On Computability and Triviality of Well Groups

The concept of well group in a special but important case captures homological properties of the zero set of a continuous map $$ on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within $$ distance r from f for a given $$ . The main drawback of the approach is that the computability of well groups was shown only when $$ or $$ . Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of $$ by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and $$ , our approximation of the $$ th well group is exact. For the second part, we find examples of maps $$ 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.