  The concept of well group in a special but important case captures
homological properties of the zero set of a continuous map $f:K\to R^n$ 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<2n-2, our approximation of the (dim K-n)th well group is
exact.
  For the second part, we find examples of maps $f,f': K\to R^n$ 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.
