Essential Laminations Research Articles

Tight Contact Structures on Laminar Free Hyperbolic Three-Manifolds

Whether every hyperbolic 3-manifold admits a tight contact structure or not is an open question. Many hyperbolic 3-manifolds contain taut foliations and taut foliations can be perturbed to tight contact structures. The first examples of hyperbolic 3-manifolds without taut foliations were constructed by Roberts, Shareshian, and Stein, and infinitely many of them do not even admit essential laminations as shown by Fenley. In this paper, we construct tight contact structures on a family of 3-manifolds including these examples. These contact structures are described by contact surgery diagrams and their tightness is proved using the contact invariant in Heegaard Floer homology.

Laminar free hyperbolic 3-manifolds

The purpose of the article is to prove that there are infinitely many closed hyperbolic 3-manifolds which do not admit essential laminations. The manifolds are obtained by Dehn surgery on torus bundles over the circle. This gives a definitive negative answer to a fundamental question posed by Gabai and Oertel when they introduced essential laminations. The proof is obtained by analysing group actions on on trees and showing that certain 3-manifold groups only have trivial actions on trees. There are corollaries concerning the existence of Reebless foliations and pseudo-Anosov flows.

Promoting essential laminations

We show that every co--orientable taut foliation F of an orientable, atoroidal 3-manifold admits a transverse essential lamination. If this transverse lamination is a foliation G, the pair F,G are the unstable and stable foliation respectively of an Anosov flow. Otherwise, F admits a pair of transverse very full genuine laminations. In the second case, M satisfies the weak geometrization conjecture - either its fundamental group contains Z+Z or it is word-hyperbolic. Moreover, if M is atoroidal, the mapping class group of M is finite, and any automorphism homotopic to the identity is isotopic to the identity.

Essential laminations and branched surfaces in the exteriors of links

In the same paper, by using the result above, he proposes a procedure for determining whether a given manifold contains an essential lamination or not. However the procedure does not work, at present, since (1) there is not a practical algorithm for determining whether a given branched surface is essential or not, and (2) there does not exist an algorithm for determining whether a given branched surface fully carries a lamination or not. The purpose of this paper is to try to carry out the procedure to the exteriors of links given by diagrams, by using various techniques in knot and link theory, and 3dimensional topology. In fact, we give a definition of standard position (with respect to a diagram of a given link) for branched surfaces contained in the exterior of links in section 2, which is a natural generalization of standard position of closed incompressible surfaces defined by W. Menasco [M1]. In section 3, we apply the result of [B], to show that any essential lamination in a link exterior can be deformed into one carried by an essential branched surface in standard position with respect to a given diagram. In section 4, we study about branched surfaces in standard positions with respect to alternating diagrams, and give a sufficient condition for the branched surfaces to be incompressible and Reebless, and possess indecomposable exteriors (for the definitions of these terms, see section 2). In [O], U.Oertel studied some fundamental properties of affine laminations in 3-manifolds. In section 5, we give a necessary and sufficient condition for a given branched surface in standard position to fully carry affine laminations, by using

Contact structures on elliptic $3$-manifolds

We show that an oriented elliptic 3-manifold admits a universally tight positive contact structure if and only if the corresponding group of deck transformations on S 3 (after possibly conjugating by an isometry) preserves the standard contact structure. We also relate universally tight contact structures on 3-manifolds covered by S 3 to the isomorphism SO(4) = (SU(2) ◊ SU(2))/±1. The main tool used is equivariant framings of 3-manifolds. A contact structure on a 3-dimensional manifold M is a smooth, totally non- integrable tangent plane field, i.e., a tangent plane field locally of the form = ker( ) for a 1-form such that ^ d is everywhere non-degenerate. We shall assume that M is oriented. We say is positive if the orientation on M agrees with that induced by the volume form ^ d . Observe that the orientation of ^d does not depend on the sign of , and is thus determined by (even though = ker( ) only locally). A central role in understanding 3-dimensional manifolds has been played by co- dimension one structures - surfaces, foliations and laminations - in these manifolds. Without additional conditions such structures always exist, and are of not much consequence. However, the presence of essential co-dimension one structures - incompressible surfaces, taut foliations and essential laminations, leads to deep

