A proof of Reidemeister-Singer's theorem by Cerf's methods

Heegaard splittings and Heegaard diagrams of a closed 3-manifold M are translated into the language of Morse functions with Morse-Smale pseudo-gradients defined on M. We make use in a very simple setting of techniques which Jean Cerf developed for solving a famous pseudo-isotopy problem. In passing, we show how to cancel the supernumerary local extrema in a generic path of functions when dim M>2. The main tool that we introduce is an elementary swallow tail lemma which could be useful elsewhere.