Combinatorial structures of the space of gradient vector fields on compact surfaces

TL;DR

This paper explores the topology of the space of gradient vector fields on compact surfaces, revealing complex structures such as non-contractible components using combinatorics and homotopy theory.

Contribution

It provides the first detailed description of the combinatorial and topological structures of the space of gradient vector fields on surfaces.

Findings

The space contains a non-contractible connected component.

This component is weakly homotopy equivalent to a bouquet of two 2-spheres.

The study uses combinatorics and simple homotopy theory to analyze the space.

Abstract

Gradient vector fields are fundamental objects from both theoretical and practical perspectives, since various phenomena can be modeled within this framework. The ``moduli space'' of such vector fields provides the foundation for describing these phenomena. However, little is known about the topology of the space of gradient vector fields. For instance, it remains unknown whether a connected component of this space can fail to be simply connected. This paper aims to lay the foundation for describing the possible generic time evolution of gradient vector fields on surfaces, with or without constraints, under the assumption that no creation or annihilation of singular points occurs, by using combinatorics and simple homotopy theory. In fact, the space of gradient vector fields on a closed annulus contains a non-contractible connected component, which is weakly homotopy equivalent to a bouquet of two two-dimensional spheres.