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

Tomoo Yokoyama

TL;DR

This work investigates the topology of the moduli space of gradient vector fields on compact surfaces, revealing a non-contractible connected component in the space of gradient-flow equivalence classes on a closed annulus, weakly homotopy equivalent to a bouquet of two 2-spheres. It develops a robust combinatorial framework: a finite abstract cell complex structure stratified by codimension, invariance of index sums under perturbations, and the construction of associated neighborhoods around multi-saddle connections together with Whitehead moves to realize local transitions. Extending to Morse-Smale-like flows with possible limit cycles, the paper establishes analogous codimension-based stratifications and shows that gradient-flow components can be nontrivially shaped in higher-genus settings, with explicit analysis on $ig[ oexpand{\mathcal{G}^r_{1,2,-1}(\,\,\,\Sigma_{0,2})}ig]$. The results illuminate how generic time evolution of gradient fields on compact surfaces progresses through a finite, hierarchically organized set of combinatorial configurations, providing a foundation for describing gradient-flow dynamics without singular-point creation/annihilation and opening routes to higher-dimensional generalizations. Overall, the work demonstrates that even in gradient-restricted dynamics, the global topology of the space of flows can be rich and nontrivial, with concrete models encoding the possible transitions between gradient configurations.

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.

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

TL;DR

This work investigates the topology of the moduli space of gradient vector fields on compact surfaces, revealing a non-contractible connected component in the space of gradient-flow equivalence classes on a closed annulus, weakly homotopy equivalent to a bouquet of two 2-spheres. It develops a robust combinatorial framework: a finite abstract cell complex structure stratified by codimension, invariance of index sums under perturbations, and the construction of associated neighborhoods around multi-saddle connections together with Whitehead moves to realize local transitions. Extending to Morse-Smale-like flows with possible limit cycles, the paper establishes analogous codimension-based stratifications and shows that gradient-flow components can be nontrivially shaped in higher-genus settings, with explicit analysis on . The results illuminate how generic time evolution of gradient fields on compact surfaces progresses through a finite, hierarchically organized set of combinatorial configurations, providing a foundation for describing gradient-flow dynamics without singular-point creation/annihilation and opening routes to higher-dimensional generalizations. Overall, the work demonstrates that even in gradient-restricted dynamics, the global topology of the space of flows can be rich and nontrivial, with concrete models encoding the possible transitions between gradient configurations.

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.
Paper Structure (70 sections, 46 theorems, 55 equations, 39 figures)

This paper contains 70 sections, 46 theorems, 55 equations, 39 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notion of topology
  4. Surface graph
  5. Double of a compact surface
  6. Curves and loops
  7. Notion of dynamical systems
  8. The $C^0$-topology on the set of flows on a topological space
  9. Structural stability in a class for flows
  10. Orbit arc and local topological equivalence
  11. Double of a flow on a compact surface
  12. Gradient vector fields
  13. Flow box and transverse annulus
  14. Types of singular points
  15. Indices of isolated singular points of $C^1$-flows on manifolds
  16. ...and 55 more sections

Key Result

Theorem 1

For any $r \in \mathbb{Z} _{\geq 0} \sqcup \{ \infty \}$, the space of $C^r$ gradient vector fields with finitely many singular points without fake multi-saddles (i.e. $\partial$-$0$-saddles and $0$-saddles) or fake parabolic sectors (see Definition def:fake below) on a closed annulus whose sums of

Figures (39)

  • Figure 1: Creation of a physical boundary.
  • Figure 2: Multi-saddles.
  • Figure 3: A saddle, two $\partial$-saddles, a sink, a $\partial$-sink, a source, a $\partial$-source, and a center
  • Figure 4: Two parabolic sectors $P^-$ and $P^+$, two hyperbolic sectors $H^-$ and $H^+$ with clockwise and anti-clockwise orbit directions, and two elliptic sectors $E^+$ and $E^-$ with clockwise and anti-clockwise orbit directions, respectively.
  • Figure 5: Inner and outer tangencies.
  • ...and 34 more figures

Theorems & Definitions (135)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • ...and 125 more