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DOI: 10.4230/LIPIcs.SOCG.2015.842
URN: urn:nbn:de:0030-drops-51159
URL: http://drops.dagstuhl.de/opus/volltexte/2015/5115/
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Franek, Peter ; Krcál, Marek

On Computability and Triviality of Well Groups

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Abstract

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.

BibTeX - Entry

@InProceedings{franek_et_al:LIPIcs:2015:5115,
  author =	{Peter Franek and Marek Krc{\'a}l},
  title =	{{On Computability and Triviality of Well Groups}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{842--856},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Lars Arge and J{\'a}nos Pach},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5115},
  URN =		{urn:nbn:de:0030-drops-51159},
  doi =		{10.4230/LIPIcs.SOCG.2015.842},
  annote =	{Keywords: nonlinear equations, robustness, well groups, computation, homotopy theory}
}

Keywords: nonlinear equations, robustness, well groups, computation, homotopy theory
Seminar: 31st International Symposium on Computational Geometry (SoCG 2015)
Issue Date: 2015
Date of publication: 11.06.2015


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