Left orderability and taut foliations with one-sided branching
Bojun Zhao
本文研究了具有单侧分支的紧致定向不可约3-流形的左可排序性，证明其基本群是左可排序的。
对于一个闭合的、可定向的、不可约的3-流形M，如果M admits一个可共定向的紧致叶面分解且具有单侧分支，则我们证明π₁(M)是左可排序的。
For a closed orientable irreducible $3$-manifold $M$ that admits a co-orientable taut foliation with one-sided branching, we show that $π_1(M)$ is left orderable.
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We study the question of when cyclic branched covers of knots admit taut foliations, have left-orderable fundamental group, and are not L-spaces.
Left orderability and taut foliations with orderable cataclysm
本文研究了3维流形的左可序性，通过可定向紧致叶面和伪Anosov流的性质，证明了特定条件下的流形基群的左可序性，并探讨了Dehn填充的应用。
This paper explores the conjecture that the following are equivalent for rational homology 3-spheres: having left-orderable fundamental group, having non-minimal Heegaard Floer homology, and admitting a co-orientable taut foliation. In particular, it adds further evidence in favor of this conjecture by studying these three properties for more than 300,000 hyperbolic rational homology 3-spheres. New or much improved methods for studying each of these properties form the bulk of the paper, including a new combinatorial criterion, called a foliar orientation, for showing that a 3-manifold has a t…
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 describe notions of tautness that arise in the study of $C^0$ foliations, $C^{1,0}$ or smoother foliations, and in geometry. We give examples to show that these notions are different, and discuss how these differences impact some classical foliation results. We construct examples of smoothly taut $C^{\infty,0}$ foliations that can be $C^0$ approximated by both weakly symplectically fillable, universally tight contact structures and by overtwisted contact structures.
Tight contact structures and taut foliations
我们证明了紧叶理理论和紧切触结构理论中几个概念的等价性。在某些情况下，我们证明了紧切触结构和紧叶理的存在是等价的。
Tautly foliated 3-manifolds with no R-covered foliations
流形M的一个叶状结构称为R-覆盖的，如果它到M的泛覆盖的提升具有叶空间R。我们发现，有许多图流形，承认拉紧叶理，但不承认任何R-覆盖叶理。另一方面，我们证明了这些流形都有允许R-覆盖叶理的有限覆盖。
Left-orderability in Dehn fillings of pseudo-Anosov mapping tori
研究伪阿诺索夫映射环的Dehn填充中基本群的左序性，通过构造紧叶理并分析其分支行为，证明了所有此类填充具有左序性，并验证了L-空间猜想。
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.
Transverse fibered knots and Taut Foliations
本研究给出Honda-Kazec-Matic定理的简短替代证明，验证了满足特定条件的纤维纽结所支撑的切触结构可由紧叶状结构微扰得到，为相关几何拓扑结论提供了新的证明路径。
The Euler classes of taut foliations on graph manifolds
本文确定了闭图流形上所有切触叶状结构的欧拉类集合，并证明任意闭无球面图流形的闭对偶瑟斯顿单位球中的整系数类可虚拟实现为欧拉类，同时通过例子说明虚拟条件的必要性及实与整欧拉类为零的差异。
Almost continuous extension for taut foliations
一个双曲3-流形的绷紧叶理在几乎每个方向上都具有连续扩张性质;也就是说，对于叶理的泛覆盖的每个叶和叶中的几乎每个测地线，3-流形的泛覆盖中的射线的极限是理想边界中的一个明确定义的点。
We show that a graph manifold which is a Z-homology 3-sphere not homeomorphic to either the 3-sphere or the Poincaré homology 3-sphere admits a horizontal foliation. This combines with known results to show that the conditions of not being an L-space, of having a left-orderable fundamental group, and of admitting a co-oriented taut foliation, are equivalent for graph manifold Z-homology 3-spheres.
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.
