 
Summary: TRANSAC:TIONS OF 'THE
AXIERICAV h~IATHEh~IATICA1.SOCIETY
Volume 3Z0. Vmmber 2, February 1998. Pagei 689 715
S 00029Sl47(98)(318248
COMPARING HEEGAARD SPLITTINGS
THE BOUNDED CASE
FIYAM RUBINSTEIN AND LIARTIN SCIIARLEAIANN
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 3manifold.
This captures all irreducible splittings of nonHaken 3manifolds. 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 2handles to a (connected surface) x I
along a collection of disjoint simple closed curves on (surface) x (01, and cap
ping off any resulting 2sphere boundary components with 3balls. 'The component
(surface) x (1) of dH is denoted a+H and the surface aH  a,.H is denoted dH.
If dH = 0 then H is a handlebody. If H = a+H x I then H is called a trzvzal
