  Let M_1 and M_2 be compact, orientable 3-manifolds, and M the manifold
obtained by gluing some component F of \bdy M_1 to some component of \bdy M_2
by a homeomorphism \phi. We show that when \phi is "sufficiently complicated"
then (1) the amalgamation of low genus, unstabilized, boundary-unstabilized
Heegaard splittings of M_i is an unstabilized splitting of M, (2) every low
genus, unstabilized Heegaard splitting of M can be expressed as an amalgamation
of unstabilized, boundary-unstabilized splittings of M_i, and possibly a Type
II splitting of F \times I, and (3) if there is no Type II splitting in such an
expression then it is unique.