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. …
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, …
Taut foliations of 3-manifolds with Heegaard genus two
本文证明了具有 Heegaard 亏格二的闭可定向不可约 3-流形，若其基本群左可序，则必存在可余定向的紧叶状结构，揭示了群序性质与叶状结构之间的联系。
Let $M$ be a closed orientable irreducible $3$-manifold such that $π_1(M)$ is left orderable. (a) Let $M_0 = M - Int(B^{3})$, where $B^{3}$ is a compact $3$-ball in $M$. We have a process to produce a co-orientable Reebless foliation $\mathcal{F}$ in $M_0$ such that: (1) $\mathcal{F}$ has a transverse $(π_1(M),\mathbb{R})$ structure, (2) there exists a simple closed curve in $M$ that is co-orientably transverse to $\mathcal{F}$ and intersects every leaf of $\mathcal{F}$. More specifically, given a pair $(<,Γ)$ composed of a left-invariant order "$<$" of $π_1(M)$ and a fundamental domain $Γ$ of…
If $Y$ is a closed orientable graph manifold, we show that $Y$ admits a coorientable taut foliation if and only if $Y$ is not an L-space. Combined with previous work of Boyer and Clay, this implies that $Y$ is an L-space if and only if $π_1(Y)$ is not left-orderable.
A remark on taut foliations and Floer homology
本文证明了承认切触叶状结构的有理同调球面的约化Floer同调作为$\mathbb{F}[U]$-模存在直接$\mathbb{F}$-直和项，从而为通过Floer理论检测$L$-空间猜想反例提供了可能。
L-spaces, taut foliations and fibered hyperbolic two-bridge links
本文证明有理同调球M是纤维化双曲双桥链环上的Dehn手术时，M不是L-空间当且仅当M支持可定向切触叶状结构，并推广到双桥替换操作。
In this article, we study the Euler class of taut foliations on the Dehn fillings of a $\mathbb{Q}$-homology solid torus. We give a necessary and sufficient condition for the Euler class of a foliation transverse to the core of the filling solid torus to vanish. We apply this condition to taut foliations on Dehn fillings of hyperbolic fibered manifolds and obtain many new left-orderable Dehn filling slopes on these manifolds. For instance, we show that when $X$ is the exterior of a pretzel knot $P(-2,3,2r+1)$, $r\geq 3$, $π_1(X(α_n))$ is left-orderable for a sequence of positive slopes $α_n$ w…
Let $F$ be a leafwise hyperbolic taut foliation of a closed 3-manifold $M$ and let $L$ be the leaf space of the pullback of $F$ to the universal cover of $M$. We show that if $F$ has branching, then the natural action of $π_1(M)$ on $L$ is faithful. We also show that if $F$ has a finite branch locus $B$ whose stabilizer acts on $B$ nontrivially, then the stabilizer is an infinite cyclic group generated by an indivisible element of $π_1(M)$.
We study positive braid knots (the knots in the three-sphere realized as positive braid closures) through the lens of the L-space conjecture. This conjecture predicts that if $K$ is a non-trivial positive braid knot, then for all $r < 2g(K)-1$, the 3-manifold obtained via $r$-framed Dehn surgery along $K$ admits a taut foliation. Our main result provides some positive evidence towards this conjecture: we construct taut foliations in such manifolds whenever $r<g(K)+1$. As an application, we produce a novel braid positivity obstruction for cable knots by proving that the $(n,\pm 1)$-cable of a k…
Positive braid closures and taut foliations
该研究探讨了正编织闭包在 $S^3$ 中的紧叶面性质，证明了特定 Dehn 手术条件下可得到非 L-space 并支持 co-oriented 紧叶面。
We study the left-orderability of the fundamental groups of cyclic branched covers of links which admit co-oriented taut foliations. In particular we do this for cyclic branched covers of fibred knots in integer homology $3$-spheres and cyclic branched covers of closed braids. The latter allows us to complete the proof of the L-space conjecture for closed, connected, orientable, irreducible $3$-manifolds containing a genus $1$ fibred knot. We also prove that the universal abelian cover of a manifold obtained by generic Dehn surgery on a hyperbolic fibred knot in an integer homology $3$-sphere …
Note on Euler class 0 taut foliation on the Whitehead link exterior
本文研究 Whitehead 链环外部上余可定向 taut 叶状结构的存在性，证明对偶单位 Thurston 球中大多数格点不能实现为 Euler 类，并结合 saddle fillings、指标公式与 Dehn surgery 拓扑阻碍给出一般性研究方法。
A 3-manifold is foliar if it supports a codimension-one co-oriented taut foliation. Suppose $M$ is an oriented 3-manifold with connected boundary a torus, and suppose $M$ contains a properly embedded, compact, oriented, surface $R$ with a single boundary component that is Thurston norm minimizing in $H_2(M, \partial M)$. We define a readily recognizable type of sutured manifold decomposition, which for notational reasons we call double-diamond taut, and show that if $R$ admits a double-diamond taut sutured manifold decomposition, then every boundary slope except one is strongly realized by a c…
On 3-manifolds admitting co-orientable taut foliations, but none with vanishing Euler class
本文构造了允许共定向紧致叶面但无消失欧拉类的3维流形，揭示了其与L空间猜想及左可序基本群的关系。
L-spaces, taut foliations and the Whitehead link
证明Whitehead链环上的Dehn手术得到的有理同调球是L-空间当且仅当它不支持共定向切触叶状结构，并确定部分流形的左可序性。
Torus leaves play a crucial role in the theory of foliations. For example non-taut foliations admit a torus leaf (see the article of Goodman). In this paper, we study all the foliations near a torus leaf, and try to understand why sometimes it is taut, or non-taut (and Reebless). We focus on some crucial examples to understand non-taut and Reebless foliations; and foliations admitting a non-separating torus leaf. It relies on the study of turbulization and spiraling, with generalizations.
Motivated by conjectures relating group orderability, Floer homology, and taut foliations, we discuss a systematic and broadly applicable technique for constructing left-orders on the fundamental groups of rational homology 3-spheres. Specifically, for a compact 3-manifold $M$ with torus boundary, we give several criteria which imply that whole intervals of Dehn fillings of $M$ have left-orderable fundamental groups. Our technique uses certain representations from $π_1(M)$ into $\widetilde{\mathrm{PSL}_2 \mathbb{R}}$, which we organize into an infinite graph in $H^1(\partial M; \mathbb{R})$ ca…
For an oriented irreducible 3-manifold M with non-empty toroidal boundary, we describe how sutured Floer homology ($SFH$) can be used to determine all fibered classes in $H^1(M)$. Furthermore, we show that the $SFH$ of a balanced sutured manifold $(M,γ)$ detects which classes in $H^1(M)$ admit a taut depth one foliation such that the only compact leaves are the components of $R(γ)$. The latter had been proved earlier by the first author under the extra assumption that $H_2(M)=0$. The main technical result is that we can obtain an extremal $\text{Spin}^c$-structure $\mathfrak{s}$ (i.e., one tha…
The Gromov norm and foliations
我们定义了一个规范上的同调的叶状流形，细化和优化通常的Gromov规范上的同调。在几何拓扑中，这个范数以上半连续的方式依赖于底层的叶理，因此可以用来研究哪些叶理是其他叶理的合痕类的几何极限。我们表明，这一规范是非平凡的-即它区分某些拉紧foliations的一个给定的双曲3-流形。事实上，模一些标准的拓扑，这个规范应该精确地检测封闭的环面3-流形的拉紧叶理，那些通用覆盖在两个方向上的分支。同伦理论的改进在某些情况下提供了更多的信息，特别是，我们表明，绷紧叶理的叶空间中的通用覆盖分支在最多一个方向不能是一个序列的几何极限的同位素的一个固定的绷紧叶理的叶空间中的通用覆盖分支在两个方向。我们的技术还可以让我们产生的例子拉紧叶理，不能横向于某些测地线三角形的双曲3-流形，即使在通过一个有限的覆盖。最后，我们的范数可以推广到流形的基本群在序树上的作用，在那里它具有类似的上半连续性。
Taut foliations, left-orders, and pseudo-Anosov mapping tori
本文通过构造基本群在实数上的保向同胚作用来刻画紧叶理3-流形的横向几何，并证明八字结所有非平凡手术的基本群左可序，同时将该方法应用于 Hodgson-Weeks 普查中 44.7% 的非 L-空间有理同调球。
This paper concerns the problem of existence of taut foliations among 3-manifolds. Since the contribution of David Gabai, we know that closed 3-manifolds with non-trivial second homology group admit a taut foliations. 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 prove 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 man…
Taut foliations from knot diagrams
证明满足特定图性质的纽结是持续叶状的，从而所有非平凡手术支持余定向紧切叶理；适用于双桥纽结、pretzel和Montesinos纽结等，并推广到链接。
It has been recently conjectured by Boyer-Gordon-Watson that a closed, orientable, irreducible $3$-manifold $M$ is a Heegaard Floer $L$-space if and only if $π_1(M)$ is not left-orderable. In this article, we study this conjecture from the point of view of lattice cohomology, an invariant introduced by Némethi which is conjecturally isomorphic to the $HF^+$ version of Heegaard Floer homology. Using the invariant's combinatorial tractability as a stepping stone, we produce some interesting quite general families of negative-definite graph manifolds against which to test the Boyer-Gordon-Watson …
For closed 3-manifolds, Heegaard Floer homology is related to the Thurston norm through results due to Ozsváth and Szabó, Ni, and Hedden. For example, given a closed 3-manifold Y, there is a bijection between vertices of the HF^+(Y) polytope carrying the group Z and the faces of the Thurston norm unit ball that correspond to fibrations of Y over the unit circle. Moreover, the Thurston norm unit ball of Y is dual to the polytope of \underline{\hfhat}(Y). We prove a similar bijection and duality result for a class of 3-manifolds with boundary called sutured manifolds. A sutured manifold is essen…
