Approximation by zero-free continuous maps

TL;DR

The paper addresses when a continuous map $f:E\to\mathbb{R}^n$ that is zero-free on the interior $E^{\circ}$ can be uniformly approximated by maps that are zero-free on all of $E$, and extends this to subsets of $n$-manifolds using the fine topology. It introduces a two-lemma framework: (1) a simplicial approximation that is zero-free on each $n$-simplex, and (2) a dimension-theoretic perturbation that upgrades a zero-free approximation on a decomposition with $\dim B\le n-1$ to a globally zero-free map. The main contribution is a generalized theorem for subsets of $n$-manifolds ensuring zero-free refinements exist in the fine topology, connecting dimension theory with zero-free approximation and providing a constructive approach. This removes topological obstructions to zero-free approximation in this setting and relates to conjectures in complex-analytic approximation about preserving zero-freeness on the boundary.

Abstract

We prove that if E a subset of an n-dimensional manifold, then every continuous R^n-valued map on E that is zero-free on the interior of E can be approximated in the fine topology, and hence, in particular, in the uniform topology, by a continuous R^n-valued map that is zero-free on all of E.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3: Andersson-Gauthier AG
• Lemma 3.1
• Lemma 3.2
• proof : Proof of Theorem \ref{['theorem-generalized']}
• Lemma 4.1
• proof
• proof : Proof of Lemma \ref{['lemma1']}
• Theorem 5.1: Hurewicz-Wallman1948, III 5 C