Stabilizations of Heegaard splittings with respect to connect-sums of 3-manifolds

The purpose of this dissertation is to address a question which appears in Rob Kirby's Topology Problem List as Problem 3.91.; Let M be a 3-manifold and let F be a Heegaard splitting surface in M of genus g. Then (M, F) is said to be stabilized if it can be formed as a connect-sum of a Heegaard splitting of M of genus < g with a Heegaard splitting of S3. It is not hard to show that this property is equivalent to the existence of two disks D_A, D_B properly embedded in opposite components of $M\backslash F$ such that $6DA\cap 6DB$ is one point.; Problem 3.91 poses the following question: Let (M ₁, F₁), (M₂, F₂) be two unstabilized Heegaard splittings. Is ( M₁, F₁)#(M ₂, F₂) an unstabilized Heegaard splitting? This dissertation will address this question in the case that (M ₁, F₁) and (M₂, F₂) are strongly irreducible.