3-manifolds as viewed from the curve complex

A Heegaard diagram for a 3-manifold is regarded as a pair of simplexes in t he complex of curves on a surface and a Heegaard splitting as a pair of sub complexes generated by the equivalent diagrams. We relate geometric and com binatorial properties of these subcomplexes with topological properties of the manifold and/or the associated splitting. For example we show that for any splitting of a 3-manifold which is Seifert fibered or which contains an essential torus the subcomplexes are at a distance at most two apart in th e simplicial distance on the curve complex; whereas there are splittings in which the subcomplexes are arbitrarily far apart. We also give obstruction s, computable from a given diagram, to being Seifert fibered or to containi ng an essential torus. (C) 2001 Elsevier Science Ltd. All rights reserved.