  Let M_1 and M_2 be compact, orientable 3-manifolds with incompressible
boundary, and M the manifold obtained by gluing with a homeomorphism $\phi:\bdy
M_1 \to \bdy M_2$. We analyze the relationship between the sets of low genus
Heegaard splittings of M_1, M_2, and M, assuming the map \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.
