Topology of Ambient 3-Manifolds of Non-singular Flows with Twisted Saddle Orbit

TL;DR

This paper classifies the topology of closed orientable 3-manifolds supporting non-singular Morse-Smale flows with exactly one twisted saddle orbit, showing they are either lens spaces, connected sums involving lens spaces, or specific Seifert manifolds.

Contribution

It provides a complete topological classification of such 3-manifolds, expanding understanding beyond lens spaces to include certain Seifert manifolds.

Findings

Manifolds are either lens spaces, connected sums with lens spaces, or Seifert manifolds with three singular fibers.

Flows with a single twisted saddle orbit do not only occur on lens spaces, contrary to previous beliefs.

The classification refutes the idea that such flows only exist on simple manifolds like lens spaces.

Abstract

In the present paper, non-singular Morse-Smale flows on closed orientable 3-manifolds under the assumption that among the periodic orbits of the flow there is only one saddle one and it is twisted are considered. An exhaustive description of the topology of such manifolds is obtained. Namely, it has been established that any manifold admitting such flows is either a lens space, or a connected sum of a lens space with a projective space, or Seifert manifolds with base homeomorphic to sphere and three singular fibers. Since the latter are simple manifolds, the result obtained refutes the result that among simple manifolds, the considered flows admit only lens spaces.