Tight Planar Contact Manifolds with Vanishing Heegaard Floer Contact Invariants
TTIGHT PLANAR CONTACT MANIFOLDS WITH VANISHINGHEEGAARD FLOER CONTACT INVARIANTS
JAMES CONWAY, AMEY KALOTI, AND DHEERAJ KULKARNI
Abstract.
In this note, we exhibit infinite families of tight non-fillable contact manifoldssupported by planar open books with vanishing Heegaard Floer contact invariants. Moreover,we also exhibit an infinite such family where the supported manifold is hyperbolic. Introduction
Many techniques have been developed to determine whether a given contact 3-manifoldis tight. As each one was developed, the question arose as to whether a given property isequivalent to tightness. Fillability was the first widely-used tool to prove tightness, but tightcontact manifolds were found by Etnyre and Honda that were not fillable [8]. Ozsv´ath andSzab´o then developed Heegaard Floer theory, which very promisingly could prove tightnessin many cases where the manifold was not fillable. However, many tight contact manifoldswith vanishing Heegaard Floer contact invariant have been discovered, see [11, 12].On the side of positive results, Honda, Kazez, and Mati´c [16] proved that for contactmanifolds supported by open books with pages of genus one and connected binding, tightness isequivalent to the non-vanishing of the Heegaard Floer contact invariant. For contact manifoldssupported by open books with planar pages, the first tight but non-fillable examples camefrom applying [21, Corollary 1.7] to examples in [13, 22]. They proved tightness by showingthat the Heegaard Floer contact invariant does not vanish. The following question, however,remained.
Question . For contact manifolds supported by open books with planar pages, is tightnessequivalent to the non-vanishing of the Heegaard Floer contact invariant?We provide a negative answer to this question. In particular, we construct infinite familiesof tight contact manifolds supported by planar open books where capping off a binding com-ponent recovers a given overtwisted manifold. Since capping off a binding component of anopen book is equivalent to admissible transverse surgery on a binding component, we havethe following.
Theorem 1.2.
Given any overtwisted contact manifold ( M, ξ ) , there exist infinite families oftight, planar, non-fillable contact manifolds with vanishing Heegaard Floer contact invariant,on which admissible transverse surgery on a link in these manifolds recovers ( M, ξ ) . We hope that these examples will be useful in further exploring characterizations of tight-ness, and in particular will provide computable examples of boundary cases. Indeed, theseexamples have already been used to investigate possible new invariants of contact structurescoming from Heegaard Floer Homology being developed by Kutluhan, Mati´c, Van Horn-Morris, and Wand [20] and Baldwin and Vela-Vick [3]. These invariants are interesting whenthe Heegaard Floer contact invariant vanishes, as in the examples from Theorem 1.2.The only infinite family of tight, non-fillable contact manifolds that are hyperbolic hasbeen produced by Baldwin and Etnyre [2]. Their examples are supported by open books withpages of genus one. In addition, their construction requires throwing out a finite number a r X i v : . [ m a t h . G T ] S e p JAMES CONWAY, AMEY KALOTI, AND DHEERAJ KULKARNI of unspecified members of their infinite family that may not be hyperbolic. We producean infinite family of such manifolds with planar supporting open books, all of whom arehyperbolic. To this end, let S denote the surface shown in Figure 1. Let v = ( p, n , n , n , n )be a 5-tuple of integers, and let φ v = τ − n − α τ pβ τ n B τ n B τ n B τ n B be a diffeomorphism of S . Weshow the following. αβB B B B Figure 1.
Surface S used in constructing ( S, φ v ), with α and β curves indicated. Theorem 1.3.
Let ( M v , ξ v ) be the contact manifold supported by the open book ( S, φ v ) . Then ( M v , ξ v ) is universally tight, not fillable, has vanishing Heegaard Floer contact invariant, andis hyperbolic, for p ≥ , and n i ≥ for each i . This paper is organized as follows. In Section 2 we recall the definitions and propertiesof open books, fillability, the Heegaard Floer contact invariant, and transverse surgery. Weprove our results in Section 3.
Acknowledgements:
The authors would like to thank John Etnyre for helpful conver-sations. The first author was supported in part by NSF grant DMS-1309073. The secondauthor was supported in part by NSF grant DMS-0804820. The third author was supportedby Indo-US Virtual Institute For Mathematical and Statistical Sciences (VIMSS) for visitingGeorgia Tech; he would also like to thank the School of Mathematics at Georgia Tech fortheir hospitality during his visit. 2.
