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 ${\text{RR}}^n$, 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.