About the diffeomorphisms of the 3-sphere and a famous theorem of Cerf (${\mathrm{\Gamma }}_4=0$)

Using a rigidity property of the foliations of $S^2\times [0,1]$ that are defined by a non-vanishing closed one-form, we give a rather simple proof of a theorem due J. Cerf, going back to 1968, that the group of direct diffeomorphisms of $S^3$ is connected.