Summary: TRANSAC:TIONS OF 'THE
Volume 3Z0. Vmmber 2, February 1998. Pagei 689 715
COMPARING HEEGAARD SPLITTINGS
-THE BOUNDED CASE
FIYAM RUBINSTEIN AND LIARTIN SCI-IARLEAIANN
A B S T R A ~ ~ .In a recent paper we used Cerf theory to compare strongly irre-
ducible Heegaard splittings of the same closed irreducible orientable 3-manifold.
This captures all irreducible splittings of non-Haken 3-manifolds. One appli-
cation is a solution to the stabilization problem for such splittings: If p < q
are the genera of two s11littings, then there is a common stabilization of genus
51, + 8q - 9. Here xve show how to obtain similar results even ~vhenthe 3-
manifold has boundary.
In this paper, all manifolds are assumed to be compact and orientable A com-
presszon body H is constructed by adding 2-handles to a (connected surface) x I
along a collection of disjoint simple closed curves on (surface) x (01, and cap-
ping off any resulting 2-sphere boundary components with 3-balls. 'The component
(surface) x (1) of dH is denoted a+H and the surface aH - a,.H is denoted d-H.
If d-H = 0 then H is a handlebody. If H = a+H x I then H is called a trzvzal