In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface.
Normal surfaces are ubiquitous in the theory of Haken manifolds and their simple and rigid structure leads quite naturally to algorithms.
The hierarchy makes proving certain kinds of theorems about Haken manifolds a matter of induction.
His outline was filled in by substantive efforts by Friedhelm Waldhausen, Klaus Johannson, Geoffrey Hemion, Sergeĭ Matveev, et al.
The hierarchy played a crucial role in William Thurston's hyperbolization theorem for Haken manifolds, part of his revolutionary geometrization program for 3-manifolds.
Johannson (1979) proved that atoroidal, anannular, boundary-irreducible, Haken three-manifolds have finite mapping class groups.