Summary: PROXIMITY IN THE CURVE COMPLEX: BOUNDARY
REDUCTION AND BICOMPRESSIBLE SURFACES
ABSTRACT. Suppose N is a compressible boundary component of a
compact irreducible orientable 3-manifold M and (Q,Q) (M,M) is
an orientable properly embedded essential surface in M in which some
essential component is incident to N and no component is a disk. Let
V and Q denote respectively the sets of vertices in the curve complex
for N represented by boundaries of compressing disks and by boundary
components of Q.
Theorem: Suppose Q is essential in M, then d(V,Q) 1 -(Q).
Hartshorn showed ([Ha]) that an incompressible surface in a closed 3-
manifold puts a limit on the distance of any Heegaard splitting. An aug-
mented version of the theorem above leads to a version of Hartshorn's
result for merely compact 3-manifolds.
In a similar spirit, here is the main result:
Theorem: Suppose a properly embedded connected surface Q is in-
cident to N. Suppose further that Q is separating and compresses on both
its sides, but not by way of disjoint disks. Then either
· d(V,Q) 1 -(Q) or