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THE STRUCTURE OF THE HEEGAARD SPLITTINGS OF A SOLVMANIFOLD
 

Summary: THE STRUCTURE OF THE HEEGAARD SPLITTINGS OF A
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
` \Sigmam \Gamma1
1 0
' ;
for some m – 3 (as always is true when the trace of the monodromy is \Sigma3), 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.
1. Introduction
The study of Heegaard splittings of 3­manifolds is now nearly a century old [He].
(See also [Prz] for a translation of the relevant parts). Such a splitting is deceptively
simple to describe: a closed 3­manifold is regarded as the union of two handlebodies
glued together along their boundaries. Although any 3­manifold can be described

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics