Heegaard splittings of toroidal 3-manifolds

Abstract

This dissertation is an investigation into the Stabilization Problem for Heegaard splittings of toroidal 3-manifolds. In several situations we obtain upper bounds on the number of stabilizations needed for two Heegaard splittings of a toroidal 3-manifold to become isotopic. Two corollaries of interest are: (1) Two strongly irreducible Heegaard splittings of sufficiently large genus of a graph manifold become isotopic after at most one stabilization of the higher genus splitting, and (2) Two strongly irreducible Heegaard splittings of genus g which are obtained by Dehn twisting along a canonical torus in the JSJ decomposition of a 3-manifold become isotopic after at most 4g−4 stabilizations.