A co-oriented foliation F of an oriented 3-manifold M is taut if and only if there is a map from M to the 2-sphere whose restriction to every leaf is a branched cover.
We extend the Eliashberg-Thurston theorem on approximations of taut oriented $C^2$-foliations of 3-manifolds by both positive and negative contact structures to a large class of taut oriented $C^{1,0}$-foliations, where by $C^{1,0}$ foliation, we mean a foliation with continuous tangent plane field. These $C^{1,0}$-foliations can therefore be approximated by weakly symplectically fillable, universally tight, contact structures. This allows applications of $C^2$-foliation theory to contact topology and Floer theory to be generalized and extended to constructions of $C^{1,0}$-foliations.
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 $\mathcal{F}$ evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation $\mathcal{F'}$ such that $S$ is homologous to a union of leaves and such that the plane field of $\mathcal{F'}$ i…
A graph manifold rational homology $3$-sphere $W$ with a left-orderable fundamental group admits a co-oriented taut foliation, though it is unknown whether it admits a smooth co-oriented taut foliation. In this paper we extend the gluing theorem of arXiv:1401.7726 to graph manifold rational homology solid tori and use this to show that there are smooth foliations on the pieces of $W$ which come close to matching up on its JSJ tori. This is applied to prove that a graph manifold with left-orderable fundamental group is not an L-space.
Slope detection, taut foliations, and the relative L-space conjecture
本文提出纽结流形上相对 L-空间猜想，统一刻画斜率检测，证明 CTF-检测斜率集为有理端点闭区间的有限并集，并证明例外 CTF-检测斜率集有限，同时证实相对猜想与环面流形 L-空间猜想在 CTF 和 NLS 性质上的等价性。
Taut foliations through a contact lens
本综述研究三维空间中重叶状结构与紧接触结构的相互作用，提出从合适接触结构对构造重叶状结构的新方法，介绍相关在研工作并针对L-空间猜想给出推测性接触视角。
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…
Suppose that $\mathcal F$ is a transversely oriented, codimension one foliation of a connected, closed, oriented 3-manifold. Suppose also that $\mathcal F$ has continuous tangent plane field and is {\sl taut}; that is, closed smooth transversals to $\mathcal F$ pass through every point of $M$. We show that if $\mathcal F$ is not the product foliation $S^1\times S^2$, then $\mathcal F$ can be $C^0$ approximated by weakly symplectically fillable, universally tight, contact structures. This extends work of Eliashberg-Thurston on approximations of taut, transversely oriented $C^2$ foliations to th…
Taut foliations and contact pairs in dimension three
该研究从三维接触对构造余维一叶状结构，作为Eliashberg-Thurston定理的逆结果，刻画重叶状结构的存在性，为L-空间猜想提供新见解。
Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields is in general not path connected. Similar methods also show that the space of representations of the fundamental group of a hyperbolic surface to the group of smooth diffeomorphisms of the circle with fixed Euler class is in general not path connected. As an important step along the way we resolve the question of which universally tight contact structures on Seif…
We describe a construction which takes as an input a left order of the fundamental group of a manifold, and outputs a (singular) foliation of this manifold which is analogous to a taut foliation. We investigate this construction in detail in dimension $2$, and exhibit connections to various problems in dimension $3$.
