## Illinois Journal of Mathematics

### Four-dimensional Haken cobordism theory

#### Abstract

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 $\pi_{1}(W)$ 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 $\pi_{1}$-injective boundary of some Haken $4$-manifold.

#### Article information

Source
Illinois J. Math., Volume 60, Number 1 (2016), 1-17.

Dates
Revised: 18 August 2015
First available in Project Euclid: 21 June 2017