Homology, Homotopy and Applications

Volume 23 (2021)

Number 1

Gradient versus proper gradient homotopies

Pages: 61 – 76

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n1.a5


Piotr Bartłomiejczyk (Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Gdańsk, Poland)

Piotr Nowak-Przygodzki (Sopot, Poland)


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.


gradient map, proper map, homotopy

2010 Mathematics Subject Classification

Primary 55Q05. Secondary 55M25.

Received 20 November 2019

Received revised 13 March 2020

Accepted 18 March 2020

Published 19 August 2020