aa r X i v : . [ m a t h . S G ] F e b ADDITIVITY OF SUPPORT NORM OF TIGHT CONTACTSTRUCTURES
VERA V ´ERTESIA
BSTRACT . The support norm sn ( ξ ) of a contact structure ξ is the mini-mum of the negative Euler characteristics of the pages of the open bookssupporting ξ . In this paper we prove additivity of the support norm fortight contact structures.
1. I
NTRODUCTION
Since Giroux’s discovery [4] of the relation between open books and con-tact structures, one of the main focus of research in 3-dimensional contactgeometry has been to deduce contact invariants from the combinatoricsof their supporting open books. The most obvious such invariant is the support genus or sg ( ξ ) of a contact structure, which is the minimal possi-ble genus of open books supporting ξ . Etnyre [3] showed that the supportgenus of overtwisted contact structures is zero, and he gave obstructionsof having support genus zero in terms of the symplectic fillings of the con-tact structure. Ozsv´ath, Stipsicz and Szab ´o [10] gave another obstructionfor sg ( ξ ) = 0 . Namely, that the contact invariant in Heegard Floer ho-mology has to be reducible. These criteria give rise to examples of contact3-manifolds whose support genus is at least one. However there is no con-firmed examples of contact structures with support genus greater than one.Another combinatorial invariant of similar type for a contact structure ξ is the support norm or sn ( ξ ) , which is the minimum of the negative Eulercharacteristic of pages of open books supporting ξ . Since an open bookgives rise to a Heegaard decomposition whose genus is the Euler char-acteristic of the open book minus one, we can immediately give contact3-manifolds with arbitrarily large support norm. Heegaard decomposi-tions, that are constrcted from these open books have been also studiedby ¨Ozba ˘gcı [9], and their minimal genus was denoted by Hg ( ξ ) , where Hg ( ξ ) = sn ( ξ ) − .As always, it is important to understand how these two invariants be-have under connected sum. Since the connected sum can always be donein a small ball that intersects the binding of the open book the invari-ants are sub-additive. More precisely sg ( ξ ξ ) ≤ sg ( ξ ) + sg ( ξ ) and sn ( ξ ξ ) ≤ sn ( ξ ) + sn ( ξ ) + 1 (or Hg ( ξ ξ ) ≤ Hg ( ξ ) + Hg ( ξ ) ). The additivity of the support genus would immediately provide examples forcontact structures with arbitrarily high support genus. In general neitherthe support genus nor the support norm is additive. The connected sumof a contact structure with support genus one and an overtwisted contactstructure gives an overtwisted contact structure which has support genuszero. ¨Ozba ˘gcı [9] gave a similar example for the non-additivity of supportnorm using the homological classification of Eliashberg [2] for overtwistedcontact structures. He observes that any overtwisted contact structure onan integer homology sphere is isotopic to its connected sum with the over-twisted contact sphere ( S , ξ − ) with d = − , which has Hg ( ξ ) = 2 . Inthis paper we prove that for tight contact structures the support norm isadditive: Theorem 1.1. (additivity of Euler characteristic for tight contact structures) Let ( Y , ξ ) and ( Y , ξ ) be tight contact 3-manifolds and let ( Y, ξ ) be their contactconnected sum. Then sn ( ξ ) + sn ( ξ ) + 1 = sn ( ξ ) This statement is another fine appearance of the tight–overtwisted di-chotomy of 3-dimensional contact structures. The proof uses cut and pastearguments for open books, that is more generally phrased and understoodby the machinery, built up in a preprint of the author with Licata [7] of foliated open books . A
CKNOWLEDGMENT
I would like to thank Joan Licata for thinking a lot together about foliatedopen books. I would also like to acknowledge the help of Paolo Ghiggini,John Etnyre, Patrick Massot and Andr´as Stipsicz in listening to an earlyversion of my proof. The comments of Paolo Ghiggini and Burak ¨Ozba ˘gcıon my preprint were very valuable. During the writing of this paper I wassupported by the ANR grant “Quantum topology and contact geometry”.2. P
