Left orderability and taut foliations with one-sided branching

It is a well-established principle in manifold topology that by studying codimension-1 objects and their complementary pieces one obtains a great deal of topological and geometric information about the manifold itself. This is particularly true for 3-dimensional manifolds. Using incompressible surfaces a great many advances have been made during the last 30 years; the most spectacular results were obtained by Haken [Ha], Waldhausen [W] and Thurston [T1]. More recently, work [T2], [G14] using taut foliations in 3-manifolds has proven fruitful. Unfortunately (in some sense) most closed 3-manifolds do not contain incompressible surfaces and it is currently not known exactly which 3-manifolds have taut foliations. The purpose of this paper is to study another codimension-1 object, the lamination, which is a generalization of the incompressible surface as well as the taut foliation. We will show that the existence of an lamination in a 3-manifold M implies that M has useful properties similar to those of manifolds having either an incompressible surface or a taut foliation. We will also show that the universal cover of a manifold containing an lamination is R3. Precise definitions of terms used in the following theorem will be given in Section 1. Here we give only a rough idea. We begin with two alternative approximate definitions of essential lamination. (I) An lamination in a 3-manifold M is a lamination satisfying four conditions: The inclusion of leaves of the lamination into M should induce an injection on sTJ, the complement of the lamination should be irreducible, no leaf should be a sphere, and the lamination should be end-incompressible. To say that a lamination is end-incompressible means, roughly, that a folded leaf can be straightened using an isotopy; there are no infinite folds. (II) Alternatively

We define a norm on the homology of a foliated manifold, which refines and majorizes the usual Gromov norm on homology. This norm depends in an upper semi-continuous way on the underlying foliation, in the geometric topology. We show that this norm is non-trivial — i.e. it distinguishes certain taut foliations of a given hyperbolic 3-manifold.¶Using a homotopy-theoretic refinement, we show that a taut foliation whose leaf space branches in at most one direction cannot be the geometric limit of a sequence of isotopies of a fixed taut foliation whose leaf space branches in both directions. Our technology also lets us produce examples of taut foliations which cannot be made transverse to certain geodesic triangulations of hyperbolic 3-manifolds, even after passing to a finite cover.¶Finally, our norm can be extended to actions of fundamental groups of manifolds on order trees, where it has similar upper semi-continuity properties.

In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation F evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation F′ such that S is homologous to a union of leaves and such that the plane field of F′ is homotopic to that of F. In particular, F and F′ have the same Euler class. In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston's conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper.

Given a fibred, compact, orientable 3-manifold with single boundary component, we show that a fibration with fiber surface of negative Euler characteristic can be perturbed to yield taut foliations realizing an open interval of boundary slopes about the boundary slope of the fibration. These taut foliations extend to taut foliations in the corresponding surgery manifolds. 2000 Mathematics Subject Classification: primary 57M25; secondary 57R30.

We construct taut foliations in every closed 3-manifold obtained by $r$-framed Dehn surgery along a positive 3-braid knot $K$ in $S^3$, where $r < 2g(K)-1$ and $g(K)$ denotes the Seifert genus of $K$. This confirms a prediction of the L-space Conjecture. For instance, we produce taut foliations in every non-L-space obtained by surgery along the pretzel knot $P(-2,3,7)$, and indeed along every pretzel knot $P(-2,3,q)$, for $q$ a positive odd integer. This is the first construction of taut foliations for every non-L-space obtained by surgery along an infinite family of hyperbolic L-space knots. Additionally, we construct taut foliations in every closed 3-manifold obtained by $r$-framed Dehn surgery along a positive 1-bridge braid in $S^3$, where $r <g(K)$.

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.

Sutured instanton Floer homology was introduced by Kronheimer and Mrowka. In this paper, we prove that for a taut balanced sutured manifold with vanishing second homology, the dimension of the sutured instanton Floer homology provides a bound on the minimal depth of all possible taut foliations on that balanced sutured manifold. The same argument can be adapted to the monopole and even the Heegaard Floer settings, which gives a partial answer to one of Juhasz's conjectures. Using the nature of instanton Floer homology, on knot complements, we can construct a taut foliation with bounded depth, given some information on the representation varieties of the knot fundamental groups. This indicates a mystery relation between the representation varieties and some small depth taut foliations on knot complements, and gives a partial answer to one of Kronheimer and Mrowka's conjecture.

