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Genus two Heegaard splittings of orientable three-manifolds

J Hyam Rubinstein and Martin Scharlemann

Geometry & Topology Monographs 2 (1999) 489–553

DOI: 10.2140/gtm.1999.2.489

Erratum: Geometry & Topology Monographs 2 (1999) 577–581

arXiv: math.GT/9712262


It was shown by Bonahon–Otal and Hodgson–Rubinstein that any two genus–one Heegaard splittings of the same 3–manifold (typically a lens space) are isotopic. On the other hand, it was shown by Boileau, Collins and Zieschang that certain Seifert manifolds have distinct genus–two Heegaard splittings. In an earlier paper, we presented a technique for comparing Heegaard splittings of the same manifold and, using this technique, derived the uniqueness theorem for lens space splittings as a simple corollary. Here we use a similar technique to examine, in general, ways in which two non-isotopic genus–two Heegard splittings of the same 3–manifold compare, with a particular focus on how the corresponding hyperelliptic involutions are related.


Heegaard splitting, Seifert manifold, hyperelliptic involution

Mathematical Subject Classification

Primary: 57N10

Secondary: 57M50


Received: 10 September 1998
Revised: 8 June 1999
Published: 22 November 1999

J Hyam Rubinstein
Department of Mathematics
University of Melbourne
Victoria 3052
Martin Scharlemann
Mathematics Department
University of California
Santa Barbara CA 93106