Braiding link cobordisms and non-ribbon surfaces

TL;DR

This paper introduces a new concept of braided link cobordisms in 3-sphere cross an interval, proving any embedded surface can be isotoped into this form, with applications to studying surfaces in 4-space and 4-manifolds.

Contribution

It defines braided link cobordisms in $S^3 imes [0,1]$ and shows all properly embedded surfaces can be isotoped into this position, extending previous notions of surface braiding.

Findings

Any properly embedded oriented surface in $S^3 imes [0,1]$ can be isotoped to a braided link cobordism.

Introduces braided surfaces with caps as a generalization of Rudolph's braided surfaces.

Applications include algebraic techniques for surfaces in 4-space and constructing fibrations on 4-manifolds.

Abstract

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.