Georgia Institute of Technology（佐治亚理工学院）；Universidad Complutense de Madrid（马德里康普顿斯大学）；Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM（西班牙国家研究委员会-马德里自治大学-马德里理工大学-马德里康普顿斯大学联合数学科学研究所）
Eduardo Fernández, Javier Martínez-Aguinaga, Francisco Presas
备注：
V3: Title and abstract changed. The exposition has been streamlined. The contents are essentially unchanged, except for the addition of Theorem 1.3.1, which could already be deduced from the results in previous versions
We study spaces of tight contact structures and contactomorphism groups in dimension $3$. We introduce a microfibration method for studying spaces of convex surfaces in contact $3$-manifolds and apply it to describe spaces of convex disks and spheres in tight contact $3$-manifolds. With these ingredients in hand, we prove a parametric gluing theorem for contact handle attachments: the weak homotopy type of the space of tight contact structures is invariant under contact $0$-, $1$-, and $3$-handle attachments.
Based on these new techniques, we present a substantial number of independent applications. First, we obtain explicit descriptions of the weak homotopy type of several spaces of tight contact structures. We determine the weak homotopy type of the contactomorphism group of every standard contact handlebody, of the standard $\mathbb{S}^1\times\mathbb{S}^2$, and of every Legendrian circle bundle with non-empty boundary, partially resolving a conjecture of Giroux. Additionally, we compute the weak homotopy type of an infinite family of spaces of Legendrian and transverse unknots, proving Legendrian and transverse analogues of Hatcher's theorem for smooth unknots in the $3$-sphere.
In this article we present infinitely many 3-manifolds admitting infinitely many universally tight contact structures each with trivial Ozsvath-Szabo contact invariants. By known properties of these invariants the contact structures constructed here are non weakly symplectically fillable.
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 Ozsvath-Szabo contact invariant is a complete classification invariant for tight contact structures on small Seifert fibered 3-manifolds which are L-spaces.
We classify up to isotopy the tight contact structures on small Seifert spaces with $e_0\neq0,-1,-2$. (The first version contains on the $e_0<-2$ case.)
We prove that there is a unique real tight contact structure on the 3-ball with convex boundary up to isotopy through real tight contact structures. We also give a partial classification of the real tight solid tori with the real structure being antipodal map along longitudinal and the identity along meridional direction. For the proofs, we use the real versions of contact neighborhood theorems and the invariant convex surface theory in real contact manifolds.
We determine the closed, oriented Seifert fibered 3-manifolds which carry positive tight contact structures. Our main tool is a new non-vanishing criterion for the contact Ozsvath-Szabo invariant.
We show the existence of tight contact structures on infinitely many hyperbolic three-manifolds obtained via Dehn surgeries along sections of hyperbolic surface bundles over circle.
It has been shown by V. Colin that every tight contact 3-manifold can be written as a connected sum of prime manifolds. Here we prove that the summands in this decomposition are unique up to order and contactomorphism.
Two of the basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. We present the first such classification on an infinite family of (mostly) hyperbolic 3-manifolds: surgeries on the figure-eight knot. We also determine which of the tight contact structures are symplectically fillable and which are universally tight.
Extending our earlier results, we prove that certain tight contact structures on circle bundles over surfaces are not symplectically semi--fillable, thus confirming a conjecture of Ko Honda.
Given a contact structure on a manifold $V$ together with a supporting open book decomposition, Bourgeois gave an explicit construction for a contact structure on $V \times \mathbb{T}^2$. We prove that all such structures are universally tight in dimension $5$, independent on whether the original contact manifold is tight or overtwisted.
Let Y(r) be the closed, oriented three-manifold obtained by performing rational r-surgery on the right-handed trefoil knot in the three-sphere. Using contact surgery and the Heegaard Floer contact invariants we construct positive, tight contact structures on Y(r) for every r not equal to 1. This implies, in particular, that the oriented boundaries of the positive E6 and E7 plumbings carry positive, tight contact structures, solving a well-known open problem.