RELIMINARIES
In this section we review basic notions about contact structures, openbooks and the foliations they induce on embedded surfaces.An open book ( B, π ) of a closed oriented 3-manifold Y is a pair of anembedded 1-manifold B in Y and a map π : Y \ B → S that restricts to ϑ in some tubular neighborhood N ( B ) ∼ = B × D ( r, ϑ ) of the binding B . Theclosures of the preimages π − ( t ) are called the pages of the open book. Bythe previous condition B is the oriented boundary of each page. The Eulercharacteristic of a page of ( B, π ) is denoted by χ ( B, π ) , and the norm An abstract open book is a pair ( S, h ) of a genus g surface S with boundaryand a diffeomorphism h : S → S , called the monodromy , that is the identity DDITIVITY OF SUPPORT NORM OF TIGHT CONTACT STRUCTURES 3 near ∂S . We can recover a 3-manifold Y ( S,h ) by factoring the mapping coneof h by the extra relation ( x, t ) ∼ ( x ′ , t ) for x ∈ ∂S and t, t ′ ∈ S . An abstractopen book ( S, h ) corresponds to an (embedded) open book ( B, π ) if there isa diffeomorphism between Y and Y ( S,h ) that maps B onto the equivalenceclass of ∂S and restricts to a bundle map on Y \ B .As it is described by Ito and Kawamuro [5, 6] an open book decomposi-tion ( B, π ) induces a foliation on (generic) embedded surfaces ι : F ֒ → Y .An open book foliation F ob = F ob ( ι ) is an oriented singular foliation on F whose leaves are defined by the level-sets ( π ◦ ι ) − ( t ) . By definition , openbook foliations satisfy the following properties:- The transverse intersection points of B with ι ( F ) are exactly the el-liptic singularities of F ob (See the first two pictures of Figure 1). Anelliptic point is positive if B coorients F , and negative otherwise. Pos-itive elliptic points are sources , while negative elliptic points are sinks of F ob . The set of elliptic points on F is E = E + ∪ E − ;- The pull back e π of the map π onto F \ E is a circle-valued Morsefunction such that each critical point have different values;- The maxima and minima of e π are center singularities of F ob (See thethird and fourth pictures of Figure 1);- The saddles of e π are hyperbolic points of F ob (See Figure 2). A hyper-bolic point is positive if ∇ π coorients F , and it is negative otherwise.The set of hyperbolic points on F is H = H + ∪ H − .By using Thom transversality theorem one can prove that any surface canbe isotoped so that it admits an open book foliation. Moreover, by a furtherisotopy one can assume that F ob is circle-free: Proposition 2.1 ([5] removing circles) . Suppose that the embedding ι : F ֒ → Y has an open book foliation with respect to the open book ( B, π ) . Then there is anisotopy of the embedding ι to ι ′ such that the foliation F ob ( ι ′ ) has no circles. (cid:3) As a corollary, in such an embedding F ob has no center singularities. Inthe sequel we will always assume that F ob has no circles. This property willturn out to be essential when one relates open book foliations to character-istic foliations (See Proposition 2.3.) which is one of the main tools in thispaper.Here we describe how one can alter the foliation F ob by isotoping theembedding of F . Note that even though there are more possibilities inchanging the open book foliation even in a general open book, here we willonly concentrate on the change we need in S with the standard open book .The standard open book ( B , π ) of S is defined on S ⊂ C ( z , z ) with The original definition in [5] is weaker, and does not require π ◦ ι to be a Morse function.Later in the proof of Theorem 2.21 of [5], however the authors use this property. VERA V ´ERTESI B = { z = 0 } ∩ S and π : S \ B → S is z | z | . This open book has discpages. Proposition 2.2 ([1] Change in foliation) . Let F ob ( ι ) be the open book foliationinduced by the open book ( B , π ) on the embedding ι : F ֒ → S . Suppose thatthere is a disc D ⊂ F so that the foliation restricted to D is as on the first