1998/03/31 by Daryl Cooper, Cooper, Daryl, Martin Scharlemann +1 · 3 citations
Mathematics · #57M50 #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds #math.GT #msc:57M50
paper · pdf · doi:10.48550/arxiv.math/9803157
17 pages including 7 figures
arxiv created 1998/03/31 · openalex publication_date 1998/03/31 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We classify isotopy classes of irreducible Heegaard splittings of solvmanifolds. If the monodromy of the solvmanifold can be expressed as a 2 x 2 matrix with 0 in the lower right hand corner (as always is true when the absolute value of the trace is 3), then any irreducible splitting is strongly irreducible and of genus two. If furthermore the absolute value of the trace is 4 or greater, then any two such splittings are isotopic. If the absolute value of the trace is 3 then, up to isotopy, there are exactly two irreducible splittings, their associated hyperelliptic involutions commute, and the product of the involutions is the central involution of the solvmanifold. If the monodromy cannot be expressed as a 2 x 2 matrix with 0 in the lower right hand corner, then the splitting is weakly reducible, of genus three and unique up to isotopy.