On 4-manifolds, folds and cusps (Q372508)

Broken Lefschetz fibrations and wrinkled fibrations, which are generic maps from a 4-manifold to a surface, are pervasive in the study of 4-dimensional differential topology these days. Through the seminal works due to \textit{S. K. Donaldson} [J. Differ. Geom. 53, No. 2, 205--236 (1999; Zbl 1040.53094)] and \textit{R. E. Gompf} [J. Symplectic Geom. 2, No. 2, 177--206 (2004; Zbl 1084.53072)], these fibrations are related to symplectic geometry. The topic is also closely related to 4-dimensional invariant theories: Seiberg-Witten invariants, Donaldson-Smith invariants, and Lagrangian matching invariants. Furthermore, they relate to handlebody theory, in fact, we can see a handle diagram from these fibrations. To study these subjects is consequently a fruitful thing.NEWLINENEWLINEThis paper focuses on wrinkled fibrations and surface diagrams. A wrinkled fibration, which was anticipated by \textit{Y. Lekili} [Geom. Topol. 13, No. 1, 277--318 (2009; Zbl 1164.57006)], \textit{J. Williams} [ibid. 14, No. 2, 1015--1061 (2010; Zbl 1204.57027)], is a generic map from a 4-dimensional manifold to a surface, and the types of critical points are indefinite fold or cusp singularities. Any broken Lefschetz fibration is homotopic to a wrinkled fibration keeping the homotopy class of the map. Any closed 4-manifold admits a simple wrinkled fibration. Even if the set of critical points is nonempty, connected and contains a cusp, the map is injective on it and all fibers are connected, we have no loss of generality. This type of fibration is the main object in this paper and is called \textit{simple wrinkled fibration}. To be accurate, there are several minor other conditions. Simplified purely wrinkled fibrations are almost equal to simple wrinkled fibrations with the base space \(S^2\).NEWLINENEWLINEThe aim of the author is to give a recipe to construct simple wrinkled fibrations in terms of a decomposition of the base space as \(B=B_+\cup A\cup B_-\). The annular part \(A\) is the regular neighborhood of the critical values, and is the interesting part. One of the main theorems is a bijection between annular simple wrinkled fibrations up to equivalence and twisted surface diagrams up to equivalence. A surface diagram is a collection of circuits of curves on the fiber surface; the curves mean vanishing cycles of the fold singularities. It is a newer way to describe any 4-manifold.NEWLINENEWLINEAs application the author gives substitution of surface diagrams and a classification of genus one simple wrinkled fibrations. The substitution is done by replacing two subcircuits.