Author(s):

Franek, Peter; Krčál, Marek

Article Title: 
Persistence of zero sets

Affiliation 
IST Austria 
Abstract: 
We study robust properties of zero sets of continuous maps f: X → ℝn. Formally, we analyze the family Z< r(f) := (g1(0): g  f < r) of all zero sets of all continuous maps g closer to f than r in the maxnorm. All of these sets are outside A := (x: f(x) ≥ r) and we claim that Z< r(f) is fully determined by A and an element of a certain cohomotopy group which (by a recent result) is computable whenever the dimension of X is at most 2n  3. By considering all r > 0 simultaneously, the pointed cohomotopy groups form a persistence modulea structure leading to persistence diagrams as in the case of persistent homology or well groups. Eventually, we get a descriptor of persistent robust properties of zero sets that has better descriptive power (Theorem A) and better computability status (Theorem B) than the established well diagrams. Moreover, if we endow every point of each zero set with gradients of the perturbation, the robust description of the zero sets by elements of cohomotopy groups is in some sense the best possible (Theorem C).

Keywords: 
Cohomotopy group; computational homotopy theory; system of equations

Journal Title:

Homology, Homotopy and Applications

Volume: 
19

Issue 
2

ISSN:

15320073

Publisher:

International Press

Date Published:

20170101

Start Page: 
313

End Page:

342

URL: 

DOI: 
10.4310/HHA.2017.v19.n2.a16

Notes: 
The research leading to these results has received funding from Austrian Science Fund (FWF): M 1980, the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme (FP7/20072013) under REA grant agreement number [291734] and from the Czech Science Foundation (GACR) grant number 1514484S with institutional support RVO:67985807. The research of Marek Krcal was supported by the project number GACR 1709142S of the Czech Science Foundation. We are grateful to Sergey Avvakumov, Ulrich Bauer, Marek Filakovski, Amit Patel, Lukas Vokrinek and Ryan Budney for useful discussions and hints.

Open access: 
yes (repository) 