On diffeomorphisms of 4-dimensional 1-handlebodies

TL;DR

This paper provides a new proof of a theorem on extending boundary diffeomorphisms of 4-dimensional 1-handlebodies to the entire handlebody, and extends the result to certain 4-dimensional cobordisms, with applications to 4-manifold decompositions.

Contribution

It introduces a novel proof of Laudenbach and Poénaru's theorem and extends it to 4-dimensional compression bodies, linking Heegaard splittings and sutured structures.

Findings

New proof of boundary diffeomorphism extension theorem

Extension of theorem to 4-dimensional compression bodies

Application to relative trisection diagrams for 4-manifolds

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.