Laminations and groups of homeomorphisms of the circle

If M is an atoroidal 3-manifold with a taut foliation, Thurston showed that π1(M) acts on a circle. Here, we show that some other classes of essential laminations also give rise to actions on circles. In particular, we show this for tight essential laminations with solid torus guts. We also show that pseudo-Anosov flows induce actions on circles. In all cases, these actions can be made into faithful ones, so π1(M) is isomorphic to a subgroup of Homeo(S 1). In addition, we show that the fundamental group of the Weeks manifold has no faithful action on S 1. As a corollary, the Weeks manifold does not admit a tight essential lamination with solid torus guts, a pseudo-Anosov flow, or a taut foliation. Finally, we give a proof of Thurston’s universal circle theorem for taut foliations based on a new, purely topological, proof of the Leaf Pocket Theorem.

Pre-taut sutured manifolds and essential laminations

In 1989, D. Gabai and U. Oertel [8] introduced the concept of the essential lamination, which is a hybrid object lying between incompressible surfaces and taut foliations, and generalizing both. We say that a 3-manifold is laminar if it contains an essential lamination. An important result of [8] is that the universal covers of laminar manifolds are homeomorphic to R. This fact furnishes a strong method for studying the manifolds obtained by Dehn surgery along knots, especially concerning Property P Conjecture (nontrivial Dehn surgery on a nontrivial knot in 3 never yields a simply-connected manifold) and Cabling Conjecture (Dehn surgery on a non-cable knot cannot yield a reducible manifold). For example, see [4] for non-torus alternating knots, [3], [12] for 2-bridge knots, [17] for most algebraic knots and [9] for knots with some kind of essential tangle decompositions. We note that by [8] a 3-manifold is laminar if and only if it contains an essential branched surface (for the definition see §2), and the above authors who followed [8] obtained their results by constructing essential branched surfaces. We note that sutured manifold theory was used in [14] and [18]. One of their approaches is to construct a closed essential branched surface in the exterior ( ) of a knot and show that remains essential after any nontrivial Dehn filling along ∂ ( ) (we call such persistently essential). Then we see, by [8], that has Property P in a strong form and that the cabling conjecture is true for . (We say that a knot has strong Property P if every manifold obtained by a nontrivial Dehn surgery along has universal cover R.) It is, however, an open question whether or not every knot with strong Property P admits a persistently essential lamination in its complement. In [1], [2], M. Brittenham had a paradigm shift in proving strong Property P for knots. Instead of constructing a branched surface in the complement of a given knot, he first constructed a branched surface and then embedded a knot in its complement. More precisely, he first constructed a closed branched surface in 3 from any in-

Essential laminations in Seifert-fibered spaces: Boundary behavior

We show that an essential lamination in a Seifert-fibered space M rarely meets the boundary of M in a Reeb-foliated annulus.

Dehn surgery on arborescent links

This paper studies Dehn surgery on a large class of links, called arborescent links. It will be shown that if an arborescent link L L is sufficiently complicated, in the sense that it is composed of at least 4 4 rational tangles T ( p i / q i ) T(p_{i}/q_{i}) with all q i > 2 q_{i} > 2 , and none of its length 2 tangles are of the form T ( 1 / 2 q 1 , 1 / 2 q 2 ) T(1/2q_{1}, 1/2q_{2}) , then all complete surgeries on L L produce Haken manifolds. The proof needs some result on surgery on knots in tangle spaces. Let T ( r / 2 s , p / 2 q ) = ( B , t 1 ∪ t 2 ∪ K ) T(r/2s, p/2q) = (B, t_{1}\cup t_{2}\cup K) be a tangle with K K a closed circle, and let M = B − Int ⁡ N ( t 1 ∪ t 2 ) M = B - \operatorname {Int} N(t_{1}\cup t_{2}) . We will show that if s > 1 s>1 and p ≢ ± 1 p \not \equiv \pm 1 mod 2 q 2q , then ∂ M \partial M remains incompressible after all nontrivial surgeries on K K . Two bridge links are a subclass of arborescent links. For such a link L ( p / q ) L(p/q) , most Dehn surgeries on it are non-Haken. However, it will be shown that all complete surgeries yield manifolds containing essential laminations, unless p / q p/q has a partial fraction decomposition of the form 1 / ( r − 1 / s ) 1/(r-1/s) , in which case it does admit non-laminar surgeries.

