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2019 Removing isolated zeroes by homotopy
Adam Coffman, Jiří Lebl
Topol. Methods Nonlinear Anal. 54(1): 275-296 (2019). DOI: 10.12775/TMNA.2019.042

Abstract

Suppose that the inverse image of the zero vector by a continuous map $f\colon {\mathbb R}^n\to{\mathbb R}^q$ has an isolated point $P$. The existence of a continuous map $g$ which approximates $f$ but is nonvanishing near $P$ is equivalent to a topological property we call ``local inessentiality of zeros'', generalizing the notion of index zero for vector fields, the $q=n$ case. We consider the problem of constructing such an approximation $g$ and a continuous homotopy $F(x,t)$ from $f$ to $g$ through locally nonvanishing maps. If $f$ is a semialgebraic map, then there exists $F$ also semialgebraic. If $q=2$ and $f$ is real analytic with a locally inessential zero, then there exists a Hölder continuous homotopy $F(x,t)$ which, for $(x,t)\ne(P,0)$, is real analytic and nonvanishing. The existence of a smooth homotopy, given a smooth map $f$, is stated as an open question.

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Adam Coffman. Jiří Lebl. "Removing isolated zeroes by homotopy." Topol. Methods Nonlinear Anal. 54 (1) 275 - 296, 2019. https://doi.org/10.12775/TMNA.2019.042

Information

Published: 2019
First available in Project Euclid: 16 July 2019

zbMATH: 07131285
MathSciNet: MR4018281
Digital Object Identifier: 10.12775/TMNA.2019.042

Rights: Copyright © 2019 Juliusz P. Schauder Centre for Nonlinear Studies

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Vol.54 • No. 1 • 2019
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