[{"author":[{"full_name":"Zhechev, Stephan Y","first_name":"Stephan Y","id":"3AA52972-F248-11E8-B48F-1D18A9856A87","last_name":"Zhechev"}],"ddc":["514"],"department":[{"_id":"UlWa"}],"_id":"6681","language":[{"iso":"eng"}],"page":"104","supervisor":[{"orcid":"0000-0002-1494-0568","last_name":"Wagner","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","full_name":"Wagner, Uli","first_name":"Uli"}],"file_date_updated":"2019-08-08T06:41:37Z","year":"2019","day":"08","alternative_title":["IST Austria Thesis"],"accept":"1","publisher":"IST Austria","type":"dissertation","publication_identifier":{"issn":["2663-337X"]},"date_created":"2019-07-26T11:14:34Z","publication_status":"published","file":[{"file_id":"6771","date_updated":"2019-08-07T13:02:50Z","relation":"main_file","access_level":"open_access","request_a_copy":0,"date_created":"2019-08-07T13:02:50Z","creator":"szhechev","file_size":1464227,"open_access":1,"file_name":"Stephan_Zhechev_thesis.pdf","content_type":"application/pdf"},{"request_a_copy":0,"date_updated":"2019-08-08T06:41:37Z","relation":"source_file","access_level":"closed","file_id":"6772","file_name":"Stephan_Zhechev_thesis.tex","content_type":"application/octet-stream","creator":"szhechev","date_created":"2019-08-07T13:03:22Z","open_access":0,"file_size":303988},{"date_created":"2019-08-07T13:03:34Z","creator":"szhechev","file_size":1087004,"open_access":0,"content_type":"application/zip","file_name":"supplementary_material.zip","file_id":"6773","date_updated":"2019-08-08T06:41:37Z","relation":"supplementary_material","access_level":"closed","request_a_copy":0}],"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."}],"date_updated":"2019-12-03T15:21:55Z","related_material":{"record":[{"id":"6774","relation":"part_of_dissertation","status":"public"}]},"month":"08","oa_version":"Published Version","cc_license":"'https://creativecommons.org/licenses/by/4.0/'","status":"public","doi":"10.15479/AT:ISTA:6681","date_published":"2019-08-08T00:00:00Z","title":"Algorithmic aspects of homotopy theory and embeddability","citation":{"short":"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","ieee":"S. Y. Zhechev, *Algorithmic aspects of homotopy theory and embeddability*. IST Austria, 2019.","apa":"Zhechev, S. Y. (2019). *Algorithmic aspects of homotopy theory and embeddability*. IST Austria. https://doi.org/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."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"}]