Background
In this section we recall basic notions from contact geometry. See [10] for an overview ofthe basics of contact geometry.The fundamental dichotomy of contact manifolds is between tight and overtwisted. Acontact manifold (
M, ξ ) is called overtwisted if there exists an embedded disc D in M suchthat ∂D is tangent to the contact planes and that the framing given by the contact planesagrees with the framing given by the surface D . If ( M, ξ ) is not overtwisted, we call it tight .2.1.
Open Book Decompositions. An abstract open book decomposition is a pair ( S, φ )with the following properties. • S is an oriented compact surface with boundary called the page of the open bookdecomposition. LANAR OPEN BOOKS AND THE CONTACT INVARIANT 3 • φ : S → S is a diffeomorphism of S such that φ | ∂S is the identity. The diffeomorphism φ is called the monodromy of the open book decomposition.We construct a 3-manifold M ( S,φ ) from the mapping torus of ( S, φ ) in the following way: M ( S,φ ) = S × [0 , x, ∼ ( φ ( x ) , , ( x ∈ ∂S, t ) ∼ ( x, t (cid:48) ) . The image of boundary ∂S × { } is a fibered link in M , and is called the binding of theopen book decomposition. By abuse of notation, we call the boundary of the surface S thebinding as well.Given a 3-manifold M , an open book decomposition for M is a diffeomorphism of M with M ( S,φ ) , for some ( S, φ ). Given a contact structure ξ on M we say that the open book decom-position ( S, φ ) supports ξ if ξ is isotopic to a contact structure with a defining 1-form α suchthat: • α > ∂S . • dα is a positive area form on the interior of each page S × { t } of the open bookdecomposition.In the supported contact structure, the binding becomes a transverse link. Every open booksupports a unique contact structure up to isotopy, and every contact structure has a supportingopen book. The following foundational result of Giroux stated in [14] forms the bedrock ofthe interaction between open book decompositions and contact geometry. Theorem 2.1 (Giroux [14]) . Let M be a closed oriented 3-manifold. Then there is a one-to-one correspondence between oriented contact structures on M up to contactomorphism andopen book decompositions of M up to positive (de)stabilization, ie. plumbing the page with anannulus and adding a positive Dehn twist to the monodromy along the core of the annulus,and the reverse process.Remark . Giroux actually defined open books on a manifold M by embedded fibered knotsthat are null-homologous. This gives an abstract open book by considered the compactificationof the fibers of the fibration, along with the induced monodromy. In this setting, Girouxproved that the correspondence was up to isotopy of contact structures on M .Given a contact manifold ( M, ξ ), we define its support genus to be the minimum of thegenus g ( S ) over all open book decompositions ( S, φ ) supporting (
M, ξ ). A contact structureis planar if its support genus is zero. The following result of Etnyre describes the supportgenus of overtwisted contact structures.
Theorem 2.3 (Etnyre [6]) . If ( M, ξ ) is an overtwisted contact manifold, then ξ is planar. Given an open book (
S, φ ) with binding components B , . . . , B k , k >
1, Baldwin [1] definesan operation called capping off . Given a boundary component B i , we create a new surface S (cid:48) by gluing D to S along B i . We create a monodromy map φ (cid:48) for S (cid:48) by extending φ over theglued-in D by the identity map. This gives a new open book decomposition ( S (cid:48) , φ (cid:48) ).2.2. Fractional Dehn Twist Coefficients.
Let (
S, φ ) be an open book decomposition withbinding components B , . . . , B k . By a result of Thurston [31], the monodromy φ is freelyisotopic to a map Φ that falls into one of three categories: pseudo-Anosov, periodic, orreducible. We will define the fractional Dehn twist coefficient of φ at B i where Φ is periodicor reducible; the definition for pseudo-Anosov maps is more involved, and as we will not needit, we will omit it here. See [17, Section 4] for more details. Definition 2.4.
