On computability and triviality of well groups
LIPIcs
Franek, Peter
Krcál, Marek
ddc:510
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 > 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 < 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.
Schloss Dagstuhl - Leibniz-Zentrum für Informatik
2015
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doc-type:conferenceObject
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http://purl.org/coar/resource_type/c_5794
https://research-explorer.app.ist.ac.at/record/1510
https://research-explorer.app.ist.ac.at/download/1510/5001
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>
eng
info:eu-repo/semantics/altIdentifier/doi/10.4230/LIPIcs.SOCG.2015.842
info:eu-repo/grantAgreement/EC/FP7/291734
info:eu-repo/semantics/openAccess