Gradient versus proper gradient homotopies

TL;DR

This paper investigates the differences between gradient and proper gradient homotopy classes in the plane, providing a comprehensive classification of such maps with non-negative Brouwer degree.

Contribution

It offers a complete description of homotopy classes of gradient and proper gradient maps in Euclidean spaces, highlighting their fundamental differences.

Findings

Gradient and proper gradient homotopy classes are essentially different.

Complete classification of gradient maps with non-negative Brouwer degree.

Analysis of homotopy classes in the plane and higher dimensions.

Abstract

We compare the sets of homotopy classes of gradient and proper gradient vector fields in the plane. Namely, we show that gradient and proper gradient homotopy classifications are essentially different. We provide a complete description of the sets of homotopy classes of gradient maps from $\mathbb{R}^n$ to $\mathbb{R}^n$ and proper gradient maps from $\mathbb{R}^2$ to $\mathbb{R}^2$ with the Brouwer degree greater or equal to zero.