Some applications of Gabaiʼs internal hierarchy

Any Haken 3-manifold (possibly with boundary consisting of tori) can be transformed into a surface×S1 by a series of splitting and regluing along incompressible surfaces. This fact was proved by Gabai as an application of his sutured manifold theory. The first half of this paper provides a few technical details in the proof. In the second half of this paper, some applications of Gabaiʼs theorem to Heegaard Floer homology are given. We refine the known results about the Thurson norm and fibrations. We also give some classification results for Floer simple knots in manifolds with positive b1.