Genus two Heegaard splittings of orientable three-manifolds. (English) Zbl 0962.57013

Rubinstein, Hyam; Scharlemann, Martin

Hass, Joel (ed.) et al., 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); correction ibid. 2, 577-581 (1999).

This paper examines ways in which two non-isotopic genus-two irreducible Heegaard splittings of the same closed orientable 3-manifold compare, with a particular focus on how the corresponding hyperelliptic involutions are related. The authors, using a technique similar to one introduced by them in an earlier paper [Topology 35, No. 4, 1005-1026 (1996; Zbl 0858.57020)], list all ways in which two such Heegaard splittings compare, and show that the list is exhaustive. The comparison is only qualitative, since two Heegaard splittings with different descriptions in the paper may be isotopic. Some results concerning the corresponding hyperelliptic involutions, which present the manifold as two-fold covering of \(S^3\) branched over a three-bridge knot or link, are proved. For example, if \(M\) is an atoroidal closed orientable 3-manifold then the involutions coming from distinct Heegaard splittings necessarily commute. Moreover, they prove that two genus-two Heegaard splittings of the same manifold become equivalent after a single stabilization.
For the entire collection see [Zbl 0939.00055].

Reviewer: Michele Mulazzani (Bologna)

MSC:

57N10	Topology of general \(3\)-manifolds (MSC2010)
57M50	General geometric structures on low-dimensional manifolds

Keywords:

Seifert manifold; hyperelliptic involution

Citations:

Zbl 0858.57020