 
Summary: ALTERNATE HEEGAARD GENUS BOUNDS DISTANCE
MARTIN SCHARLEMANN AND MAGGY TOMOVA
ABSTRACT. Suppose M is a compact orientable irreducible 3manifold
with Heegaard splitting surfaces P and Q. Then either Q is isotopic to a
possibly stabilized copy of P or the distance d(P) 2genus(Q).
More generally, if P and Q are bicompressible but weakly incom
pressible connected closed separating surfaces in M then either
· P and Q can be wellseparated or
· P and Q are isotopic or
· d(P) 2genus(Q).
1. INTRODUCTION
Suppose M is an irreducible compact orientable 3manifold and P M
is a closed connected separating surface properly embedded in M. P is
bicompressible if it compresses into both complementary components A and
B. P is strongly compressible if there are compressing disks for P in A and B
which have disjoint boundaries in P. If this is not the case, then P is weakly
incompressible.
Given a closed bicompressible surface P in M, there is a natural general
ization, essentially due to Hempel [He], of the notion of strong compress
ibility. Let U,V be the sets of isotopy classes of essential simple closed
