A refined estimate for the topological degree

TL;DR

This paper improves an existing estimate for the topological degree of continuous sphere maps in dimensions two and higher, answering a previously open question for these cases.

Contribution

It refines the estimate of Bourgain, Brezis, and Nguyen for the topological degree, providing a definitive answer for dimensions $d \\ge 2$.

Findings

Sharpened estimate for topological degree in higher dimensions

Answer to Brezis's question for $d \\ge 2$

Open problem remains for $d=1$

Abstract

We sharpen an estimate of Bourgain, Brezis, and Nguyen for the topological degree of continuous maps from a sphere $\mathbb{S}^d$ into itself in the case $d \ge 2$. This provides the answer for $d \ge 2$ to a question raised by Brezis. The problem is still open for $d=1$.