Braiding non-ribbon surfaces and constructing broken Lefschetz fibrations on 4-manifolds

We define the notion of a braided link cobordism in $S^3 \times [0,1]$, which generalizes Viro's closed surface braids in $\mathbb{R}^4$. We prove that any properly embedded oriented surface $W \subset S^3 \times [0,1]$ is isotopic to a surface in this special position, and that the isotopy can be taken rel boundary when $\partial W$ already consists of closed braids. These surfaces are closely related to another notion of surface braiding in $D^2 \times D^2$, called braided surfaces with caps, which are a generalization of Rudolph's braided surfaces. We mention several applications of braided surfaces with caps, including using them to apply algebraic techniques from braid groups to studying surfaces in 4-space, as well as constructing singular fibrations on smooth 4-manifolds from a given handle decomposition.