#### Geometry & Topology Monographs 2 (1999),
Proceedings of the Kirbyfest,
paper no. 24, pages 489-553.

## Genus two Heegaard splittings of orientable three-manifolds

### Hyam Rubinstein, Martin Scharlemann

**Abstract**.
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.
**Keywords**.
Heegaard splitting, Seifert manifold, hyperelliptic involution

**AMS subject classification**.
Primary: 57N10.
Secondary: 57M50.

**E-print:** `arXiv:math.GT/9712262`

Submitted: 10 September 1998.
(Revised: 8 June 1999.)
Published: 22 November 1999.

Notes on file formats
Hyam Rubinstein, Martin Scharlemann

Department of Mathematics, University of Melbourne

Parkville, Vic 3052, Australia

Mathematics Department, University of California

Santa Barbara, CA 93106, USA

Email: rubin@ms.unimelb.edu.au, mgscharl@math.ucsb.edu

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