If Φ is periodic, then Φ n is isotopic to τ m B τ m B · · · τ m k B k for some positive integer n , and integers m , . . . , m k , where B , . . . , B k are isotopic to the boundary components of S .We define the fractional Dehn twist coefficient of φ at B i to be c ( φ, B i ) = m i /n. JAMES CONWAY, AMEY KALOTI, AND DHEERAJ KULKARNI
If Φ is reducible, then there exists some simple closed multicurve on S that is preservedby Φ. In this case, there is some maximal subsurface S (cid:48) of S containing B i on which Φ | S (cid:48) iswell defined and is either pseudo-Anosov or periodic. On this subsurface, Φ | S (cid:48) may permutesome boundary components. Take a large enough power n of Φ | S (cid:48) such that all boundarycomponents are fixed. We define the fractional Dehn twist coefficient of φ at B i to be 1 /n times the fractional Dehn twist coefficient of Φ | nS (cid:48) at B i . We will only consider reduciblemonodromies that restrict to being periodic.We will not calculate the fractional Dehn twist coefficient of a pseudo-Anosov monodromyfrom the definition, but we will use the following estimate, due to Kazez and Roberts. First,we give the following definition, which will make the theorem easier to state. Given an openbook ( S, φ ) and a boundary component B of S , we say that φ is right-veering at B if, forevery properly embedded arc α with an endpoint on B , the image φ ( α ) is either isotopic to α or to an arc right of α near B with respect to the orientation of S , after isotoping the arcsto minimize intersections. Theorem 2.5 (Kazez–Roberts [19]) . Let ( S, φ ) be an open book, and let B be a boundarycomponent of S such that φ is right-veering at B . Let α be an arc properly embedded in S ,with at least one endpoint on B . After isotoping α and φ ( α ) to minimize intersections, let i φ ( α ) denote the signed count of intersections p in the interiors of φ ( α ) and α such that theunion of the arc segments of α and φ ( α ) from B to p are contained in an annular neighborhoodof B . Then the fractional Dehn twist coefficient c ( φ, B ) ≥ i φ ( α ) . Calculating fractional Dehn twist coefficients on a planar open book allows us to concludefacts about the supported contact structure, by the following result.
Theorem 2.6 (Ito–Kawamuro [18]) . If ( S, φ ) is an open book such that S is a planar surface,and c ( φ, B ) > for every boundary component B of S , then ( S, φ ) supports a tight contactstructure. In addition, if φ is pseudo-Anosov, we can conclude that the supported contact structureis universally tight. Although the result was originally stated for connected binding, Baldwinand Etnyre [2] noted that it works for multiple binding components as well. Note that theformulation here is simpler than the actual theorem proved by Colin and Honda, but issufficient for our needs. Theorem 2.7 (Colin–Honda [4]) . Let ( S, φ ) be an open book with pseudo-Anosov monodromy φ . If c ( φ, B ) ≥ for every boundary component B of S , then the contact manifold supportedby ( S, φ ) is universally tight. Hyperbolic Manifolds.
To prove Theorem 1.3, we wish to know how to constructhyperbolic manifolds. We will use the following theorem of Ito and Kawamuro.
Theorem 2.8 (Ito–Kawamuro [17]) . If ( S, φ ) is an open book decomposition of M , and c ( φ, B ) > for every boundary component B of S , then: • M is toroidal if and only if φ is reducible. • M is hyperbolic if and only if φ is pseudo-Anosov. • M is Seifert fibered if and only if φ is periodic. To use Theorem 2.8, we will construct pseudo-Anosov monodromies, using the followingconstruction of Penner, based on work of Thurston.
Theorem 2.9 (Penner [30]) . Let Γ and Γ be two multicurves which fill a surface S (ie.after being put in minimal position, their complement is a collection of discs and annuli), andlet φ be a product of positive Dehn twists on the elements of Γ and negative Dehn twists onthe elements of Γ , where each curve in Γ ∪ Γ appears at least once with non-zero exponentin φ . Then φ is pseudo-Anosov. LANAR OPEN BOOKS AND THE CONTACT INVARIANT 5
Fillability.
When discussing 4-manifold fillings X of a 3-manifold M , we will alwaysassume that ∂X = M with the same orientation. For contact manifolds ( M, ξ ), there arethree main types of fillings: weak fillings , strong fillings , and Stein fillings . Definition 2.10.
