{"citation":{"ieee":"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>","ista":"Zhechev SY. 2019. Algorithmic aspects of homotopy theory and embeddability. Institute of Science and Technology Austria.","short":"S.Y. Zhechev, Algorithmic Aspects of Homotopy Theory and Embeddability, Institute of Science and Technology Austria, 2019.","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>.","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>.","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>"},"supervisor":[{"first_name":"Uli","orcid":"0000-0002-1494-0568","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli","last_name":"Wagner"}],"type":"dissertation","date_created":"2019-07-26T11:14:34Z","has_accepted_license":"1","article_processing_charge":"No","date_updated":"2023-09-07T13:10:36Z","month":"08","title":"Algorithmic aspects of homotopy theory and embeddability","abstract":[{"lang":"eng","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."}],"year":"2019","ddc":["514"],"publication_identifier":{"issn":["2663-337X"]},"publisher":"Institute of Science and Technology Austria","status":"public","degree_awarded":"PhD","oa_version":"Published Version","department":[{"_id":"UlWa"}],"date_published":"2019-08-08T00:00:00Z","publication_status":"published","day":"08","author":[{"first_name":"Stephan Y","full_name":"Zhechev, Stephan Y","last_name":"Zhechev","id":"3AA52972-F248-11E8-B48F-1D18A9856A87"}],"doi":"10.15479/AT:ISTA:6681","alternative_title":["ISTA Thesis"],"_id":"6681","oa":1,"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","image":"/images/cc_by.png"},"related_material":{"record":[{"status":"public","id":"6774","relation":"part_of_dissertation"}]},"language":[{"iso":"eng"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","file_date_updated":"2020-07-14T12:47:37Z","page":"104","file":[{"file_name":"Stephan_Zhechev_thesis.pdf","access_level":"open_access","relation":"main_file","file_id":"6771","checksum":"3231e7cbfca3b5687366f84f0a57a0c0","date_created":"2019-08-07T13:02:50Z","content_type":"application/pdf","creator":"szhechev","file_size":1464227,"date_updated":"2020-07-14T12:47:37Z"},{"date_updated":"2020-07-14T12:47:37Z","file_size":303988,"creator":"szhechev","content_type":"application/octet-stream","date_created":"2019-08-07T13:03:22Z","checksum":"85d65eb27b4377a9e332ee37a70f08b6","file_id":"6772","relation":"source_file","access_level":"closed","file_name":"Stephan_Zhechev_thesis.tex"},{"file_id":"6773","checksum":"86b374d264ca2dd53e712728e253ee75","date_created":"2019-08-07T13:03:34Z","file_name":"supplementary_material.zip","relation":"supplementary_material","access_level":"closed","file_size":1087004,"date_updated":"2020-07-14T12:47:37Z","content_type":"application/zip","creator":"szhechev"}]}