About the diffeomorphisms of the 3-sphere and a famous theorem of Cerf ($\Gamma_4 = 0$)

TL;DR

This paper provides a simplified proof of Cerf's theorem that the group of orientation-preserving diffeomorphisms of the 3-sphere is connected, using a rigidity property of certain foliations.

Contribution

It introduces a new, simpler proof of Cerf's theorem leveraging foliation rigidity, enhancing understanding of diffeomorphism groups of 3-spheres.

Findings

The group of orientation-preserving diffeomorphisms of S^3 is connected.

A rigidity property of foliations defined by non-vanishing closed one-forms is key.

The proof simplifies previous approaches to Cerf's theorem.

Abstract

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.