Genus Two Heegaard Splittings of Orientable Three-manifolds
نویسندگان
چکیده
It is shown in [BoO] that any two genus one Heegaard splittings of the same manifold (typically a lens space) are isotopic. On the other hand, it is shown in [BCZ] that certain Seifert manifolds have distinct genus two Heegaard splittings (see also Section 3 here). In [RS] we presented a technique for comparing Heegaard splittings of the same manifold and, using this technique, derive the Bonahon-Otal uniqueness theorem for lens space splittings as a simple corollary. The intent of this paper is to use a similar technique to examine, in general, ways in which two non-isotopic genus two Heegard splittings of the same 3-manifold are related.
منابع مشابه
Genus Two Heegaard Splittings of Orientable Three-manifolds Hyam Rubinstein and Martin Scharlemann
Contents 1. Introduction 2 2. Cabling handlebodies 3 3. Seifert examples of multiple Heegaard splittings 6 4. Other examples of multiple Heegaard splittings 8 4.1. Cablings 9 4.2. Double cablings 10 4.3. Non-separating tori 11 4.4. K 4 examples 13 5. Essential annuli in genus two handlebodies 16 6. Canonical tori in Heegaard genus two manifolds 20 7. Longitudes in genus 2 handlebodies – some te...
متن کاملConnected sums of unstabilized Heegaard splittings are unstabilized
Let M1 and M2 be closed, orientable 3–manifolds. Let Hi denote a Heegaard surface in Mi . We prove that if H1#H2 comes from stabilizing a lower genus splitting of M1#M2 then one of H1 or H2 comes from stabilizing a lower genus splitting. This answers a question of C Gordon [9, Problem 3.91]. We also show that every unstabilized Heegaard splitting has a unique expression as the connected sum of ...
متن کاملincompressible surfaces and stabilizations of Heegaard splittings
We describe probably the simplest 3-manifold which contains closed separating incompressible surfaces of arbitrarily large genus. Two applications of this observation are given. (1) For any closed, orientable 3-manifold M and any integer m> 0, a surgery on a link inM of at most 2m+1 components will provide a closed, orientable, irreducible 3-manifold containing m disjoint, non-parallel, separat...
متن کاملHeegaard Splittings of Branched Coverings of S 3
This paper concerns itself with the relationship between two seemingly different methods for representing a closed, orientable 3-manifold: on the one hand as a Heegaard splitting, and on the other hand as a branched covering of the 3-sphere. The ability to pass back and forth between these two representations will be applied in several different ways: 1. It will be established that there is an ...
متن کامل