This paper concerns the problem of existence of taut foliations among 3-manifolds. From the work of David Gabai we know that a closed 3-manifold with non-trivial second homology group admits a taut foliation. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite different if they are integral or rational but non-integral homology 3-spheres. Concerning integral homology 3-spheres, we can see that all but the 3-sphere and the Poincaré 3-sphere admit a taut foliation. Concerning non-integral homology 3-spheres, we prove there are infinitely many that admit a taut foliation, and infinitely many without a taut foliation. Moreover, we show that the geometries do not determine the existence of taut foliations on non-integral Seifert fibered homology 3-spheres.

This article proves that the parity of the number of Klein-bottle leaves in a smooth cooriented taut foliation is invariant under smooth deformations within taut foliations, provided that every Klein-bottle leaf involved in the counting has non-trivial linear holonomy.

We show the equivalence of several notions in the theory of taut foliations and the theory of tight contact structures. We prove equivalence, in certain cases, of existence of tight contact structures and taut foliations.

Let $M$ be a fibered 3-manifold with multiple boundary components. We show that the fiber structure of $M$ transforms to closely related transversely oriented taut foliations realizing all rational multislopes in some open neighborhood of the multislope of the fiber. Each such foliation extends to a taut foliation in the closed 3-manifold obtained by Dehn filling along its boundary multislope. The existence of these foliations implies that certain contact structures are weakly symplectically fillable.

We generalize the main results from the author's paper in Geom. Topol. 4 (2000), 457–515 and from Thurston's eprint math.GT/9712268 to taut foliations with one-sided branching. First constructed by Meigniez, these foliations occupy an intermediate position between ℝ-covered foliations and arbitrary taut foliations of 3-manifolds. We show that for a taut foliation $$F$$ with one-sided branching of an atoroidal 3-manifold M, one can construct a pair of genuine laminations Λ± of M transverse to $$F$$ with solid torus complementary regions which bind every leaf of $$F$$ in a geodesic lamination. These laminations come from a universal circle, a refinement of the universal circles proposed by Thurston (unpublished), which maps monotonely and π1(M)-equivariantly to each of the circles at infinity of the leaves of $$\tilde F$$ , and is minimal with respect to this property. This circle is intimately bound up with the extrinsic geometry of the leaves of $$\tilde F$$ . In particular, let $$\tilde F$$ denote the pulled-back foliation of the universal cover, and co-orient $$\tilde F$$ so that the leaf space branches in the negative direction. Then for any pair of leaves of $$\tilde F$$ with μλ, the leaf λ is asymptotic to μ in a dense set of directions at infinity. This is a macroscopic version of an infinitesimal result from Thurston and gives much more drastic control over the topology and geometry of $$F$$ , than is achieved by him. The pair of laminations Λ± can be used to produce a pseudo-Anosov flow transverse to $$F$$ which is regulating in the nonbranching direction. Rigidity results for Λ± in the ℝ-covered case are extended to the case of one-sided branching. In particular, an ℝ-covered foliation can only be deformed to a foliation with one-sided branching along one of the two laminations canonically associated to the ℝ-coveredfoliation constructed in Geom. Topol. 4 (2000), 457–515, and these laminations become exactly the laminations Λ± for the new branched foliation. Other corollaries include that the ambient manifold is δ-hyperbolic in the sense of Gromov, and that a self-homeomorphism of this manifold homotopic to the identity is isotopic to the identity.

Bill Thurston proved that taut foliations of hyperbolic 3‐manifolds have Euler classes of norm at most one, and conjectured that any integral second cohomology class of norm equal to one is realized as the Euler class of some taut foliation. Recent work of the second author, joint with David Gabai, has produced counterexamples to this conjecture. Since tight contact structures exist whenever taut foliations do and their Euler classes also have norm at most one, it is natural to ask whether the Euler class one conjecture might still be true for tight contact structures. In this paper, we show that the previously constructed counterexamples for Euler classes of taut foliations in Mehdi Yazdi [Acta Math. 225 (2020) no. 2, 313–368] are in fact realized as Euler classes of tight contact structures. This provides some evidence for the Euler class one conjecture for tight contact structures.

We show that for any nontrivial knot in $S^3$, there is an open interval containing zero such that a Dehn surgery on any slope in this interval yields a 3-manifold with taut foliations. This generalizes a theorem of Gabai on zero frame surgery.

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.