Four-dimensional Haken cobordism theory

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Cobordism of Haken n-manifolds is defined by a Haken (n+1)-manifold W whose boundary has two components, each of which is a closed Haken n-manifold. In addition, the inclusion map of the fundamental group of each boundary component to π₁(W) is injective. In this paper we prove that there are 4-dimensional Haken cobordisms whose boundary consists of any two closed Haken 3-manifolds. In particular, each closed Haken 3-manifold is the π₁-injective boundary of some Haken 4-manifold.