On codimension one foliations with Morse singularities on three-manifolds

A closed, connected oriented three-manifold supporting a codimension one oriented smooth foliation with Morse singularities having more centers than saddles and without saddle connections is diffeomorphic to the three-sphere. The use of the Reeb Stability theorem in place of the Poincaré–Bendixson theorem paves the way to a three-dimensional version, for foliations with singularities of Morse type, of a classical result of Haefliger. Finally, we give an example of a codimension one C ∞ foliation in the closed ball B ¯ 4 ⊂ R 4 , with only one singularity which is of saddle type 2–2 and transverse to the boundary S 3 = ∂ B 4 .