#### Volume 15, issue 6 (2015)

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Braiding link cobordisms and non-ribbon surfaces

### Mark C Hughes

Algebraic & Geometric Topology 15 (2015) 3707–3729
##### Abstract

We define the notion of a braided link cobordism in ${S}^{3}×\left[0,1\right]$, which generalizes Viro’s closed surface braids in ${ℝ}^{4}$. We prove that any properly embedded oriented surface $W\subset {S}^{3}×\left[0,1\right]$ 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 ${D}^{2}×{D}^{2}$, 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.