# On Computability and Triviality of Well Groups

Discrete & Computational Geometry, Jun 2016

The concept of well group in a special but important case captures homological properties of the zero set of a continuous map $f:K\rightarrow {\mathbb {R}}^n$ 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 $n=1$. 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 ${\mathbb {R}}^n$ by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and $\dim K<2n-2$, our approximation of the $(\dim \ K-n)$th well group is exact. For the second part, we find examples of maps $f,f':K\rightarrow {\mathbb {R}}^n$ 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.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2Fs00454-016-9794-2.pdf

Peter Franek, Marek Krčál. On Computability and Triviality of Well Groups, Discrete & Computational Geometry, 2016, 126-164, DOI: 10.1007/s00454-016-9794-2