Applications
The hierarchy makes proving certain kinds of theorems about Haken manifolds a matter of induction. One proves the theorem for 3-balls. Then one proves that if the theorem is true for pieces resulting from a cutting of a Haken manifold, that it is true for that Haken manifold. The key here is that the cutting takes place along a surface that was very "nice", i.e. incompressible. This makes proving the induction step feasible in many cases.
Haken sketched out a proof of an algorithm to check if two Haken manifolds were homeomorphic or not. His outline was filled in by substantive efforts by Waldhausen, Johannson, Hemion, Matveev, et al. Since there is an algorithm to check if a 3-manifold is Haken (cf. Jaco-Oertel), the basic problem of recognition of 3-manifolds can be considered to be solved for Haken manifolds.
Waldhausen (1968) proved that closed Haken manifolds are topologically rigid: roughly, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism (for the case of boundary, a condition on peripheral structure is needed). So these three-manifolds are completely determined by their fundamental group. In addition, Waldhausen proved that the fundamental groups of Haken manifolds have solvable word problem; this is also true for virtually Haken manifolds.
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. This result can be recovered from the combination of Mostow rigidity with Thurston's geometrization theorem.
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