Does anyone know of papers that mention the classification of non-singular Morse-Smale (NMS) flows up to topological equivalency? I am particularly interested in the flows on manifolds of dimension 3. But all I can find so far is the paper by Bin Yu on the classification of the NMS flows defined on the three-sphere.

## 1 Answer

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Generally, if you hope a very clean list (like topological classification of surfaces) to completely classify NMS flows on 3-manifolds (even in three sphere) up to topological equivalence. It seems hopeless. One reason is that heteroclinic trajectories connecting saddle orbits will lead a complete list quite wild.

If you reduce the requirement, there are several directions:

- if you only need to classify up to indexed links of periodic orbits, Wada give a nice algorithm to deal with the question in three sphere;
- if you don't need a clean list, there exists an attempt to consider complete classification. But the list seems as complicated as NMS flows themselves;
- if we don't care "heteroclinic trajectories connecting saddle orbits" (Certainly this is very small subset of all NMS flows), we can do some discussions, for instance, see here.

More intersting discussions of NMS flows on three manifolds always are connected with the other topics. The following directions are what I know:

- Similar the connection of gradient flows and handle decompositions, Asimov connect NMS flows with round handle decompositions. And J. Morgan (1979) use it to nearly detect which 3-manifolds admitting NMS flows. And Yano considered for a given 3-manifold, which homotopy classes of nonsingular vector fields admit MS flows.
- People use NMS flows to represent the homotopy classes of nonsingular vector fields, see here.
- there are intersting connection between the periodic orbits of some NMS flows and integrable Hamiltonian systems, see here.
- People use some parts of NMS flows to construct more complicated system, for instance, Anosov flows on 3-manifolds, see here and here.

classificationhere? Like a list? $\endgroup$