On 4-Manifolds, Folds and Cusps

TL;DR

This paper explores simple wrinkled fibrations on 4-manifolds, relating them to handle decompositions and surface diagrams, and classifies all genus one cases, advancing understanding of 4-manifold structures.

Contribution

It introduces a combinatorial approach to simple wrinkled fibrations and classifies all genus one cases, linking fibrations to handle decompositions and surface diagrams.

Findings

Handle decompositions similar to Lefschetz fibrations

Surface diagrams encode cut-and-paste operations

Complete classification of genus one 4-manifolds with simple wrinkled fibrations

Abstract

We study simple wrinkled fibrations, a variation of the simplified purely wrinkled fibrations introduced by Williams, and their combinatorial description in terms of surface diagrams. We show that simple wrinkled fibrations induce handle decompositions on their total spaces which are very similar to those obtained from Lefschetz fibrations. The handle decompositions turn out to be closely related to surface diagrams and we use this relationship to interpret some cut-and-paste operations on 4-manifolds in terms of surface diagrams. This, in turn, allows us classify all closed 4-manifolds which admit simple wrinkled fibrations of genus one, the lowest possible fiber genus.