Genus two Heegaard splittings of orientable three-manifolds

Details Genus two Heegaard splittings of orientable three-manifolds Genus two Heegaard splittings of orientable three-manifolds

1997-12-24

arxiv.org/abs/math/9712262v3

Geom. Topol. Monogr. 2 (1999), 489-553

Mathematics

Geometric Topology

65 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon2/paper24.abs.html

Scientific paper

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.

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