On computability and triviality of well groups
The concept of well groups in a special case captures homological properties of the zero set of a continuous map \(f: K\rightarrow {\mathbb R}^n\) 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 \(n=1\). The contribution of the authors to the theory of well groups in this paper is twofold: on the one hand, the computability issue is improved, and on the other hand, a range of examples is presented where the well groups are incomplete invariants (that is, fail to capture certain important robust properties of the zero set). For the first part, the authors identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of \({\mathbb R}^n\) by \(f\). In other words, well groups can be algorithmically approximated from below. When \(f\) is smooth and dim\(K<2n-2\), the approximation developed of the (dim \(K-n\))th well group is exact. For the second part, the authors find examples of maps \(f,f': K\rightarrow {\mathbb R}^n\) with all well groups isomorphic but whose perturbations have different zero sets. Finally, a possible replacement of the well groups of vector-valued maps is discussed by an invariant of a better descriptive power and computability status.
scientific article; zbMATH DE number 6621061
scientific article