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

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.