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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
Full Text: DOI arXiv
[1] J Berge, A classification of pairs of disjoint nonparallel primitives in the boundary of a genus two handlebody · arxiv:0910.3038
[2] J Berge, A closed orientable 3-manifold with distinct distance three genus two Heegaard splittings, to appear
[3] J Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (2001) 631 · Zbl 0985.57014 · doi:10.1016/S0040-9383(00)00033-1
[4] Y N Minsky, The classification of punctured-torus groups, Ann. of Math. \((2)\) 149 (1999) 559 · Zbl 0939.30034 · doi:10.2307/120976 · www.math.princeton.edu · eudml:120365
[5] H Rubinstein, M Scharlemann, Genus two Heegaard splittings of orientable three-manifolds, Geom. Topol. Monogr. 2, Geom. Topol. Publ., Coventry (1999) 489 · Zbl 0962.57013 · doi:10.2140/gtm.1999.2.489 · emis:journals/UW/gt/GTMon2/paper24.abs.html
[6] M Scharlemann, Berge’s distance 3 pairs of genus 2 Heegaard splittings · Zbl 1226.57033 · doi:10.1017/S0305004111000223 · arxiv:1002.4887
[7] M Scharlemann, Genus two Heegaard splittings: an omission, Geom. Topol. Monogr. 2, Geom. Topol. Publ., Coventry (1999) 577 · Zbl 0962.57013 · doi:10.2140/gtm.1999.2.489
[8] M Scharlemann, A Thompson, Heegaard splittings of \((\mathrm{surface})\times I\) are standard, Math. Ann. 295 (1993) 549 · Zbl 0814.57010 · doi:10.1007/BF01444902 · eudml:165057
[9] A Thompson, The disjoint curve property and genus 2 manifolds, Topology Appl. 97 (1999) 273 · Zbl 0935.57022 · doi:10.1016/S0166-8641(98)00063-7
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