Essential laminations and Kneser normal form

Essential laminations and Kneser normal form

ESSENTIAL LAMINATIONS, EXCEPTIONAL SEIFERT-FIBERED SPACES, AND DEHN FILLING

We show how essential laminations can be used to provide an improvement on (some of) the results of the 2π-Theorem; at most 20 Dehn fillings on a hyperbolic 3-manifold with boundary a torus T can yield a reducible manifold, finite π1 manifold, or exceptional Seifert-fibered space. Recent work of Wu allows us to add toroidal manifolds to this list, as well.

Dehn surgery on arborescent knots and links—A survey

In this survey, we present some recent results about Dehn surgeries on arborescent knots and links. Arborescent links are also known as algebraic links [1, 2] and star links [3]. The set of arborescent knots and links is a large class, including all 2-bridge links and Montesinos links. They have been studied by many people, see e.g. [4,2,5,3,6, 7]. We will give some definitions below. The reader is referred to [8, 9] for more detailed background material for 3-manifold topology, to [2,1,4,10] for arborescent tangles and links, to [11] for hyperbolic manifolds, and to [12] for essential laminations and branched surfaces.

Affine laminations and their stretch factors

A lamination L embedded in a manifold M is an ane lamination if its lift L to the universal cover ~ M of M is a measured lamination and each covering translation multiplies the measure by a given by a homomorphism, called the stretch homomorphism, from 1(M) to the positive real numbers. There is a method for analyzing precisely the set of ane laminations carried by a given branched manifold B embedded in M. The notion of the \stretch factor of an ane lamination is a generalization of the notion of the stretch of a pseudo-Anosov map. The same method that serves to analyze the ane laminations carried by B also allows calculation of stretch factors. Ane laminations occur commonly as essential 2-dimensional laminations in 3-manifolds. We shall describe some examples. In particular, we describe ane essential laminations which represent classes in real 2-dimensional homology with twisted coecients. The rst version of this paper was written in 1988, at about the same time as the paper [HO1], which is a study of spaces of ane laminations in surfaces. Since then, there has been a renewed interest in ane laminations, see for example [L] and [Ca]. The current version includes a new section on ane 2-laminations in 3-manifolds. These arise, for example, as realizations of classes in the real homology with twisted coecients of a 3-manifold. More general constructions show that ane essential laminations are, in some sense, abundant in 3-manifolds, see [GO] for background on essential laminations.

Sutured manifold hierarchies, essential laminations, and Dehn surgery

x0. Introduction A compact orientable surface F with nonnegative Euler characteristic is either a sphere, a disk, a torus, or an annulus. If a 3-manifold M contains such an essential surface, then it is said to be reducible, @-reducible, toroidal, or annular, respectively. Any such surface can be used to decompose the manifold further into simpler manifolds. We say that M is a simple manifold if it has no such surfaces. A simple manifold is expected to have a nice geometric structure. If M has nonempty boundary, then the Geometrization Theorem of Thurston for Haken manifolds says that M with boundary tori removed admits a nite volume hyperbolic structure with totally geodesic boundary. When M has no boundaries, Thurston’s Geometrization Conjecture asserts that M is either hyperbolic, or is a Seifert b er space with orbifold a sphere with at most 3 cone points. Suppose T is a torus boundary component of M. We use M( ) to denote the manifold obtained by Dehn lling on T so that the slope on T bounds a disk in the Dehn lling solid torus. When M = E(K) is the exterior of a knot K in S 3 , denote M( ) by K( ), and call it the manifold obtained by surgery on the knot K. It is well known that if M is simple then there are only nitely many Dehn llings on T which produce non simple manifolds. If M( 1) and M( 2) are non simple manifolds, then the geometric intersection number between 1 and 2, denoted by ( 1; 2), is proved to be at most 8 by Gordon [Gor1]. There are 10 dieren t cases, according to the types of nonnegative Euler characteristic surfaces in M( i). In many cases, the upper bounds for ( 1; 2) have been established, see Table 0.1.

