**Title:** Constructing broken fibrations on 4-dimensional manifolds and surface braids

**Abstract:** In this talk I will introduce the notion of an open book decomposition of a closed 3-manifold M, which is a way to decompose M into a collection of 2-dimensional “pages” with a 1-dimensional “binding.” These decompositions are closely related to contact structures on 3-manifolds, by a celebrated theorem of Giroux. I will sketch an elementary proof due to Alexander showing that any closed 3-manifold M admits such a decomposition, which involves working with closed braids in 3-space.

Moving one dimension higher, I will show that Alexander’s techniques can also be applied to smooth 4-manifolds. In this setting the structures constructed are broken Lefschetz fibrations, which are closely related to near-symplectic structures on 4-manifolds. This extension will require us to work with braided surfaces, which are a higher dimensional analogue of classical braids in 3-space. I will discuss some interesting aspects of the theory of braided surfaces, including some open problems.