On diffeomorphisms of 4-dimensional 1-handlebodies

TL;DR

This work addresses the problem of extending boundary diffeomorphisms of a $4$-dimensional $1$-handlebody to its interior and extends the framework to $4$-dimensional compression bodies in the relative setting for trisections. It uses the uniqueness of minimal-genus Heegaard splittings of double handlebodies (Carvalho–Oertel) and Cerf's isotopy results for the $3$-ball, along with a strong Haken theorem for sutured Heegaard splittings, to produce a streamlined, relative Laudenbach–Poénaru-type proof and a classification of double compression bodies; it then applies these ideas to relative multisection diagrams and the relative trisection framework. The paper proves a relative extension theorem for $P$-based hyper compression bodies under the no-$S^2$ condition on $P$ and shows how this implies the uniqueness of relative multisection diagrams, linking to the relative trisection framework. It also identifies a fundamental obstruction: if $P$ contains $S^2$, the uniqueness and extension properties fail, demonstrated by a family of manifolds with identical trisection diagrams but distinct $H_2$-groups, highlighting the limits of the diagrammatic approach.

Abstract

We give a new proof of Laudenbach and Poénaru'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.

Figures (3)

• Figure 1: Schematic of a multisection
• Figure 2: Heegaard diagram for $C_{i-1}\cup C_i$
• Figure 3: Decomposition of the surface $\Sigma$ and schematic of the construction of the manifolds $X_n$

Theorems & Definitions (47)

• Theorem : Theorem \ref{['th:LPrel']}
• Theorem : Theorem \ref{['th:doublecompHSfull']}
• theorem 2.1: Carvalho--Oertel
• theorem 2.2: Cerf
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• theorem 2.5: Laudenbach--Poénaru
• proof