Removing Isolated Zeroes by Homotopy

TL;DR

This paper studies the conditions under which isolated zeros of continuous maps can be removed via homotopies, introducing the concept of 'locally inessential' zeros and exploring their properties in various contexts.

Contribution

It generalizes the notion of index for vector fields to higher dimensions and establishes conditions for constructing homotopies that remove isolated zeros.

Findings

Every isolated zero with trivial homotopy group is locally inessential.

Existence of semialgebraic homotopies for semialgebraic maps.

Construction of Hölder continuous homotopies for real analytic maps with inessential zeros.

Abstract

Suppose that the inverse image of the zero vector by a continuous map $f:{\mathbb R}^n\to{\mathbb R}^q$ has an isolated point $P$. There is a local obstruction to removing this isolated zero by a small perturbation, generalizing the notion of index for vector fields, the $q=n$ case. The existence of a continuous map $g$ which approximates $f$ but is nonvanishing near $P$ is equivalent to a topological property we call "locally inessential," and for dimensions $n$, $q$ where $\pi_{n-1}(S^{q-1})$ is trivial, every isolated zero is locally inessential. We consider the problem of constructing such an approximation $g$, and show that there exists a continuous homotopy from $f$ to $g$ through locally nonvanishing maps. If $f$ is a semialgebraic map, then there exists such a homotopy which is also semialgebraic. For $q=2$ and $f$ real analytic with a locally inessential isolated zero, there exists a H\"older 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.