Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds

Abstract

Let M 1 and M 2 be compact, orientable 3-manifolds with incompressible boundary, and M the manifold obtained by gluing with a homeomorphism $${\phi : {\partial}M_1 \to {\partial}M_2}$$ . We analyze the relationship between the sets of low genus Heegaard splittings of M 1, M 2, and M, assuming the map $${\phi}$$ is “sufficiently complicated”. This analysis yields counter-examples to the Stabilization Conjecture, a resolution of the higher genus analogue of a conjecture of Gordon, and a result about the uniqueness of expressions of Heegaard splittings as amalgamations.