Summary: THE STRUCTURE OF THE HEEGAARD SPLITTINGS OF A
DARYL COOPER AND MARTIN SCHARLEMANN
Abstract. We classify isotopy classes of Heegaard splittings of solvmanifolds. If
the monodromy of the solvmanifold can be expressed as
` \Sigmam \Gamma1
for some m – 3 (as always is true when the trace of the monodromy is \Sigma3), then
any irreducible splitting is strongly irreducible and of genus two. If m – 4 any two
such splittings are isotopic. If m = 3 then, up to isotopy, there are exactly two
irreducible splittings, their associated hyperelliptic involutions commute, and their
product is the central involution of the solvmanifold.
If the monodromy cannot be expressed in the form above then the splitting is
weakly reducible, of genus three and unique up to isotopy.
The study of Heegaard splittings of 3manifolds is now nearly a century old [He].
(See also [Prz] for a translation of the relevant parts). Such a splitting is deceptively
simple to describe: a closed 3manifold is regarded as the union of two handlebodies
glued together along their boundaries. Although any 3manifold can be described