Ozsvath-Szabo contact invariants are a powerful way to prove tightness of contact structures but they are known to vanish in the presence of Giroux torsion. In this paper we construct, on infinitely many manifolds, infinitely many isotopy classes of universally tight torsion free contact structures whose Ozsvath-Szabo invariant vanishes. We also discuss the relation between these invariants and an invariant on T^3 and construct other examples of new phenomena in Heegaard--Floer theory. Along the way, we prove two conjectures of K Honda, W Kazez and G Matic about their contact topological quant…
We investigate the line between tight and overtwisted for surgeries on fibred transverse knots in contact 3-manifolds. When the contact structure $ξ_K$ is supported by the fibred knot $K \subset M$, we obtain a characterisation of when negative surgeries result in a contact structure with non-vanishing Heegaard Floer contact class. To do this, we leverage information about the contact structure $ξ_{\overline{K}}$ supported by the mirror knot $\overline{K} \subset -M$. We derive several corollaries about the existence of tight contact structures, L-space knots outside $S^3$, non-planar contact …
The support norm $sn(ξ)$ of a contact structure $ξ$ is the minimum of the negative Euler characteristics of the pages of the open books supporting $ξ$. In this paper we prove additivity of the support norm for tight contact structures.
We employ cut and paste contact topological techniques to classify some tight contact structures on the closed, oriented genus-2 surface times the interval. A boundary condition is specified so that the Euler class of the of the contact structure vanishes when evaluated on each boundary component. We prove that there exists a unique, non-product tight contact structure in this case.
We classify tight contact structures on the small Seifert fibered 3--manifold M(-1; r_1, r_2, r_3) with r_i in (0,1) and r_1, r_2 \geq 1/2. The result is obtained by combining convex surface theory with computations of contact Ozsvath--Szabo invariants. We also show that some of the tight contact structures on the manifolds considered are nonfillable, justifying the use of Heegaard Floer theory.
Let p and n be positive integers with p>1, and let E(p,n) be the oriented 3-manifold obtained by performing pn(p-1)-1 surgery on a positive torus knot of type (p, pn+1). We prove that E(2,n) does not carry tight contact structures for any n, while E(p,n) carries tight contact structures for any n and any odd p. In particular, we exhibit the first infinite family of closed, oriented, irreducible 3-manifolds which do not support tight contact structures. We obtain the nonexistence results via standard methods of contact topology, and the existence results by using a quite delicate computation of…
We show that the canonical contact structure on the link of a normal complex singularity is universally tight. As a corollary we show the existence of closed, oriented, atoroidal 3-manifolds with infinite fundamental groups which carry universally tight contact structures that are not deformations of taut (or Reebless) foliations. This answers two questions of Etnyre.
We study weak versus strong symplectic fillability of some tight contact structures on torus bundles over the circle. In particular, we prove that almost all of these tight contact structures are weakly, but not strongly symplectically fillable. For the 3-torus this theorem was established by Eliashberg.
In this article we conjecture a 4-dimensional characterization of tightness: a contact structure is tight if and only if a slice-Bennequin inequality holds for smoothly embedded surfaces in Yx[0,1]. An affirmative answer to our conjecture would imply an analogue of the Milnor conjecture for torus knots: if a fibered link L induces a tight contact structure on Y then its fiber surface maximize Euler characteristic amongst all surfaces in Yx[0,1] with boundary L. We provide evidence for both conjectures by proving them for contact structures with non-vanishing Ozsváth-Szabó contact invariant. We…
We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of the contact structures are identified with hypertrees in a certain hypergraph. Using earlier work, this establishes a connection between contact topology and the Homfly polynomial. We also show that the contact invariants of our tight contact structures form a basis for sutured F…
Dehn surgery on a knot determines a dual knot in the surgered manifold, the core of the filling torus. We consider duals of knots in $S^3$ that have a lens space surgery. Each dual supports a contact structure. We show that if a universally tight contact structure is supported, then the dual is in the same homology class as the dual of a torus knot.
It was proven in the first author's paper "Contact 3-manifolds twenty years since J. Martinet's work" (Ann. Inst. Fourier, 42(1992), 165--192) that any tight contact structure on the 3-sphere is diffeomorphic to the standard one. It was also claimed there without a proof that similar methods could be used to prove a multi-parametric version: the space of tight contact structures on $S^3$, fixed at a point, is contractible. We prove this result in the current paper.
We exhibit an infinite family of tight contact structures with the property that none of the supporting open books minimizes the genus and maximizes the Euler characteristic of the page simultaneously, answering a question of Baldwin and Etnyre in arXiv:0910.5021 .
Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact structures for every r not= 2g_s(K)-1, where g_s(K) is the slice genus of K. This implies, in particular, that the Brieskorn spheres -Sigma(2,3,4) and -Sigma(2,3,3) carry tight, positive contact structures. As an application of our main result we show that for each m in N there exists a Seifert fibered rational homology 3-sphere M_m carrying at lea…
We characterize L-spaces which are Seifert fibered over the 2-sphere in terms of taut foliations, transverse foliations and transverse contact structures. We give a sufficient condition for certain contact Seifert fibered 3-manifolds with e_0=-1 to have nonzero contact Ozsvath--Szabo invariants. This yields an algorithm for deciding whether a given small Seifert fibered L-space carries a contact structure with nonvanishing contact Ozsvath--Szabo invariant. As an application, we prove the existence of tight contact structures on some 3-manifolds obtained by integral surgery along a positive tor…
We define the reduced Khovanov homology of an open book (S,h), and we identify a distinguished "contact element" in this group which may be used to establish the tightness or non-fillability of contact structures compatible with (S,h). Our construction generalizes the relationship between the reduced Khovanov homology of a link and the Heegaard Floer homology of its branched double cover. As an application, we give combinatorial proofs of tightness for several contact structures which are not Stein-fillable. Lastly, we investigate a comultiplication structure on the reduced Khovanov homology o…
In this note we introduce the (homologically essential) arc complex of a surface as a tool for studying properties of open book decompositions and contact structures. After characterizing destabilizability in terms of the essential translation distance of the monodromy of an open book we given an application of this result to show that there are planer open books of the standard contact structure on the 3-sphere with 5 (or any number larger than 5) boundary components that do not destabilize. We also show that any planar open book with 4 or fewer boundary components does destabilize.
Using the relation between Khovanov homology and the Heegaard Floer homology of branched double covers, we show how Khovanov homology can be used to establish tightness of branched double covers of certain transverse knots. We give examples of several infinite families of knots whose branched covers are tight for Khovanov-homological reasons, and show that some of these branched covers are not Stein fillable.
Given a contact structure on a manifold $V$ together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on $V \times \mathbb{T}^2$. We prove that all such structures are universally tight in dimension $5$, independent on whether the original contact manifold is itself tight or overtwisted. In arbitrary dimensions, we provide obstructions to the existence of strong symplectic fillings of Bourgeois manifolds. This gives a broad class of new examples of weakly but not strongly fillable contact $5$-manifolds, as well as the first examples of w…
We prove gluing theorems for tight contact structures. In particular, we rederive (as special cases) gluing theorems due to Colin and Makar-Limanov, and present an algorithm for determining whether a given contact structure on a handlebody is tight. As applications, we construct a tight contact structure on a genus 4 handlebody which becomes overtwisted after Legendrian -1 surgery, and study certain Legendrian surgeries on T^3.
We prove that an infinite family of virtually overtwisted tight contact structures discovered by Honda on certain circle bundles over surfaces admit no symplectic semi-fillings. The argument uses results of Mrowka, Ozsvath and Yu on the translation-invariant solutions to the Seiberg-Witten equations on cylinders and the non-triviality of the Kronheimer-Mrowka monopole invariants of symplectic fillings.
We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.
We classify positive tight contact structures, up to isotopy fixing the boundary, on the manifolds $N=M(D^{2}; r_1, r_2)$ with minimal convex boundary of slope $s$ and Giroux torsion 0 along $\partial N$, where $r_1,r_2\in (0,1)\cap\mathbb{Q}$, in the following cases: (1) $s\in(-\infty, 0)\cup[2, +\infty)$; (2) $s\in[0, 1)$ and $r_1,r_2\in [1/2,1)$; (3) $s\in[1, 2)$ and $r_1,r_2\in(0,1/2)$; (4) $s=\infty$ and $r_1=r_2=1/2$. We also classify positive tight contact structures, up to isotopy fixing the boundary, on $M(D^2;1/2,1/2)$ with minimal convex boundary of arbitrary slope and Giroux torsio…
This paper describes a characterization of tightness of closed contact 3-manifolds in terms of supporting open book decompositions. The main result is that tightness of a closed contact 3-manifold is preserved under Legendrian surgery.
