{"article_processing_charge":"No","related_material":{"record":[{"id":"6774","status":"public","relation":"part_of_dissertation"}]},"citation":{"ista":"Zhechev SY. 2019. Algorithmic aspects of homotopy theory and embeddability. Institute of Science and Technology Austria.","mla":"Zhechev, Stephan Y. <i>Algorithmic Aspects of Homotopy Theory and Embeddability</i>. Institute of Science and Technology Austria, 2019, doi:<a href=\"https://doi.org/10.15479/AT:ISTA:6681\">10.15479/AT:ISTA:6681</a>.","short":"S.Y. Zhechev, Algorithmic Aspects of Homotopy Theory and Embeddability, Institute of Science and Technology Austria, 2019.","ama":"Zhechev SY. Algorithmic aspects of homotopy theory and embeddability. 2019. doi:<a href=\"https://doi.org/10.15479/AT:ISTA:6681\">10.15479/AT:ISTA:6681</a>","apa":"Zhechev, S. Y. (2019). <i>Algorithmic aspects of homotopy theory and embeddability</i>. Institute of Science and Technology Austria. <a href=\"https://doi.org/10.15479/AT:ISTA:6681\">https://doi.org/10.15479/AT:ISTA:6681</a>","chicago":"Zhechev, Stephan Y. “Algorithmic Aspects of Homotopy Theory and Embeddability.” Institute of Science and Technology Austria, 2019. <a href=\"https://doi.org/10.15479/AT:ISTA:6681\">https://doi.org/10.15479/AT:ISTA:6681</a>.","ieee":"S. Y. Zhechev, “Algorithmic aspects of homotopy theory and embeddability,” Institute of Science and Technology Austria, 2019."},"title":"Algorithmic aspects of homotopy theory and embeddability","ddc":["514"],"publication_identifier":{"issn":["2663-337X"]},"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","short":"CC BY (4.0)","image":"/images/cc_by.png"},"date_published":"2019-08-08T00:00:00Z","oa":1,"abstract":[{"text":"The first part of the thesis considers the computational aspects of the homotopy groups πd(X) of a topological space X. 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.\r\nHowever, 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). We present an algorithm that, given a simply connected space X, computes πd(X) and represents its elements as simplicial maps from suitable triangulations 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,\r\nwe 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 S d on which the map is defined, is exponential in size(X).\r\nIn the second part of the thesis, we prove that the following question is algorithmically undecidable for d < ⌊3(k+1)/2⌋, k ≥ 5 and (k, d) ̸= (5, 7), which covers essentially everything outside the meta-stable range: Given a finite simplicial complex K of dimension k, decide whether there exists a piecewise-linear (i.e., linear on an arbitrarily fine subdivision of K) embedding f : K ↪→ Rd of K into a d-dimensional Euclidean space.","lang":"eng"}],"year":"2019","day":"08","language":[{"iso":"eng"}],"author":[{"last_name":"Zhechev","first_name":"Stephan Y","id":"3AA52972-F248-11E8-B48F-1D18A9856A87","full_name":"Zhechev, Stephan Y"}],"page":"104","file_date_updated":"2020-07-14T12:47:37Z","doi":"10.15479/AT:ISTA:6681","department":[{"_id":"UlWa"}],"oa_version":"Published Version","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","type":"dissertation","date_updated":"2023-09-07T13:10:36Z","status":"public","degree_awarded":"PhD","_id":"6681","file":[{"date_updated":"2020-07-14T12:47:37Z","creator":"szhechev","access_level":"open_access","file_size":1464227,"date_created":"2019-08-07T13:02:50Z","file_name":"Stephan_Zhechev_thesis.pdf","checksum":"3231e7cbfca3b5687366f84f0a57a0c0","content_type":"application/pdf","relation":"main_file","file_id":"6771"},{"file_name":"Stephan_Zhechev_thesis.tex","checksum":"85d65eb27b4377a9e332ee37a70f08b6","file_id":"6772","relation":"source_file","content_type":"application/octet-stream","file_size":303988,"date_updated":"2020-07-14T12:47:37Z","access_level":"closed","creator":"szhechev","date_created":"2019-08-07T13:03:22Z"},{"content_type":"application/zip","file_id":"6773","relation":"supplementary_material","checksum":"86b374d264ca2dd53e712728e253ee75","file_name":"supplementary_material.zip","date_created":"2019-08-07T13:03:34Z","access_level":"closed","creator":"szhechev","date_updated":"2020-07-14T12:47:37Z","file_size":1087004}],"month":"08","date_created":"2019-07-26T11:14:34Z","publication_status":"published","alternative_title":["ISTA Thesis"],"publisher":"Institute of Science and Technology Austria","supervisor":[{"full_name":"Wagner, Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","first_name":"Uli","orcid":"0000-0002-1494-0568","last_name":"Wagner"}]}