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