Multisections of surface bundles and bundles over the circle

TL;DR

This paper introduces multisections as a generalization of Heegaard splittings and trisections, demonstrating their existence in certain high-dimensional manifolds like surface bundles and fiber bundles over the circle, with explicit constructions.

Contribution

It proves that surface bundles and certain fiber bundles admit multisections, expanding the class of manifolds known to have such decompositions, with explicit examples and constructions.

Findings

Surface bundles admit multisections.

Fiber bundles over the circle with multisected fibers admit multisections.

Explicit constructions of multisections provided.

Abstract

A multisection is a decomposition of a manifold into 1-handlebodies, where each subcollection of the pieces intersects along a 1-handlebody except the global intersection which is a closed surface. These generalizations of Heegaard splittings and Gay-Kirby trisections were introduced by Ben Aribi, Courte, Golla and the author, who proved in particular that any 5-manifold admits such a multisection. In arbitrary dimension, we show that two classes of manifolds admit multisections: surface bundles and fiber bundles over the circle whose fiber itself is multisected. We provide explicit constructions, with examples.