On 4-manifolds, folds and cusps

In the past decade the notion of Lefschetz fibrations, which roughly characterize symplectic
4-manifolds, has been generalized (by various authors) to so called broken Lefschetz fibrations
which are now known to exist on all closed, orientable 4-manifolds. During this process,
Williams obtained an existence result for maps to the 2-sphere with a rather special
singularity structure, which we call simple wrinkled fibrations. Although these maps look
rather different from Lefschetz fibrations, they also admit combinatorial descriptions in
terms of certain curves configurations on a surface which are known as surface diagrams.
In Part I will give an introduction to simple wrinkled fibrations and surface diagrams.
We will then indicate how to go back and forth between both notions using handlebody
theory. Using this we can discuss how two simple operations on surface diagrams
correspond to well known operations on 4-manifolds and show how they can be
used to classify simple wrinkled fibrations of genus 1 (which is the lowest possible genus).
In Part II we will go into some more details in a slightly more general setting. If time
permits, we will describe how some homotopy invariants are encoded  in surface diagrams.