## Topological Methods in Nonlinear Analysis

### Removing isolated zeroes by homotopy

#### 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 54, Number 1 (2019), 275-296.

Dates
First available in Project Euclid: 16 July 2019

https://projecteuclid.org/euclid.tmna/1563242564

Digital Object Identifier
doi:10.12775/TMNA.2019.042

#### Citation

Coffman, Adam; Lebl, Jiří. Removing isolated zeroes by homotopy. Topol. Methods Nonlinear Anal. 54 (2019), no. 1, 275--296. doi:10.12775/TMNA.2019.042. https://projecteuclid.org/euclid.tmna/1563242564

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