Danny Calegari
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.
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We construct one dimensional foliations which are subfoliations of two dimensional foliations in 3-manifolds. The subfoliation is by quasigeodesics in each two dimensional leaf, but it is not funnel: not all quasigeodesics share a common ideal point in most leaves. This article answers the question whether the funnel property is a consequence of the leafwise quasigeodesic property or not.
We study the group of leafwise holomorphic smooth automorphisms of Reeb components of leafwise complex foliation which are obtained by a certain Hopf construction. In particular, in the case where the boundary holonomy is infinitely tangent to the identity, we determine the structure of the group of leafwise holomorphic automorphisms.
We review the standard Hopf construction of Reeb components with leafwise complex structure and almost determine the group of leafwise holomorphic smooth automorphisms for Reeb components of certain type in the case of complex leaf dimension one. In particular, it contains an infinite dimensional vector space.
The aim of this paper is to show that Lawson's foliation on the 5-sphere admits a smooth leafwise symplectic structure. The main part of the construction is to show that the Fermat type cubic surface admits an end-periodic symplectic structure.
Mitsumatsu constructed leafwise symplectic structures of certain codimension one foliations of the 5-sphere. This inspired the present author to improve his result on convergence of contact structure to foliation. We describe convergence of contact strcture to leafwise symplectic foliation by means of confoliation equipped with a certain 2-form. Such a 2-form appears in the works of Nunes da Costa and Petalidou on twisted Jacobi structures and partially relates to weak domination due to Massot, Niederkruger and Wendl. We also construct leafwise symplectic foliations on the product of the 4-sph…
Geometric conditions are given so that the leafwise reduced cohomology is of infinite dimension, specially for foliations with dense leaves on closed manifolds. The main new definition involved is the intersection number of subfoliations with "appropriate coefficients". The leafwise reduced cohomology is also described for homogeneous foliations with dense leaves on closed nilmanifolds.
This is a note of the author's lectures at "Advanced courses in Foliation" in the research program "Foliation", which was held at the Centre de Recerca Mathematica in the May of 2010. In this note, we discuss about the relationship between deformation of actions of Lie groups and the leafwise cohomology of the orbit foliation.
Intersection of transverse foliations in 3-manifolds: Hausdorff leafspace implies leafwise quasi-geodesic
本文研究闭3维流形中两横截二维叶状结构相交所产生的一维叶状结构，证明其叶状拟测地性质等价于万有覆盖中诱导叶状结构具有 Hausdorff 叶空间。
The automorphisms group of the 3-dimensional Reeb component with complex leaves is computed in the case where the component is obtained by the Hopf construction and the holonomy of the boundary leaf is not tangent to the identity to the infinite order. Combined with a previous work, for 3-dimensional Reeb components obtained by the Hopf construction, we have an almost complete description of the groups of leafwise holomorphic smooth automorphisms.
A leafwise Hodge decomposition was proved by Sanguiao for Riemannian foliations of bounded geometry. Its proof is explained again in terms of our study of bounded geometry for Riemannian foliations. It is used to associate smoothing operators to foliated flows, and describe their Schwartz kernels. All of this is extended to a leafwise version of the Novikov differential complex.
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.
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。
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.
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.
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。
A foliation of a manifold M is called R-covered if its lift to the universal cover of M has space of leaves R. We show that there are many graph manifolds which admit taut foliations, but which do not admit any R-covered foliations. On the other hand, we show that these manifolds all have finite covers admitting R-covered foliations.
We study the question of when cyclic branched covers of knots admit taut foliations, have left-orderable fundamental group, and are not L-spaces.
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.
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.
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。
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…
Left orderability and taut foliations with one-sided branching
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。
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。
A taut foliation of a hyperbolic 3-manifold has the continuous extension property for leaves in almost every direction; that is, for each leaf of the universal cover of the foliation and almost every geodesic ray in the leaf, the limit of the ray in the universal cover of the 3-manifold is a well-defined point in the ideal boundary.
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, …
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 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。
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。
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。
A contact foliation is a foliation endowed with a leafwise contact structure. In this remark we explain a turbulisation procedure that allows us to prove that tightness is not a homotopy invariant property for contact foliations.
The idea of Lichnerowicz or Morse-Novikov cohomology groups of a manifold has been utilized by many researchers to study important properties and invariants of a manifold. Morse-Novikov cohomology is defined using the differential $d_ω=d+ω\wedge$, where $ω$ is a closed $1$-form. We study Morse-Novikov cohomology relative to a foliation on a manifold and its homotopy invariance and then extend it to more general type of forms on a Riemannian foliation. We study the Laplacian and Hodge decompositions for the corresponding differential operators on reduced leafwise Morse-Novikov complexes. In the…
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。
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…
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…
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。
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。
Every noncompact surface is a leaf of a minimal foliation
本文证明任意非紧定向曲面均可作为闭3维流形中极小叶状结构的叶，并构造可同时包含可数多种开曲面拓扑的双曲例子。
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…
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, and can therefore be used to study the question of which foliations arise as geometric limits of isotopy classes of other foliations. We show that this norm is non-trivial -- i.e. it distinguishes certain taut foliations of a given hyperbolic 3-manifold. In fact, modulo some standard conjectures, this norm should precisely detect amongst taut foliations of closed atoroida…
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。
We show that every topological surface lamination of a 3-manifold M is isotopic to one with smoothly immersed leaves. This carries out a project proposed by Gabai in [Problems in foliations and laminations, AMS/IP Stud. Adv. Math. 2.2 1--33]. Consequently any such lamination admits the structure of a Riemann surface lamination, and therefore useful structure theorems of Candel [Uniformization of surface laminations, Ann. Sci. Ecole Norm. Sup. 26 (1993) 489--516] and Ghys [Dynamique et geometrie complexes, Panoramas et Syntheses 8 (1999)] apply.
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。
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…
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。
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…
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