Nonsingular Morse-Smale flows of n-manifolds with attractor-repeller dynamics

TL;DR

This paper classifies nonsingular Morse-Smale flows with attractor-repeller dynamics on n-manifolds, revealing topological constraints, classifications, and existence results for various manifolds including tori, Klein bottles, and lens spaces.

Contribution

It provides a comprehensive topological classification of such flows on n-manifolds, identifying specific manifolds that admit these flows and their equivalence classes.

Findings

Only tori and Klein bottles can host these flows in 2D.

Specific counts of flow classes on lens spaces and spheres.

The only non-orientable n-manifold admitting such flows is the twisted I-bundle over (n-1)-sphere.

Abstract

In the present paper the exhaustive topological classification of nonsingular Morse-Smale flows of $n$-manifolds with two limit cycles is presented. Hyperbolicity of periodic orbits implies that among them one is attracting and another is repelling. Due to Poincare-Hopf theorem Euler characteristic of closed manifold $M^n$ which admits the considered flows is equal to zero. Only torus and Klein bottle can be ambient manifolds for such flows in case of $n=2$. Authors established that there exist exactly two classes of topological equivalence of such flows of torus and three of the Klein bottle. There are no constraints for odd-dimensional manifolds which follow from the fact that Euler characteristic is zero. However, it is known that orientable $3$-manifold admits a flow of considered class if and only if it is a lens space. In this paper, it is proved that up to topological equivalence each of $\mathbb S^3$ and $\mathbb RP^3$ admit one such flow and other lens spaces two flows each. Also, it is shown that the only non-orientable $n$-manifold (for $n>2$), which admits considered flows is the twisted I-bundle over $(n-1)$-sphere. Moreover, there are exactly two classes of topological equivalence of such flows. Among orientable $n$-manifolds only the product of $(n-1)$-sphere and the circle can be ambient manifold of a considered flow and the flows are split into two classes of topological equivalence.