Irreducible Heegaard Splittings Research Articles

On Heegaard splittings of mapping torus and induced by open book decompositions

In this paper, we give a sufficient condition for a keen weakly reducible and unstabilized Heegaard splitting of mapping torus and a sufficient condition for an irreducible, unstabilized and strongly irreducible Heegaard splitting induced by open book decomposition.

Strongly irreducible Heegaard splittings of hyperbolic 3-manifolds

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 finite collection of strongly irreducible Heegaard surfaces $S_i$ and incompressible surfaces $K_j$ such that any strongly irreducible Heegaard surface is a Haken sum $S_i + \sum_j n_j K_j$, up to one-sided associates of the Heegaard surfaces.

Finiteness of mapping class groups: locally large strongly irreducible Heegaard splittings

By Namazi and Johnson’s results, for any distance at least 4 Heegaard splitting, its mapping class group is finite. In contrast, Namazi showed that for a weakly reducible Heegaard splitting, its mapping class group is infinite; Long constructed an irreducible Heegaard splitting where its mapping class group contains a pseudo anosov map. Thus it is interesting to know that for a strongly irreducible but distance at most 3 Heegaard splitting, when its mapping class group is finite. In [19], Qiu and Zou introduced the definition of a locally large distance 2 Heegaard splitting. Extending their definition into a locally large strongly irreducible Heegaard splitting, we proved that its mapping class group is finite. Moreover, for a toroidal 3-manifold which admits a locally large distance 2 Heegaard splitting in [19], we prove that its mapping class group is finite.

Decomposing Heegaard splittings along separating incompressible surfaces in 3-manifolds

In this paper, by putting a separating incompressible surface in a 3-manifold into Morse position relative to the height function associated to a strongly irreducible Heegaard splitting, we show that an incompressible subsurface of the Heegaard splitting can be found, by decomposing the 3-manifold along the separating surface. Further if the Heegaard surface is of Hempel distance at least 4, then there is a pair of such subsurfaces on both sides of the given separating surface. This gives a particularly simple hierarchy for the 3-manifold.

Effective finiteness of irreducible Heegaard splittings of non-Haken 3-manifolds

The main result is a short effective proof of Tao Li’s theorem that a closed non-Haken hyperbolic 3-manifold N has at most finitely many irreducible Heegaard splittings. Along the way we show that N has finitely many branched surfaces of pinched negative sectional curvature carrying all closed index-≤1 minimal surfaces. This effective result, together with the sequel with Daniel Ketover, solves the classification problem for Heegaard splittings of non-Haken hyperbolic 3-manifolds.

Reducible handle additions to weakly reducible Heegaard splittings

Let V∪SW be an irreducible Heegaard splitting of M and F a component of ∂−V. A simple closed curve J in F is called reducible if MJ=VJ⋃SW is reducible, where MJ is obtained by attaching a 2-handle to M along J. In this paper, we give a sufficient condition for the diameter of the set of reducible curves in F to be bounded in C(F) for the keen weakly reducible Heegaard splittings. Moreover, an upper bound of the diameter of the set of the reducible curves in F is given under some circumstance for the weakly reducible Heegaard splittings.

Unstabilized and uncritical self-amalgamation along essential subsurfaces

Suppose V ∪SW is a strongly irreducible Heegaard splitting of a compact connected orientable 3-manifold M and F1 and F2 are pairwise disjoint homeomorphic essential subsurfaces in ∂_V. In this paper, we give a suffcient condition such that the self-amalgamation of V ∪SW along F1 and F2 is unstabilized and uncritical.

A sufficient condition for a self-amalgamation of a Heegaard splitting to be unstabilized

In this paper, we will give a sufficient condition for self-amalgamation of a strongly irreducible Heegaard splitting along any essential subsurface to be unstabilized.

High distance Heegaard splittings via Dehn twists

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 when using a high power of a suitably chosen Dehn twist. In certain cases, we can then determine the exact distance of the resulting splitting. These results can be seen as a natural extension of work by A. Casson and C. Gordon in 1987 regarding strongly irreducible Heegaard splittings.

Are large distance Heegaard splittings generic?

In a previous paper we introduced a notion of genericity for countable sets of curves in the curve complex of a surface S, based on the Lebesgue measure on the space of projective measured laminations in S. With this definition we prove that for each fixed g > 1 the set of irreducible genus g Heegaard splittings of high distance is generic, in the set of all irreducible Heegaard splittings. Our definition of genericity is different and more intrinsic then the one given via random walks.

ON (DISK, ANNULUS) PAIRS OF HEEGAARD SPLITTINGS THAT INTERSECT IN ONE POINT

Let <TEX>$M=H_1{\cup}_SH_2$</TEX> be a Heegaard splitting of a 3-manifold M, D be an essential disk in <TEX>$H_1$</TEX> and A be an essential annulus in <TEX>$H_2$</TEX>. Suppose D and A intersect in one point. First, we show that a Heegaard splitting admitting such a (D, A) pair satisfies the disjoint curve property, yet there are infinitely many examples of strongly irreducible Heegaard splittings with such (D, A) pairs. In the second half, we obtain another Heegaard splitting <TEX>$M=H'_1{\cup}_{S'}H'_2$</TEX> by removing the neighborhood of A from <TEX>$H_2$</TEX> and attaching it to <TEX>$H_1$</TEX>, and show that <TEX>$M=H'_1{\cup}_{S'}H'_2$</TEX> also has a (D, A) pair with <TEX>$|D{\cap}A|=1$</TEX>.

