Related papers: Many Haken Heegaard splittings

We construct examples of closed non-Haken hyperbolic 3-manifolds with a Heegaard splitting of arbitrarily large distance.

A famous example of Casson and Gordon shows that a Haken 3-manifold can have an infinite family of irreducible Heegaard splittings with different genera. In this paper, we prove that a closed non-Haken 3-manifold has only finitely many…

We define a notion of Hempel distance for one-sided Heegaard splittings and show that the existence of alternate surfaces restricts distance for one-sided splittings in a manner similar to Hartshorn's and Scharlemann-Tomova's results for…

We examine three key conjectures in 3-manifold theory: the virtually Haken conjecture, the positive virtual b_1 conjecture and the virtually fibred conjecture. We explore the interaction of these conjectures with the following seemingly…

Casson and Gordon gave the rectangle condition for strong irreducibility of Heegaard splittings [1]. We give a parity condition for irreducibility of Heegaard splittings of irreducible manifolds. As an application, we give examples of…

We show that sub-surfaces of a Heegaard surface for which the relative Hempel distance of the splitting is sufficiently high have to appear in any Heegaard surface of genus bounded by half that distance.

We provide two constructions of hyperbolic metrics on 3-manifolds with Heegaard splittings that satisfy certain topological conditions, which both apply to random Heegaard splittings with asymptotic probability 1. These constructions…

We consider 3-manifolds given as Heegaard splittings $M=H^-\cup_\Sigma H^+$ with the aim to describe the hyperbolic metric of $M$ under topological conditions on the splitting guaranteeing that the manifold is hyperbolic. In particular,…

We give a necessary and sufficient condition for the fundamental group of the space of Heegaard splittings of an irreducible $3$-manifold to be finitely generated. The condition is exactly the conclusion of the thick isotopy lemma proved by…

We use thin position of Heegaard splittings to give a new proof of Haken's Lemma that a Heegaard surface of a reducible manifold is reducible and of Scharlemann's ``Strong Haken Theorem'': a Heegaard surface for a 3-manifold may be isotoped…

Haken showed that the Heegaard splittings of reducible 3-manifolds are reducible, that is, a reducing 2-sphere can be found which intersects the Heegaard surface in a single simple closed curve. When the genus of the "interesting" surface…

In 2001, J. Hempel proved the existence of Heegaard splittings of arbitrarily high distance by using a high power of a pseudo-Anosov map as the gluing map between two handlebodies. We show that lower bounds on distance can also be obtained…

Colding and Gabai have given an effective version of Li's theorem that non-Haken hyperbolic 3-manifolds have finitely many irreducible Heegaard splittings. As a corollary of their work, we show that Haken hyperbolic 3-manifolds have a…

Non-isotopic Heegaard splittings of non-minimal genus were known previously only for very special 3-manifolds. We show in this paper that they are in fact a wide spread phenomenon in 3-manifold theory: We exhibit a large class of knots and…

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…

A manifold which admits a reducible genus-$2$ Heegaard splitting is one of the $3$-sphere, $S^2 \times S^1$, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is…

Suppose T is a Heegaard splitting surface for a compact orientable 3-manifold M, and S is a reducing sphere for M. In 1968 Haken showed that there is then also a reducing sphere S* for the Heegaard splitting. That is, S* is a reducing…

We construct infinitely many linearly independent quasi-homomorphisms on the mapping class group of a Riemann surface with genus at least two which vanish on a handlebody subgroup. As a corollary, we disprove a conjecture of Reznikov on…

In this paper we show that for a given 3-manifold and a given Heegaard splitting there are finitely many preferred decomposing systems of $3g - 3$ disjoint essential disks. These are characterized by a combinatorial criterion which is a…

A gap in a paper of Rubinstein-Scharlemann is explored: new examples are found of closed orientable 3-manifolds with possibly multiple genus 2 Heegaard splittings. Properties common to all the examples in the original paper are not…