Braiding link cobordisms and non-ribbon surfaces

Mark C Hughes
2015 Algebraic and Geometric Topology  
We define the notion of a braided link cobordism in S^3 × [0,1], which generalizes Viro's closed surface braids in R^4. We prove that any properly embedded oriented surface W ⊂ S^3 × [0,1] is isotopic to a surface in this special position, and that the isotopy can be taken rel boundary when ∂ 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
more » ... 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.
doi:10.2140/agt.2015.15.3707 fatcat:xp2242fo2fhahdny3urzr2gfey