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Heegaard splittings of Haken manifolds have bounded distance
 Pacific J. Math
"... Given a Heegaard splitting and an incompressible surface S and a Heegaard splitting of an irreducible manifold, I shall use a generalization of Haken’s lemma proved by Kobayashi in order to define a pair of simple closed curves on the splitting surface such that each bounds a disc in one of the hand ..."
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Given a Heegaard splitting and an incompressible surface S and a Heegaard splitting of an irreducible manifold, I shall use a generalization of Haken’s lemma proved by Kobayashi in order to define a pair of simple closed curves on the splitting surface such that each bounds a disc in one of the handlebodies of the splitting. By modifying the proof of Kobayashi’s lemma, I shall show that the sequence of boundary compressions used to isotope S places a bound on the distance between these two simple closed curves in the complex of curves. This will then place a bound on the distance of the Heegaard splitting. 1. Introduction. Let Σ be a closed, orientable surface of genus g ≥ 2. Associated with Σ is a “curve complex ” C(Σ) that has been defined by Harvey [4]. A vertex of this complex is an isotopy class of essential simple closed curves on Σ. Two vertices are joined by an edge if the corresponding isotopy classes have
Heegaard splittings and virtually Haken Dehn filling II
, 2006
"... We use Heegaard splittings to give a criterion for a tunnel number one knot manifold to be nonfibered and to have large cyclic covers. We also show that such a knot manifold (satisfying the criterion) admits infinitely many virtually Haken Dehn fillings. Using a computer, we apply this criterion to ..."
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We use Heegaard splittings to give a criterion for a tunnel number one knot manifold to be nonfibered and to have large cyclic covers. We also show that such a knot manifold (satisfying the criterion) admits infinitely many virtually Haken Dehn fillings. Using a computer, we apply this criterion to the 2 generator, nonfibered knot manifolds in the cusped Snappea census. For each such manifold M, we compute a number c(M), such that, for any n> c(M), the nfold cyclic cover of M is large. 1
MANIFOLDS ADMITTING BOTH STRONGLY IRREDUCIBLE AND WEAKLY REDUCIBLE MINIMAL GENUS
, 812
"... Abstract. We construct infinitely many manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Both closed manifolds and manifolds with boundary tori are constructed. The pioneering work of Casson and Gordon [1] shows that a minimal genus Heegaard splitt ..."
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Abstract. We construct infinitely many manifolds admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Both closed manifolds and manifolds with boundary tori are constructed. The pioneering work of Casson and Gordon [1] shows that a minimal genus Heegaard splitting of an irreducible, nonHaken 3manifold is necessarily strongly irreducible; by contrast, Haken [2] showed that a minimal genus (indeed, any) Heegaard splitting of a composite 3manifold is necessarily reducible, and hence weakly reducible. The following question of Moriah [8] is therefore quite natural: Question 1 ([8], Question 1.2). Can a 3manifold M have both weakly reducible and strongly irreducible minimal genus Heegaard splittings? We answer this question affirmatively: Theorem 2. There exist infinitely many closed, orientable 3manifolds of Heegaard genus 3, each admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Theorem 2 is proved in Section 2. In Remark 7 we offer a strategy to generalize Theorem 2 to construct examples of genus g, for each g ≥ 3; it is easy to see that no such examples can exist if g < 3. In Section 3 we give examples of manifolds with one, two or three torus boundary components, each admitting both strongly irreducible and weakly reducible minimal genus Heegaard splittings. Moreover,
BETWEEN LOWER AND HIGHER DIMENSIONS (in the work of Terry Lawson)
"... There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had ..."
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There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly welldeveloped theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental
RANK AND GENUS OF 3MANIFOLDS
"... 2. Heegaard splittings and amalgamation 779 3. Annulus sum 781 4. The construction of X 794 ..."
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2. Heegaard splittings and amalgamation 779 3. Annulus sum 781 4. The construction of X 794