Non-isotopic Heegaard splittings of Seifert fibered spaces

We find a geometric invariant of isotopy classes of strongly irreducible Heegaard splittings of toroidal 3-manifolds. Combining this invariant with a theorem of R Weidmann, proved here in the appendix, we show that a closed, totally orientable Seifert fibered space M has infinitely many isotopy classes of Heegaard splittings of the same genus if and only if M has an irreducible, horizontal Heegaard splitting, has a base orbifold of positive genus, and is not a circle bundle. This characterizes precisely which Seifert fibered spaces satisfy the converse of Waldhausen's conjecture.

Sweepouts of amalgamated 3–manifolds

We show that if two 3-manifolds with toroidal boundary are glued via a `sufficiently complicated' map then every Heegaard splitting of the resulting 3-manifold is weakly reducible. Additionally, if Z is a manifold obtained by gluing X and Y, two connected small manifolds with incompressible boundary, along a closed surface F. Then the genus g(Z) of Z is greater than or equal to 1/2(g(X)+g(Y)-2g(F)). Both results follow from a new technique to simplify the intersection between an incompressible surface and a strongly irreducible Heegaard splitting.

Heegaard surfaces and measured laminations, II: Non-Haken 3–manifolds

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 irreducible Heegaard splittings, up to isotopy. This is much stronger than the generalized Waldhausen conjecture. Another immediate corollary is that for any irreducible non-Haken 3–manifold M M , there is a number N N such that any two Heegaard splittings of M M are equivalent after at most N N stabilizations.

Locally unknotted spines of Heegaard splittings

We show that under reasonable conditions, the spines of the handlebodies of a strongly irreducible Heegaard splitting will intersect a closed ball in a graph which is isotopic into the boundary of the ball. This is in some sense a generalization of the results by Scharlemann on how a strongly irreducible Heegaard splitting surface can intersect a ball.

Surface bundles versus Heegaard splittings

This paper studies Heegaard splittings of surface bundles via the curve complex of the fibre. The translation distance of the monodromy is the smallest distance it moves any vertex of the curve complex. We prove that the translation distance is bounded above in terms of the genus of any strongly irreducible Heegaard splitting. As a consequence, if a splitting surface has small genus compared to the translation distance of the monodromy, then the splitting is standard.

CLOSED ESSENTIAL SURFACES AND WEAKLY REDUCIBLE HEEGAARD SPLITTINGS IN MANIFOLDS WITH BOUNDARY

We show that there are infinitely many two component links in S3 whose complements have weakly reducible and irreducible non-minimal genus Heegaard splittings, yet the construction given in the theorem of Casson and Gordon does not produce an essential closed surface. The situation for manifolds with a single boundary component is still unresolved though we obtain partial results regarding manifolds with a non-minimal genus weakly reducible and irreducible Heegaard splitting.

Heegaard splittings of graph manifolds

Let M be a totally orientable graph manifold with characteristic submanifold T and let M = V cup_S W be a Heegaard splitting. We prove that S is standard. In particular, S is the amalgamation of strongly irreducible Heegaard splittings. The splitting surfaces S_i of these strongly irreducible Heegaard splittings have the property that for each vertex manifold N of M, S_i cap N is either horizontal, pseudohorizontal, vertical or pseudovertical.

Local detection of strongly irreducible Heegaard splittings via knot exteriors

Let H 1∪ Σ H 2 be a strongly irreducible Heegaard splitting of a 3-manifold M other than S 3, and X a 3-dimensional submanifold of M such that: (1) X is homeomorphic to the exterior of a non-trivial knot in S 3, and (2) there is a compressing disk, say D X , of ∂X such that ∂D X is a meridian curve of X. Suppose that ∂X∩ Σ consists of a non-empty collection of simple closed curves which are essential in both ∂X and Σ. Then we show that: (1) the closure of some component of Σ⧹ ∂X is an annulus and is parallel to an annulus in ∂X, and (2) each component of Σ∩ X is a (possibly boundary parallel) meridional annulus.

Critical Heegaard surfaces

In this paper we introduce critical surfaces, which are described via a 1-complex whose definition is reminiscent of the curve complex. Our main result is that if the minimal genus common stabilization of a pair of strongly irreducible Heegaard splittings of a 3-manifold is not critical, then the manifold contains an incompressible surface. Conversely, we also show that if a non-Haken 3-manifold admits at most one Heegaard splitting of each genus, then it does not contain a critical Heegaard surface. In the final section we discuss how this work leads to a natural metric on the space of strongly irreducible Heegaard splittings, as well as many new and interesting open questions.