Stabilizing and destabilizing Heegaard splittings of sufficiently complicated 3-manifolds
arXiv:1201.3438
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 $ϕ:\bdy M_1 \to \bdy M_2$. We analyze the relationship between the sets of low genus Heegaard splittings of M_1, M_2, and M, assuming the map ϕis "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.
This is the final version, to appear in Mathematische Annalen. This paper combines the results in arxiv:0903.1692, 0903.1695, and 0904.0485, and replaces those three papers