{"license":"https://creativecommons.org/licenses/by/4.0/","page":"842 - 856","oa_version":"Published Version","tmp":{"image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"oa":1,"has_accepted_license":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","abstract":[{"lang":"eng","text":"The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f from K to R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within L_infty distance r from f for a given r &gt; 0. The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K &lt; 2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps f, f' from K to R^n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status. "}],"intvolume":"        34","publist_id":"5667","year":"2015","language":[{"iso":"eng"}],"day":"11","date_updated":"2023-02-21T17:02:57Z","doi":"10.4230/LIPIcs.SOCG.2015.842","ec_funded":1,"conference":{"end_date":"2015-06-25","start_date":"2015-06-22","name":"SoCG: Symposium on Computational Geometry","location":"Eindhoven, Netherlands"},"scopus_import":1,"date_created":"2018-12-11T11:52:26Z","file_date_updated":"2020-07-14T12:44:59Z","date_published":"2015-06-11T00:00:00Z","project":[{"_id":"25681D80-B435-11E9-9278-68D0E5697425","grant_number":"291734","call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme"}],"title":"On computability and triviality of well groups","citation":{"short":"P. Franek, M. Krcál, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015, pp. 842–856.","apa":"Franek, P., &#38; Krcál, M. (2015). On computability and triviality of well groups (Vol. 34, pp. 842–856). Presented at the SoCG: Symposium on Computational Geometry, Eindhoven, Netherlands: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.842\">https://doi.org/10.4230/LIPIcs.SOCG.2015.842</a>","mla":"Franek, Peter, and Marek Krcál. <i>On Computability and Triviality of Well Groups</i>. Vol. 34, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015, pp. 842–56, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.842\">10.4230/LIPIcs.SOCG.2015.842</a>.","ista":"Franek P, Krcál M. 2015. On computability and triviality of well groups. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 34, 842–856.","ieee":"P. Franek and M. Krcál, “On computability and triviality of well groups,” presented at the SoCG: Symposium on Computational Geometry, Eindhoven, Netherlands, 2015, vol. 34, pp. 842–856.","ama":"Franek P, Krcál M. On computability and triviality of well groups. In: Vol 34. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2015:842-856. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.842\">10.4230/LIPIcs.SOCG.2015.842</a>","chicago":"Franek, Peter, and Marek Krcál. “On Computability and Triviality of Well Groups,” 34:842–56. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. <a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.842\">https://doi.org/10.4230/LIPIcs.SOCG.2015.842</a>."},"author":[{"full_name":"Franek, Peter","id":"473294AE-F248-11E8-B48F-1D18A9856A87","first_name":"Peter","last_name":"Franek"},{"full_name":"Krcál, Marek","id":"33E21118-F248-11E8-B48F-1D18A9856A87","last_name":"Krcál","first_name":"Marek"}],"month":"06","related_material":{"record":[{"relation":"later_version","status":"public","id":"1408"}]},"volume":34,"publication_status":"published","quality_controlled":"1","file":[{"file_id":"5001","date_created":"2018-12-12T10:13:19Z","relation":"main_file","file_size":623563,"date_updated":"2020-07-14T12:44:59Z","creator":"system","content_type":"application/pdf","checksum":"49eb5021caafaabe5356c65b9c5f8c9c","access_level":"open_access","file_name":"IST-2016-503-v1+1_32.pdf"}],"_id":"1510","status":"public","ddc":["510"],"alternative_title":["LIPIcs"],"type":"conference","pubrep_id":"503","department":[{"_id":"UlWa"},{"_id":"HeEd"}]}