The Structure of the Heegaard Splittings of a Solvmanifold

SOLVMANIFOLD 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 m 1 1 0 ; for some m 3 (as always is true when the trace of the monodromy is 3), 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.