[{"ddc":["514"],"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"file_date_updated":"2020-07-14T12:47:37Z","doi":"10.15479/AT:ISTA:6681","alternative_title":["IST Austria Thesis"],"date_updated":"2020-07-14T23:11:15Z","oa_version":"Published Version","status":"public","date_published":"2019-08-08T00:00:00Z","publication_identifier":{"issn":["2663-337X"]},"publisher":"IST Austria","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","type":"dissertation","language":[{"iso":"eng"}],"month":"08","publication_status":"published","supervisor":[{"orcid":"0000-0002-1494-0568","full_name":"Wagner, Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","last_name":"Wagner","first_name":"Uli"}],"oa":1,"has_accepted_license":"1","page":"104","year":"2019","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."}],"title":"Algorithmic aspects of homotopy theory and embeddability","date_created":"2019-07-26T11:14:34Z","_id":"6681","author":[{"full_name":"Zhechev, Stephan Y","first_name":"Stephan Y","id":"3AA52972-F248-11E8-B48F-1D18A9856A87","last_name":"Zhechev"}],"day":"08","citation":{"apa":"Zhechev, S. Y. (2019). *Algorithmic aspects of homotopy theory and embeddability*. IST Austria. https://doi.org/10.15479/AT:ISTA:6681","short":"S.Y. Zhechev, Algorithmic Aspects of Homotopy Theory and Embeddability, IST Austria, 2019.","ieee":"S. Y. Zhechev, *Algorithmic aspects of homotopy theory and embeddability*. IST Austria, 2019.","chicago":"Zhechev, Stephan Y. *Algorithmic Aspects of Homotopy Theory and Embeddability*. IST Austria, 2019. https://doi.org/10.15479/AT:ISTA:6681.","ama":"Zhechev SY. *Algorithmic Aspects of Homotopy Theory and Embeddability*. IST Austria; 2019. doi:10.15479/AT:ISTA:6681","mla":"Zhechev, Stephan Y. *Algorithmic Aspects of Homotopy Theory and Embeddability*. IST Austria, 2019, doi:10.15479/AT:ISTA:6681.","ista":"Zhechev SY. 2019. Algorithmic aspects of homotopy theory and embeddability, IST Austria, 104p."},"file":[{"date_created":"2019-08-07T13:02:50Z","file_id":"6771","content_type":"application/pdf","date_updated":"2020-07-14T12:47:37Z","relation":"main_file","access_level":"open_access","file_size":1464227,"creator":"szhechev","file_name":"Stephan_Zhechev_thesis.pdf","checksum":"3231e7cbfca3b5687366f84f0a57a0c0"},{"content_type":"application/octet-stream","date_created":"2019-08-07T13:03:22Z","file_id":"6772","date_updated":"2020-07-14T12:47:37Z","relation":"source_file","file_size":303988,"creator":"szhechev","checksum":"85d65eb27b4377a9e332ee37a70f08b6","file_name":"Stephan_Zhechev_thesis.tex","access_level":"closed"},{"relation":"supplementary_material","access_level":"closed","file_name":"supplementary_material.zip","checksum":"86b374d264ca2dd53e712728e253ee75","creator":"szhechev","file_size":1087004,"file_id":"6773","date_created":"2019-08-07T13:03:34Z","content_type":"application/zip","date_updated":"2020-07-14T12:47:37Z"}],"department":[{"_id":"UlWa"}],"related_material":{"record":[{"relation":"part_of_dissertation","id":"6774","status":"public"}]}}]