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arXiv2013

Four-dimensional Haken cobordism theory

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 $π_1(W)$ 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 $π_1$-injective boundary of some Haken 4-manifold.

math.GT↗

arXiv2011

The universal covering space of a Haken $n$--manifold

We define the class of Haken $n$--manifolds as a generalisation of Haken 3--manifolds. We prove that the interior of the universal covering of a Haken $n$--manifold is $\RR^n$, which generalises a result of Waldhausen. The techniques used allow us to provide a new proof of Waldhausen's universal cover theorem for Haken 3--manifolds.

math.GT↗