Simplified broken Lefschetz fibrations and trisections of 4-manifolds

TL;DR

This paper introduces methods to simplify generic maps of 4-manifolds into surfaces, enabling better decompositions and establishing a correspondence between broken Lefschetz fibrations and trisections, thus advancing understanding of 4-manifold topology.

Contribution

It presents a topological simplification technique for maps of 4-manifolds and links two important decompositions, providing new tools and insights in 4-manifold topology.

Findings

Simplified models of 4-manifolds via broken Lefschetz fibrations and trisections.

Established a correspondence between these two decompositions.

Produced new examples of 4-manifold decompositions.

Abstract

Shapes of four dimensional spaces can be studied effectively via maps to standard surfaces. We explain, and illustrate by quintessential examples, how to simplify such generic maps on 4-manifolds topologically, in order to derive simple decompositions into much better understood manifold pieces. Our methods not only allow us to produce various interesting families of examples, but also to establish a correspondence between simplified broken Lefschetz fibrations and simplified trisections of closed, oriented 4-manifolds.