Mathematics Colloquia and Seminars

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We classify isotopy classes of irreducible Heegaard splittings of solvmanifolds. If the monodromy of the solvmanifold can be expressed as a matrix with a zero in the right hand bottom corner and a plus or minus m, m no less than 3, in the upper left (as always is true when the trace of the monodromy is plus or minus 3), then any irreducible splitting is strongly irreducible and of genus two. If m is no less than 4 then 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.