A contact manifold (
M, ξ ) is weakly fillable if M is the oriented boundaryof a symplectic manifold ( X, ω ) and ω | ξ > strongly fillable if in addition, there is a vector field v pointing transversely out of X along M such that ξ is isotopic to ι v ω and L v ω = ω .To define Stein fillability, we first define Stein domains. Definition 2.11. A Stein domain ( X, J, f ) is an open complex manifold (
X, J ) with a properstrictly plurisubharmonic function f : X → [0 , ∞ ). In this case, d ( df ◦ J ) defines a symplecticform on X . A contact manifold ( M, ξ ) is
Stein fillable if M is the regular level set of f forsome Stein domain ( X, J, f ) and ξ is isotopic to T M ∩ J ( T M ).If (
M, ξ ) is Stein fillable, then (
M, ξ ) is strongly fillable, and if (
M, ξ ) is strongly fillable,then it is weakly fillable. For planar contact manifolds, Niederkr¨uger and Wendl have shownthat the reverse implications hold.
Theorem 2.12 (Niederkr¨uger–Wendl [26]) . If ( M, ξ ) is a planar contact manifold, then thefollowing are equivalent. • ( M, ξ ) is weakly fillable. • ( M, ξ ) is strongly fillable. • ( M, ξ ) is Stein fillable. Eliashberg [5] and Gromov [15] showed that if (
M, ξ ) is fillable, then ξ is tight.2.5. Heegaard Floer contact invariant.
Another effective way to detect tightness is withtools from Heegaard Floer homology, defined by Ozsv´ath and Szab´o [27, 28]. We briefly recallthe construction here. For any 3-manifold M , Ozsv´ath and Szab´o defined a set of invariants,the simplest of which is the so-called hat theory, which takes the form of a vector space (cid:100) HF ( M ) over Z / Z . Given a contact structure ξ on M , Ozsv´ath and Szab´o associate anelement c ( ξ ) ∈ (cid:100) HF ( − M ), see [29]. We call c ( ξ ) the Heegaard Floer contact element .Ozsv´ath and Szab´o proved the following properties of the contact element. • If (
M, ξ ) is overtwisted, then c ( ξ ) = 0. • If (
M, ξ ) is Stein fillable, then c ( ξ ) (cid:54) = 0. • If ( M (cid:48) , ξ (cid:48) ) is obtained from ( M, ξ ) by Legendrian surgery, ie. surgery on a Legendrianknot with framing one less than the contact framing, then c ( ξ ) (cid:54) = 0 implies that c ( ξ (cid:48) ) (cid:54) = 0.Theorem 2.12 allows us to conclude that in the case of planar contact manifolds, any ofthree fillability conditions (weak, strong, or Stein) implies that c ( ξ ) (cid:54) = 0.Baldwin tracks the contact invariant under the capping off operation, and concludes thefollowing. Theorem 2.13 (Baldwin [1]) . Let ( S, φ ) be an open book decomposition of ( M, ξ ) , where ∂S has more than one component. Let ( S (cid:48) , φ (cid:48) ) be the open book obtained by capping off aboundary component of S , and let ( M (cid:48) , ξ (cid:48) ) be the contact manifold it supports. If c ( ξ ) (cid:54) = 0 ,then c ( ξ (cid:48) ) (cid:54) = 0 . Transverse Surgery.
Given a framed transverse knot K in a contact manifold ( M, ξ ),a neighborhood of K is contactomorphic to a standard neighborhood, ie. the solid torusneighborhood S r = { r ≤ r } of the z -axis in R / ( z ∼ z + 1) with contact structureker(cos r dz + r sin r d θ ), where K gets mapped to the z -axis. The boundary ∂S r has a JAMES CONWAY, AMEY KALOTI, AND DHEERAJ KULKARNI characteristic foliation given by parallel linear leaves, of slope − cot r /r in the co-ordinatesystem on ∂S r given by (cid:0) ∂∂z , ∂∂θ (cid:1) . The following exposition of surgery on transverse knots isdue to Baldwin and Etnyre [2], based off of work of Martinet [25], Lutz [23, 24], and Gay [9]. Definition 2.14.
