3 D ec 1 99 7 3 - MANIFOLDS AS VIEWED FROM THE CURVE COMPLEX

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. 0. Introduction. Throughout S will denote a closed, connected, oriented surface of genus g ≥ 2. The curve complex of S, denoted C(S), will be the complex whose vertices are the isotopy classes of essential simple closed curves in S, and where distinct vertices x0, x1, . . . , xk determine a k-simplex of C(S) if they are represented by pairwise disjoint simple closed curves. If we fix a hyperbolic metric on S, then each isotopy class contains a unique geodesic. Moreover two isotopy classes have disjoint representatives if and only if their geodesic representatives are disjoint. We will thus always think of vertices as being geodesics and will use the same notation for a simplex of C(S), the corresponding collection of mutually exclusive simple closed curves in S, and their union as a subset of S. A simplex X of C(S) determines a compression body VX = S × [0, 1] ∪X×1 2− handles ∪ 3− handles obtained by attaching 2-handles along the components of X × 1 and filling in any resulting 2-sphere boundary components with 3-cells. S × 0 is called the outer boundary of VX and is naturally identified with S. A pair X, Y of simplexes of C(S) determine a (Heegaard) splitting (S;VX , VY ) of a 3-manifold MX,Y = VX ∪S VY Research at MSRI is supported in part by NSF grant DMS-9022140 Typeset by AMS-TEX 1