Franek, Peter ;
Krcál, Marek
On Computability and Triviality of Well Groups
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 < 2n2, our approximation of the (dim Kn)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 = {842856},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897835},
ISSN = {18688969},
year = {2015},
volume = {34},
editor = {Lars Arge and J{\'a}nos Pach},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5115},
URN = {urn:nbn:de:0030drops51159},
doi = {10.4230/LIPIcs.SOCG.2015.842},
annote = {Keywords: nonlinear equations, robustness, well groups, computation, homotopy theory}
}
2015
Keywords: 

nonlinear equations, robustness, well groups, computation, homotopy theory 
Seminar: 

31st International Symposium on Computational Geometry (SoCG 2015)

Issue date: 

2015 
Date of publication: 

2015 