Given a neighborhood N of K contactomorphic to S r , let 0 ≤ r < r besuch that − cot r /r is a rational number p/q ( q may be 0). Removing the interior of S r from S r , we are left with a manifold where the characteristic foliation on the boundary is bycurves of slope p/q . We quotient ∂ ( M \ S r ) by the S -action of translation along the leavesto get M (cid:48) . It can be shown that this is a manifold, and that the contact structure on M \ S r descends to M (cid:48) in a well-defined manner. We define admissible transverse p/q -surgery on K to be ( M (cid:48) , ξ (cid:48) ), where ξ (cid:48) is the induced contact structure. Notice that M (cid:48) is topologically p/q -surgery on K , with respect to the framing of K . Remark . In general, this construction will depend on the choice of neighborhood of K .In addition, there are infinitely many r corresponding to the rational number p/q , although agiven neighborhood of K will contain only finitely many of them. In all cases under discussionhere, the neighborhood used and the particular choice of r will be clear from the situation.A transverse knot K in the binding of an open book ( S, φ ) has a framing induced bythe page, called the page framing . Baldwin and Etnyre show [2] that if K is not the onlyboundary component of S , then there exists a standard neighborhood of K , where the slopeof the characteristic foliation on the boundary is (cid:15) >
0, measured with respect to the pageframing. They then show that capping off an open book is an admissible transverse surgeryon the binding component being capped off. Note that the proof in [2] extends to allownon-intersecting neighborhoods of multiple boundary components. The advantage of thisextension is that now capping off multiple boundary components can be seen as admissiblesurgery on a link.
Proposition 2.16 (Baldwin–Etnyre [2]) . Let ( S, φ ) be an open book with binding compo-nents B , . . . , B k , where k ≥ . Then there exist pairwise disjoint standard neighborhoods N , . . . , N k − of B , . . . , B k − , where N i is contactomorphic to S (cid:15) i for some (cid:15) i > , withrespect to the page framing. Proofs of results
We start with a construction that will be used throughout the proofs of our examples.
Construction . Let (
S, φ ) be an open book, and let B be a boundary component of S ,shown on the left side of Figure 2. We attach the surface shown on the right side of Figure 2,by identifying the boundary components labeled B . The new surface S (cid:48) has replaced B with m boundary components, for some integer m ≥
2. We label the new boundary componentsby B (cid:48) , . . . , B (cid:48) m . The new open book decomposition has monodromy φ (cid:48) = φ ◦ τ − n m B τ n B (cid:48) · · · τ n m B (cid:48) m ,where φ is extended by the identity to be defined on all of S (cid:48) . Note that after capping off theboundary components B (cid:48) , . . . , B (cid:48) m − we get the original open book decomposition ( S, φ ). SeeFigure 2 for a graphical representation of the construction.
Lemma 3.2.
The fractional Dehn twist coefficient c ( φ (cid:48) , B (cid:48) i ) = n i for each i = 1 , . . . , m .Proof. Let γ be a simple closed curve on S (cid:48) isotopic to B . Then γ is fixed by φ (cid:48) . Thus φ (cid:48) is areducible monodromy, and to calculate the fractional Dehn twist coefficient at B (cid:48) i , we need tofind the maximal subsurface S (cid:48)(cid:48) of S (cid:48) containing B (cid:48) i on which φ (cid:48) restricts to a pseudo-Anosovor periodic monodromy. This subsurface can be identified with the surface added to S tomake S (cid:48) , namely, the right-hand side of Figure 2. Any larger subsurface would contain B as LANAR OPEN BOOKS AND THE CONTACT INVARIANT 7 n B (cid:48) n B (cid:48) n B (cid:48) n m − B (cid:48) m − n m B (cid:48) m n m B B . . . . . .. . .. . .. . .. . . ...
Figure 2.
Construction 3.1 on a boundary component B . The red curves onthe right are positive Dehn twists, and the blue curves on the left, parallel to B , are negative Dehn twists.a non-boundary-parallel curve, and the monodromy restricted to the larger subsurface wouldbe reducible.Now, restricted to S (cid:48)(cid:48) , φ (cid:48) is periodic, and is in fact already isotopic to a product of boundaryDehn twists. Thus, the count of Dehn twists at each boundary component is the fractionalDehn twist coefficient, and the lemma follows. (cid:3) We now prove Theorem 1.2, which is a corollary of Construction 3.1.
Proof of Theorem 1.2.
