Author(s):

Franek, Peter; Krčál, Marek

Title: 
On computability and triviality of well groups

Title Series: 
LIPIcs

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 f from 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 < 2n2, our approximation of the (dim Kn)th well group is exact. For the second part, we find examples of maps f, f' from 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.

Keywords: 
robustness; well groups; Homotopy theory; Nonlinear equations; computation

Conference Title:

SoCG: Symposium on Computational Geometry

Volume: 
34

Conference Dates:

June 22  25, 2015

Conference Location:

Eindhoven, Netherlands

ISBN:

9781450325943

Publisher:

ACM

Date Published:

20150101

Start Page: 
842

End Page:

856

Copyright Statement: 
CCBY

Sponsor: 
This research was supported by institutional support RVO:67985807 and by the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/20072013) under REA grant agreement n◦[291734].

URL: 

DOI: 
10.4230/LIPIcs.SOCG.2015.842

Notes: 
We are grateful to Ryan Budnay, Martin Čadek, Marek Filakovský, Tom Goodwillie, Amit Patel, Martin Tancer, Lukáš Vokřínek and Uli Wagner for useful discussions.

Open access: 
yes (OA journal) 