Flows on 3-Manifolds without Periodic Orbits

Abstract: The topology of a manifold determines if a flow has fixed points, as stated in the Poincaré-Hopf theorem, but if a manifold (oriented) admits a flow without fixed points it also admits a flow without periodic orbits. If the manifold has dimension 2, this assertion is almost trivial. In dimension 4 and higher, it follows from a construction of F. W. Wilson from 1974. The difficult case is dimension 3, where such examples were constructed by K. Kuperberg in 1994.
The question of when a flow has a periodic orbit is an active subject of research and I will try to present the latest results, together with the open questions.