer, 2018, pp. 177–231, doi:10.1007/s41468-018-0021-5. short: M. Filakovský, P. Franek, U. Wagner, S.Y. Zhechev, Journal of Applied and Computational Topology 2 (2018) 177–231. date_created: 2019-08-08T06:47:40Z date_published: 2018-12-01T00:00:00Z date_updated: 2023-09-07T13:10:36Z day: '01' ddc: - '514' department: - _id: UlWa doi: 10.1007/s41468-018-0021-5 file: - access_level: open_access checksum: cf9e7fcd2a113dd4828774fc75cdb7e8 content_type: application/pdf creator: dernst date_created: 2019-08-08T06:55:21Z date_updated: 2020-07-14T12:47:40Z file_id: '6775' file_name: 2018_JourAppliedComputTopology_Filakovsky.pdf file_size: 1056278 relation: main_file file_date_updated: 2020-07-14T12:47:40Z has_accepted_license: '1' intvolume: ' 2' issue: 3-4 language: - iso: eng month: '12' oa: 1 oa_version: Published Version page: 177-231 project: - _id: 25F8B9BC-B435-11E9-9278-68D0E5697425 call_identifier: FWF grant_number: M01980 name: Robust invariants of Nonlinear Systems - _id: 3AC91DDA-15DF-11EA-824D-93A3E7B544D1 call_identifier: FWF name: FWF Open Access Fund publication: Journal of Applied and Computational Topology publication_identifier: eissn: - 2367-1734 issn: - 2367-1726 publication_status: published publisher: Springer quality_controlled: '1' related_material: record: - id: '6681' relation: dissertation_contains status: public status: public title: Computing simplicial representatives of homotopy group elements tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 2 year: '2018' ...