Non-isotopic Heegaard splittings of Seifert fibered spaces. With an appendix by R.Weidmann (Q2492034)

The recent proof of the Waldhausen Conjecture [see \textit{T. Li}, ``Heegaard surfaces and measured laminations: the Waldhausen conjecture'', preprint arXiv:math.GT/0408198] establishes that a \(3\)-manifold \(M\) admits infinitely many non-isotopic Heegaard splittings of some genus only if \(M\) contains an incompressible torus. The authors are interested in the converse of this statement. The only known examples of \(3\)-manifolds that admit infinitely many non-isotopic Heegaard splittings of the same genus are given by \textit{K. Morimoto} and \textit{M. Sakuma} in [Math. Ann. 289, 143--167 (1991; Zbl 0697.57002)]. In the paper under review, the authors find a geometric invariant of isotopy classes of strongly irreducible Heegaard splittings of toroidal \(3\)-manifolds. Combining this invariant with a theorem of R. Weidmann (which states that irreducible Heegaard splittings of circle bundles are unique, see the appendix), they prove that a closed totally orientable Seifert fibered space \(M\) has infinitely many isotopy classes of Heegaard splittings of the same genus if and only if \(M\) has at least one irreducible horizontal Heegaard splitting, has a base orbifold with positive genus, and is not a circle bundle. This gives a complete characterization of closed totally orientable Seifert fibered spaces that satisfy the converse of the Waldhausen conjecture.