[{"title":"Algorithmic aspects of homotopy theory and embeddability","language":[{"iso":"eng"}],"_id":"6681","date_updated":"2020-01-21T13:21:43Z","date_created":"2019-07-26T11:14:34Z","creator":{"login":"szhechev","id":"3AA52972-F248-11E8-B48F-1D18A9856A87"},"page":"104","related_material":{"record":[{"status":"public","relation":"part_of_dissertation","id":"6774"}]},"status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","citation":{"chicago":"Zhechev, Stephan Y. *Algorithmic Aspects of Homotopy Theory and Embeddability*. IST Austria, 2019. https://doi.org/10.15479/AT:ISTA:6681.","ista":"Zhechev SY. 2019. Algorithmic aspects of homotopy theory and embeddability, IST Austria, 104p.","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","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.","short":"S.Y. Zhechev, Algorithmic Aspects of Homotopy Theory and Embeddability, IST Austria, 2019."},"accept":"1","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_published":"2019-08-08T00:00:00Z","alternative_title":["IST Austria Thesis"],"publisher":"IST Austria","supervisor":[{"last_name":"Wagner","orcid":"0000-0002-1494-0568","id":"36690CA2-F248-11E8-B48F-1D18A9856A87","first_name":"Uli","full_name":"Wagner, Uli"}],"day":"08","cc_license":"cc_by","ddc":["514"],"file_date_updated":"2019-08-08T06:41:37Z","author":[{"full_name":"Zhechev, Stephan Y","last_name":"Zhechev","id":"3AA52972-F248-11E8-B48F-1D18A9856A87","first_name":"Stephan Y"}],"publication_identifier":{"issn":["2663-337X"]},"oa_version":"Published Version","_version":30,"month":"08","department":[{"tree":[{"_id":"ResearchGroups"},{"_id":"IST"}],"_id":"UlWa"}],"type":"dissertation","doi":"10.15479/AT:ISTA:6681","file":[{"access_level":"open_access","request_a_copy":0,"file_size":1464227,"open_access":1,"file_id":"6771","content_type":"application/pdf","relation":"main_file","creator":"szhechev","date_created":"2019-08-07T13:02:50Z","date_updated":"2019-08-07T13:02:50Z","file_name":"Stephan_Zhechev_thesis.pdf"},{"access_level":"closed","file_size":303988,"request_a_copy":0,"open_access":0,"file_id":"6772","relation":"source_file","content_type":"application/octet-stream","date_created":"2019-08-07T13:03:22Z","creator":"szhechev","date_updated":"2019-08-08T06:41:37Z","file_name":"Stephan_Zhechev_thesis.tex"},{"request_a_copy":0,"file_size":1087004,"access_level":"closed","open_access":0,"file_id":"6773","date_created":"2019-08-07T13:03:34Z","creator":"szhechev","content_type":"application/zip","relation":"supplementary_material","file_name":"supplementary_material.zip","date_updated":"2019-08-08T06:41:37Z"}],"year":"2019","publication_status":"published"}]