In this paper, we study strong symplectic fillability and Stein fillability of some tight contact structures on negative parabolic and negative hyperbolic torus bundles over the circle. For the universally tight contact structure with twisting $π$ in $S^1$-direction on a negative parabolic torus bundle, we completely determine its strong symplectic fillability and Stein fillability. For the universally tight contact structure with twisting $π$ in $S^1$-direction on a negative hyperbolic torus bundle, we give a necessary condition for it being strongly symplectically fillable. For the virtually…
We propose in this paper a method for studying contact structures in 3-manifolds by means of branched surfaces. We explain what it means for a contact structure to be carried by a branched surface embedded in a 3-manifold. To make the transition from contact structures to branched surfaces, we first define auxiliary objects called sigma-confoliations and pure contaminations, both generalizing contact structures. We study various deformations of these objects and show that the sigma-confoliations and pure contaminations obtained by suitably modifying a contact structure remember the contact str…
We study contact structures compatible with genus one open book decompositions with one boundary component. Any monodromy for such an open book can be written as a product of Dehn twists around dual non-separating curves in the once-punctured torus. Given such a product, we supply an algorithm to determine whether the corresponding contact structure is tight or overtwisted. We rely on Ozsv{á}th-Szab{ó} Heegaard Floer homology in our construction and, in particular, we completely identify the $L$-spaces with genus one, one boundary component, pseudo-Anosov open book decompositions. Lastly, we r…
We study the effect of surgery on transverse knots in contact 3-manifolds. In particular, we investigate the effect of such surgery on open books, the Heegaard Floer contact invariant, and tightness. The overarching theme of this paper is to show that in many contexts, surgery on transverse knots is more natural than surgery on Legendrian knots. Besides reinterpreting surgery on Legendrian knots in terms of transverse knots, our main results on are in two complementary directions: conditions under which inadmissible transverse surgery (\textit{cf.\@} positive contact surgery on Legendrian knot…
This paper begins the study of relations between Riemannian geometry and global properties of contact structures on 3-manifolds. In particular we prove an analog of the sphere theorem from Riemannian geometry in the setting of contact geometry. Specifically, if a given three dimensional contact manifold (M,ξ) admits a complete compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure ξis tight; in particular, the contact structure pulled back to the universal cover is the standard contact structure on S^3. We also describe geometric conditions in dime…
Let V be a closed 3-manifold. In this paper we prove that the homotopy classes of plane fields on V that contain tight contact structures are in finite number and that, if V is atoroidal, the isotopy classes of tight contact structures are also in finite number.
We produce the first examples of closed, tight contact 3-manifolds which become overtwisted after performing admissible transverse surgeries. Along the way, we clarify the relationship between admissible transverse surgery and Legendrian surgery. We use this clarification to study a new invariant of transverse knots - namely, the range of slopes on which admissible transverse surgery preserves tightness - and to provide some new examples of knot types which are not uniformly thick. Our examples also illuminate several interesting new phenomena, including the existence of hyperbolic, universall…
We prove that any small enough neighborhood of a closed contact submanifold is always tight under a mild assumption on its normal bundle. The non-existence of $C^0$--small positive loops of contactomorphisms in general overtwisted manifolds is shown as a corollary.
We use the Ozsváth-Szabó contact invariants to distinguish between tight contact structures obtained by Legendrian surgeries on stabilized Legendrian links in tight contact 3-manifolds. We also discuss the implication of our result on the tight contact structures on the Brieskon homology spheres $-Σ(2,3,6n-1)$.
We show that an oriented elliptic 3-manifold admits a universally tight positive contact structure iff the corresponding group of deck transformations on $S^3$ preserves a standard contact structure pointwise. We also relate univerally tight contact structures on 3-manifolds covered by $S^3$ to the exceptional isomorphism $SO(4)=(SU(2)\times SU(2))/{\pm 1}$. The main tool used is equivariant framings of 3-manifolds.
Using contact surgery we define families of contact structures on certain Seifert fibered three-manifolds. We prove that all these contact structures are tight using contact Ozsath-Szabo invariants. We use these examples to show that, given a natural number n, there exists a Seifert fibered three-manifold carrying at least n pairwise non-isomorphic tight, not fillable contact structures.
