Haken manifold

Summary

A Haken manifold is a compact irreducible 3-manifold that contains a two-sided incompressible surface. Such a manifold can be cut along incompressible surfaces again and again until only balls remain, a process called a Haken hierarchy. This makes Haken manifolds the class where induction arguments work. Many results, such as Waldhausen's topological rigidity and Thurston's hyperbolization, were first proved for them.