Multiple genus 2 Heegaard splittings: a missed case. (English) Zbl 1232.57020

It was shown independently in [M. Boileau and J.-P. Otal, C. R. Acad. Sci., Paris, Sér. I 303, 19–22 (1986; Zbl 0596.57010)] and [C. Hodgson and J. H. Rubinstein, in: Knot theory and manifolds, Proc. Conf., Vancouver/Can. 1983, Lect. Notes Math. 1144, 60–96 (1985;Zbl 0605.57022)] that any two genus-one Heegaard decompositions of a given 3-manifold are isotopic. In contrast, some Seifert manifolds have distinct genus-two Heegaard splittings. In [Geom. Topol. Monogr. 2, 489–553 (1999; Zbl 0962.57013)], H. Rubinstein and M. Scharlemann gave a near-complete list of ways in which multiple splittings could be constructed.
In the paper reviewed here, the authors fill a gap in this earlier work by describing a final class of examples omitted from the list of Rubinstein and Scharlemann. The examples have some interesting new properties not shared by the examples in Rubinstein and Scharlemann’s paper. In particular, at least some of the new examples are of Hempel distance 3, whereas all of the examples in the original work were of Hempel distance 2.


57N10 Topology of general \(3\)-manifolds (MSC2010)
57M15 Relations of low-dimensional topology with graph theory
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