Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds

Let M1 and M2 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 M1, M2, 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.

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