Universal circles for Anosov foliations
本文研究Anosov流与叶面的关系，揭示了通用圆构造的多样性及不同方法间的非 conjugate 性。
Ziggurats, taut foliations, and contact structures
该研究探讨带环面边界的紧致三维流形上横叶状结构的几何分布，通过推广相关定理建立叶状结构与接触结构的双向对应，运用接触几何方法研究金字形神塔的结构性质。
Left-orderability of Groups Acting on Bifoliated Planes
本文证明保持双叶平面两种叶状结构定向的忠实群作用必为左可序群，并通过叶空间端点与无穷远边界圆子集的等同给出两类忠实作用等价的条件。
Taut ideal triangulations of 3-manifolds
3-流形的紧理想三角剖分是具有额外组合结构的拓扑理想三角剖分：在每个理想2-单形上选择横方向，满足两个简单条件。本文的目的是证明，拉紧理想三角剖分是非常常见的，他们的行为是非常相似的一个拉紧叶理。例如，通过研究紧致理想三角剖分中的法向曲面，给出了Gabai关于三维球面纽结的奇异亏格与其亏格相等的结果的一个新的证明。
Laminations and groups of homeomorphisms of the circle
如果M是一个具有紧叶理的环面3-流形，Thurston证明了pi_1（M）作用在一个圆上.在这里，我们表明，一些其他类别的基本层压也引起行动的圈子。特别是，我们表明这是紧密的基本层压固体环面肠。我们还表明，伪Anosov流引起的行动圈。在所有情况下，这些作用都可以被转化为忠实作用，所以pi_1（M）同构于Homeo（S^1）的子群。此外，我们还证明了Weeks流形的基本群在S^1上没有忠实作用。作为一个推论，威克斯流形不承认一个紧密的基本层压，一个伪阿诺索夫流，或一个紧张的叶理。最后，我们给出了一个证明瑟斯顿的通用圆定理拉紧叶理的基础上一个新的，纯拓扑，叶袋定理的证明。
Co-orientable taut foliations in Dehn fillings of pseudo-Anosov mapping tori with co-orientation-reversing monodromy
研究了伪Anosov映射筒Dehn填充后可定向紧致叶面的存在性，证明了特定斜率下填充后存在可定向紧致叶面，并给出了相关例子。
We show that the properties of admitting a co-oriented taut foliation and having a left-orderable fundamental group are equivalent for rational homology $3$-sphere graph manifolds and relate them to the property of not being a Heegaard-Floer L-space. This is accomplished in several steps. First we show how to detect families of slopes on the boundary of a Seifert fibred manifold in four different fashions - using representations, using left-orders, using foliations, and using Heegaard-Floer homology. Then we show that each method of detection determines the same family of detected slopes. Next…
Topological invariance of Liouville structures for taut foliations and Anosov flows
研究闭定向三维流形上紧叶状结构相关的刘维尔结构的拓扑不变性，通过特定方法证明其为拓扑不变量，还涉及阿诺索夫流弱叶状结构及轨道等价情况，证明结合平滑方案和沃格尔结果，附录用于构造新例子助于分类。
R-covered foliations of hyperbolic 3-manifolds
我们生产的例子拉紧foliations的双曲3-流形是R-覆盖，但不均匀-即叶空间的普遍覆盖是R，但对叶不包含在有界的邻居。这否定了Thurston的猜想“三流形、叶理和圆I”（math.GT/9712268）。我们进一步证明了这些叶理可以通过闭曲面选择为C^0接近叶理。我们的建设强调的重要性，存在的横向调节向量场和锥场的R-覆盖叶理。最后，我们讨论了扰动任意R覆盖叶理的影响。
Foliations Transverse to Triangulations of 3-Manifolds
我们调查的组合类似物，在正常的表面，拉紧和横向测量（余维1）3流形的叶理。我们建立了某些组合结构的存在性，先验弱于相应的叶理的存在性，足以保证流形满足某些性质，例如不可约性。我们的组合结构的有限性，使我们能够使我们的结果定量的性质，并有（粗）几何后果的流形。此外，我们的技术给出了一个简单的组合证明诺维科夫定理。
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