The structure of a solvmanifold’s Heegaard splittings. (English) Zbl 0948.57015

A solvmanifold \(M_L\) is the mapping cylinder of an orientation preserving homeomorphism \(L\) on the torus with \(|\text{trace}(L)|>2\). The authors classify irreducible Heegaard splittings (into two compression bodies) of solvmanifolds: (1) If \( |\text{trace}(L)|>3 \) then any two irreducible Heegaard splittings of \(M_L\) are isotopic. Furthermore the splitting is of genus two (resp. three) if it is strongly irreducible (resp. weakly reducible). (2) If \( |\text{trace}(L)|\) = 3 then \(M_L\) has precisely two isotopy classes of irreducible Heegaard splittings. These are strongly irreducible and of genus two.


57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
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