  A Heegaard diagram for a 3-manifold is regarded as a pair of simplexes in the
complex of curves on a surface and a Heegaard splitting as a pair of
subcomplexes generated by the equivalent diagrams. We relate geometric and
combinatorial 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 the
simplicial distance on the curve complex; whereas there are splittings in which
the subcomplexes are arbitrarily far apart. We also give obstructions,
computable from a given diagram, to being Seifert fibered or to containing an
essential torus.
