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. Hyperbolicity of periodic orbits implies that among them one is attracting and another is repelling. Due to Poincaré-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 S 3 and 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.