Some applications of Gabai's internal hierarchy

TL;DR

This paper discusses Gabai's theorem on transforming Haken 3-manifolds into surface times circle, provides technical proof details, and explores applications to Heegaard Floer homology, including Thurston norm and knot classification.

Contribution

It offers detailed proof insights of Gabai's theorem and applies it to refine results in Heegaard Floer homology and knot classification in 3-manifolds.

Findings

Refined Thurston norm estimates

Classification of Floer simple knots

Technical proof details of Gabai's theorem

Abstract

Any Haken 3--manifold (possibly with boundary consisting of tori) can be transformed into a $\mathrm{surface}\times S^1$ 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 Gabai's 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 $b_1$.