Braiding link cobordisms and non-ribbon surfaces

Abstract

We define the notion of a braided link cobordism in $S^3\times \left[0,1\right]$, which generalizes Viro’s closed surface braids in ${ℝ}^4$. We prove that any properly embedded oriented surface $W\subset S^3\times \left[0,1\right]$ 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.