Author(s):

Franek, Peter; Krčál, Marek

Article Title: 
On computability and triviality of well groups

Affiliation 
IST Austria 
Abstract: 
The concept of well group in a special but important case captures homological properties of the zero set of a continuous map (Formula presented.) on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within (Formula presented.) distance r from f for a given (Formula presented.). The main drawback of the approach is that the computability of well groups was shown only when (Formula presented.) or (Formula presented.). 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 (Formula presented.) by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and (Formula presented.), our approximation of the (Formula presented.)th well group is exact. For the second part, we find examples of maps (Formula presented.) 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.

Keywords: 
robustness; well groups; Computational topology; Homotopy theory; Nonlinear equations; Obstruction theory

Journal Title:

Discrete & Computational Geometry

Volume: 
56

Issue 
1

ISSN:

01795376

Publisher:

Springer

Date Published:

20160701

Start Page: 
126

End Page:

164

Copyright Statement: 
CCBY 4.0

URL: 

DOI: 
10.1007/s0045401697942

Notes: 
Open access funding provided by Institute of Science and Technology (IST Austria). This research was supported by Austrian Science Fund (FWF): M 1980, by the Czech Science Foundation (GACR) Grant Number 1514484S with institutional support RVO:67985807, and by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007–2013) under REA grant agreement n [291734].

Open access: 
yes (OA journal) 