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.