On diffeomorphisms of 4-dimensional 1-handlebodies

Abstract

We give a new proof of Laudenbach and Po\'enaru's theorem, which states that any diffeomorphism of the boundary of a 4-dimensional 1-handlebody extends to the whole handlebody. Our proof is based on the cassification of Heegaard splittings of double handlebodies and a result of Cerf on diffeomorphisms of the 3-ball. Further, we extend this theorem to 4-dimensional compression bodies, namely cobordisms between 3-manifolds constructed using only 1-handles: when the negative boundary is a product of a compact surface by interval, we show that every diffeomorphism of the positive boundary extends to the whole compression body. This invlolves a strong Haken theorem for sutured Heegaard splittings and a classification of sutured Heegaard splittings of double compression bodies. Finally, we show how this applies to the study of relative trisection diagrams for compact 4-manifolds.