On computability and triviality of well groups Conference Paper

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 < 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' 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: 978-1-4503-2594-3
Publisher: ACM  
Date Published: 2015-01-01
Start Page: 842
End Page: 856
Copyright Statement: CC-BY
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 (FP7/2007-2013) under REA grant agreement n◦[291734].
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)
IST Austria Authors