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

TL;DR

This paper provides a new proof of Reidemeister-Singer's theorem using Cerf's methods, translating Heegaard splittings into Morse functions and introducing an elementary swallow tail lemma to simplify the process.

Contribution

It introduces a novel proof technique for Reidemeister-Singer's theorem based on Cerf's pseudo-isotopy methods and simplifies the handling of local extrema in Morse functions.

Findings

Established a new proof of Reidemeister-Singer's theorem

Developed an elementary swallow tail lemma for Morse functions

Demonstrated cancellation of supernumerary extrema in generic function paths

Abstract

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.