Let (
M, ξ ) be an overtwisted contact manifold. By Theorem 2.3, (
M, ξ )is supported by a planar open book decomposition (
S, φ ). Apply Construction 3.1 to eachboundary component of S , with each m ≥ n i ≥
2, to get another planar open bookdecomposition ( S (cid:48) , φ (cid:48) ). Denote the contact manifold supported by this open book decompo-sition by ( M (cid:48) , ξ (cid:48) ). Since the fractional Dehn twists coefficient c ( φ (cid:48) , B (cid:48) ) ≥ B (cid:48) of S (cid:48) by Lemma 3.2, Theorem 2.6 implies that ξ (cid:48) is tight. Note that aftercapping off all but one boundary component on each added surface, we recover ( S, φ ). Since(
S, φ ) is overtwisted, the Heegaard Floer contact invariant c ( ξ ) = 0, so Theorem 2.13 impliesthat the contact invariant c ( ξ (cid:48) ) = 0. Since S (cid:48) is planar, Theorem 2.12 implies that ( M (cid:48) , ξ (cid:48) )is not weakly, strongly, or Stein fillable. Let L be the transverse link given by the boundarycomponents of S (cid:48) capped off to recover ( S, φ ). Proposition 2.16 gives us a neighborhood of L such that we can realize the capping off operation as admissible surgery on each componentof the link. Thus, admissible surgery on L in ( M (cid:48) , ξ (cid:48) ) will recover ( M, ξ ). Finally, note thatwe have infinitely many choices of ( M (cid:48) , ξ (cid:48) ), as for each boundary component of S (cid:48) , we haveinfinitely many choices of m and n i . (cid:3) We now turn to the proof of Theorem 1.3.
Proof of Theorem 1.3.
First note that if Γ = { α } and Γ = { β } , where α and β are as inFigure 1, then Γ and Γ are multicurves that fill the surface S . Thus by Theorem 2.9, thediffeomorphism φ (cid:48) = τ − n − α τ pβ is pseudo-Anosov, as p ≥ n ≥
6. Similarly, composing φ (cid:48) with a product of boundary parallel Dehn twists does not change the free isotopy classof φ (cid:48) , thus φ v is also pseudo-Anosov. Let γ and γ be the arcs shown in Figure 3. Onecan check that i φ v ( γ ) near B and B is n − n − i φ v ( γ ) near B and B . Thus, by Theorem 2.5, the fractional Dehn twist coefficient c ( φ, B i ) ≥ n i − ≥ >
4. By Theorem 2.8, the contact manifold ( M v , ξ v ) supportedby ( S, φ v ) is hyperbolic. By Theorem 2.7, ξ v is universally tight. Note that capping off JAMES CONWAY, AMEY KALOTI, AND DHEERAJ KULKARNI γ γ B B B B Figure 3. S with γ and γ indicated. B results in an open book supporting an overtwisted contact structure, as the monodromyaround B consists of a negative Dehn twist. Thus, similarly to the proof of Theorem 1.2,( M v , ξ v ) has c ( ξ v ) = 0 and is not fillable, by Theorem 2.13 and Theorem 2.12.We now want to show that there are infinitely many distinct examples as we range overvarious values of v . To do this, we compute the order of the group H ( M v ; Z ) and show thatit attains infinitely many values.0 0 0 B − B − B − B − α β − Figure 4.
The knots involved in a surgery diagram for M v .To get a surgery diagram for M v , we use the following fact: a simple closed curve on thepage of an open book corresponds to a knot in the 3-manifold described by the open book,and a right-handed (resp. left-handed) Dehn twist about this curve corresponds to a surgeryon the knot with framing − M v using Figure 4. The 0-framedunknots come from treating the page as a disc bounded by B with three 1-handles attached.The surgeries on the knots labeled B , . . . , B correspond to the boundary-parallel Dehn twists LANAR OPEN BOOKS AND THE CONTACT INVARIANT 9 in the monodromy; take n i copies of B i , and do surgery with framing − α and β correspond to the α and β curves on the page of the open-book (see Figure 1); take n + 1 copies of α and p copies of β , and do surgery with framing 1 on the copies of α andwith framing − β . All copies are push-offs using the Seifert framing of theknot in S , and the surgery framings are with respect to this same Seifert framing.Now, after blowing down the ± L v = n + n + p n + p n n + p n + p − − n − n − . The order of H ( M v ; Z ) is given by the absolute value of the determinant of the linking matrix L v . Indeed, if we let v k = (6 , , , ,
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