On Computability and Triviality of Well Groups

0459972 - ÚI 2017 RIV US eng J - Journal Article
Franek, Peter - Krčál, M.
On Computability and Triviality of Well Groups.
Discrete & Computational Geometry. Roč. 56, č. 1 (2016), s. 126-164. ISSN 0179-5376. E-ISSN 1432-0444
R&D Projects: GA ČR GA15-14484S
Keywords : nonlinear equations * robustness * well groups * computational topology * obstruction theory * homotopy theory
Subject RIV: BA - General Mathematics
Impact factor: 0.724, year: 2016

The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f: K -> R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within distance r from f for a given r>0 in the max-norm. 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’: K -> 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.
Permanent Link: http://hdl.handle.net/11104/0260117