Celebratio Mathematica

H. Rubinstein and M. Scharlemann: “Comparing Heegaard splittings of non-Haken 3-manifolds,” Topology 35 : 4 (October 1996), pp. 1005–1026. MR 1404921 Zbl 0858.57020 article

Cerf theory can be used to compare two strongly irreducible Heegaard splittings of the same closed orientable 3-manifold. Any two splitting surfaces can be isotoped so that they intersect in a non-empty collection of curves, each of which is essential in both splitting surfaces. More generally, there are interesting isotopies of the splitting surfaces during which this intersection property is preserved. As sample applications we give new proofs of Waldhausen’s theorem that Heegaard splittings of \( \mathbb{S}^3 \) are standard, and of Bonahon and Otal’s theorem that Heegaard splittings of lens spaces are standard. We also present a solution to the stabilization problem for irreducible non-Haken 3-manifolds: If \( p \leq q \) are the genera of two splittings of such a manifold, then there is a common stabilization of genus \( 5p + 8q - 9 \).

H. Rubinstein and M. Scharlemann: “Transverse Heegaard splittings,” Mich. Math. J. 44 : 1 (1997), pp. 69–83. MR 1439669 Zbl 0907.57013 article

H. Rubinstein and M. Scharlemann: “Comparing Heegaard splittings: The bounded case,” Trans. Am. Math. Soc. 350 : 2 (1998), pp. 689–715. MR 1401528 Zbl 0892.57009 article

H. Rubinstein and M. Scharlemann: “Genus two Heegaard splittings of orientable three-manifolds,” pp. 489–553 in Proceedings of the KirbyFest (Berkeley, CA, 22–26 June 1998). Edited by J. Hass and M. Scharlemann. Geometry & Topology Monographs 2. International Press (Cambridge, MA), 1999. Papers dedicated to Rob Kirby on the occasion of his 60th birthday. MR 1734422 Zbl 0962.57013 ArXiv 9712262 incollection

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

J. Hyam Rubinstein	Related	Volume
Joel Hass	Related
Robion C. Kirby	Related	Volume

Geometry and topology down under: A conference in honour of Hyam Rubinstein (Melbourne, Australia, 11–22 July 2011). Edited by C. D. Hodgson, W. H. Jaco, M. G. Scharlemann, and S. Tillmann. Contemporary Mathematics 597. American Mathematical Society (Providence, RI), 2013. MR 3202515 Zbl 1272.57002 book

J. Hyam Rubinstein	Related	Volume
William H. Jaco	Related
Craig David Hodgson	Related
Stephan Tillmann	Related

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