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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.

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M50 General geometric structures on low-dimensional manifolds
##### Keywords:
Heegaard splitting; solvmanifold
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