Constructing essential laminations in 2-bridge knot surgered 3-manifolds

A nontrivial knot that can be drawn with only two relative maxima in the vertical direction is called a 2-bridge knot, and one that can be drawn on a torus is called a torus knot. Loosely speaking, a lamination in a manifold M is a foliation of M, except that it can have a nonempty open complement in M, and very loosely speaking, the lamination is essential if each leaf of it L is incompressible, i.e. inclusion of L into M induces an injective homomorphism from 1(L) into 1(M). Our main result is: Theorem 2. Every 3-manifold obtained by surgery on a nontorus 2-bridge knot admits essential laminations. Some immediate corollaries are that these manifolds are covered by R 3 , and have innite fundamental group. So Property P is true for non-torus 2-bridge knots, i.e. surgery on these knots never yields a homotopy 3-sphere, or a counterexample to the Poincare conjecture. We use general techniques which are not specic to 2-bridge knots to nd and explicitly construct these laminations. In trying to understand 3-manifolds with the hope of eventually classifying them, as with 2-manifolds, one approach that has turned out to be fruitful is to study objects of codimension one in them, more specically, incompressible surfaces, Reebless foliations, and essential laminations. For a 3-manifoldM containing an incompressible surface, with some extra hypotheses, Waldhausen proved: The universal cover of M is R 3 , the topology of M is determined by its fundamental group, and homotopic homeomorphisms of M are isotopic. Similar theorems for manifolds containing Reebless foliations were proven by Haefliger, Novikov, Palmeira, Rosenberg, and others. The essential lamination was developed in the 1980’s as a generalization of the incompressible surface and the Reebless foliation, which themselves qualify as essential laminations. In fact they are just extreme cases of essential laminations: At one end we have compact properly embedded surfaces,

Essential laminations in 𝐼-bundles

We show that, with a few familiar exceptions, every essential lamination in an interval-bundle over a closed surface can be isotoped to lie everywhere transverse to the I I -fibers of the bundle.

Full laminations in 3-manifolds

Essential laminations in 3-manifolds, introduced in [GO], are a natural generalization of both incompressible surfaces and foliations without Reeb components. One of the main reasons why essential laminations are of particular interest is that many manifolds which contain no incompressible surfaces contain essential laminations. Indeed, there is some evidence that almost all irreducible manifolds with infinite fundamental group contain essential laminations [B1;D;GM;GO;H;N;R], the only known exceptions being Seifert manifolds [B2]. So essential laminations are potentially a quite powerful tool for studying non-Haken 3-manifolds.

Essential laminations and Dehn surgery on 2-bridge knots

In this paper we show that any nontrivial Dehn surgery on a nontorus 2-bridge knot produces a manifold which is covered by R 3. In particular, this manifold is irreducible and has infinite fundamental group. (As a consequence, it is also clear that 2-bridge knots satisfy property P, although this was shown previously by Takahashi (1981) using more algebraic techniques.) The result is a consequence of showing that such a manifold is laminar, that is it contains an essential lamination. We accomplish this by constructing in the exterior of each nontorus 2-bridge knot an essential lamination which remains essential in all manifolds produced by nontrivial Dehn filling. We call an essential lamination with this property persistent. The examples of essential laminations produced in this way are of interest since they tend support the conjecture that “most” manifolds with infinite fundamental group contain essential laminations.

Essential laminations and Haken normal form

Essential laminations and Haken normal form

Essential laminations and Haken normal form: Laminations with no holonomy

Essential laminations and Haken normal form: Laminations with no holonomy