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Berge’s distance 3 pairs of genus 2 Heegaard splittings. (English) Zbl 1226.57033
J. Berge [A classification of pairs of disjoint nonparallel primitives in the boundary of a genus two handlebody, preprint] gave a criterion for a Heegaard splitting of genus two of a \(3\)-manifold to be of Hempel distance at least 3, and discovered an example of multiple Heegaard splittings of the same \(3\)-manifold which have distance 3. Unfortunately, this implies that the listing of all possible ways, by H. Rubinstein and M. Scharlemann [Proceedings of the Kirbyfest, Berkeley, CA, USA, June 22-26, 1998. Warwick: University of Warwick, Institute of Mathematics, Geom. Topol. Monogr. 2, 489–553 (1999; Zbl 0962.57013)], in which a \(3\)-manifold might have distinct genus two Heegaard splittings is incomplete, because all examples of multiple Heegaard splittings appearing there have distance at most two. For details, see J. Berge and M. Scharlemann [Algebr. Geom. Topol. 11, No. 3, 1781–1792 (2011; Zbl 1232.57020)].
In the paper under review, the author describes a general way of constructing further examples of multiple Heegaard splittings with distance 3. Such examples are obtained as the result of Dehn surgery on a specific \(4\)-component link in \((S^1\times S^2)\sharp (S^1\times S^2)\).

MSC:
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
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[1] DOI: 10.1016/S0166-8641(98)00063-7 · Zbl 0935.57022 · doi:10.1016/S0166-8641(98)00063-7
[2] DOI: 10.1016/S0040-9383(00)00033-1 · Zbl 0985.57014 · doi:10.1016/S0040-9383(00)00033-1
[3] DOI: 10.2140/gtm.1999.2.489 · doi:10.2140/gtm.1999.2.489
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