# Generic path in the space of vector fields on the orientable surface

Who knows, does a theorem of such form exist:

Any one-parameter family of vector fields on the orientable surface may be slightly perturbed such that 1) all fields in the family except finite number are Morse-Smale; 2) at each non-Morse-Smale field a bifurcation from some list occurs (bifurcations are considered modulo topological equivalence).

• Where does this question arise from? – Stefan Kohl Jul 20 '14 at 15:38
• A one-parameter family of vector field is an horizontal vector field on the product $\Sigma\times I$ where $\Sigma$ is the surface $I$ is an interval, and horizontal'' means tangent to the slices $\Sigma\times\{t\}$. Now, did you try to apply classical genericity results on this horizontal vector field (and then projects the obtained vector field to an horizontal one)? – Benoît Kloeckner Jul 20 '14 at 15:49
• Stefan, I have a family of embedded surfaces of certain type into a manifold with a contact structure and I want to prove that contact folitations on the initial surface and the terminal one are related by some moves. So I need much control over foliation on each surface in the family. – Maxim Prasolov Jul 20 '14 at 19:13
• Benoit, I do not know which genericity results are to apply in 3d-case. For example, Morse-Smale vector fields are no longer dense on 3-manifolds. – Maxim Prasolov Jul 20 '14 at 19:29
• @MaximPrasolov: your last comments answers mine perfectly. Considering the previous one, it rings a bell; did you try to apply the methods of Emmanuel Giroux? I guess you know about them, but just in case I think that the most relevant paper is "Structures de contact en dimension trois et bifurcations des feuilletages de surfaces." [Contact structures in dimension three and bifurcations of surface foliations] Invent. Math. 141 (2000), no. 3, 615–689. It is in French but there should be some accounts of these works in english, at least I know who to ask. – Benoît Kloeckner Jul 21 '14 at 11:48