picturein Figure 3. Then there is an isotopy of the embedding ι to ι ′ (or ι ′′ ) that is constantaway from D such that the open book foliation F ob ( ι ′ ) is the second (or the third)picture on D . (cid:3) These changes are achieved by exchanging the π -values of the hyperbolicsingularities.A contact structure ξ on an oriented 3-manifold Y is a planefield given asthe kernel of a 1-form α for which α ∧ dα is a volume form. The character-istic foliation F ξ = F ξ ( ι ) on F corresponding to the embedding ι : F ֒ → Y is an oriented singular foliation given by the dual of the 1-form ι ∗ α . Theleaves of this foliation are given by the pull-back of the integral of the ori-ented linefield T p ι ( F ) ∩ ξ p . The singular points of F ξ are the points where T p ι ( F ) = ξ p , and one can assign signs to them depending on whether theorientation of T p ι ( F ) and ξ p agree (positive) or disagree (negative). Positiveelliptic points are sources, while negative ones are sinks.F IGURE
1. Singularities of open book foliations. The firstrow depicts the embedding ι : F ֒ → Y with respect to theopen book. While the second row depicts the foliation F ob on F . The first and second pictures are intersections of thebinding with ι ( F ) . The third and fourth pictures depict aminimum and a maximum of π ◦ ι The surface F is depictedyellow when its orientation is counterclockwise, and it is or-ange when its orientation is clockwise. DDITIVITY OF SUPPORT NORM OF TIGHT CONTACT STRUCTURES 5 F IGURE
2. Hyperbolic singularities of open book foliations.The function π is assumed to be given by the vertical coordi-nate. The surface F is depicted yellow when its orientationis counterclockwise, and it is orange when its orientationis clockwise. The first picture is a positive hyperbolic sin-gularity, while the second picture is a negative hyperbolicsingularity.F IGURE
3. The foliation on the disc D . The elliptic pointsare denoted by hollow circles with signs, while the hyper-bolic points are the intersections of the arcs on the Figure.All the non-depicted leaves are regular on D. We further as-sume that the sign of the hyperbolic points are all the same.Later we will use a graph associated to an oriented singular foliation F with compact leaves, only elliptic and hyperbolic singular points and withthe extra information of “sign” for hyperbolic singularities. The vertices ofthe graph G ++ = G ++ ( F ) are the positive elliptic points (sources) of F , andthe edges connect those positive elliptic points that lie on the ends of stablesepartrices of the same poisitive hyperbolic point. See Figure 4An open book ( B, π ) supports a contact structure ξ = ker α if α is positiveon B and dα pulls back as a volume form to the pages. The relation be-tween open book foliations and characteristic foliations is described in thefollowing. We say that two oriented singular foliations on a surface F are topologically conjugate if there is a homomorphism of the surface that takesone foliation to the other. For foliations with compact leaves this means that VERA V ´ERTESI F IGURE
4. The construction of the graph G ++ (right) from afoliation (left).the combinatorial information of which elliptic points do the separatricesof the hyperbolic points go is the same. Proposition 2.3 ([5] connection of open book foliations and characteristicfoliations) . Let ( B, π ) be an open book of the 3-manifold Y . Suppose that foran embedding ι : F ֒ → Y the open book foliation has no circles. Then there is acontact structure ξ supported by the open book ( B, π ) such that F ξ and F ob aretopologically conjugate. Remark 2.4.
In most (maybe all) papers about braid- or open book folia-tions, the authors treated these singular foliations merely as a collection ofleaves with some singularities with no additional structure. On the otherhand Giroux [4] defined characteristic foliations as the dual of the pullbackof the contact form to the surface. Thus characteristic foliations have anextra structure of a singular vector field up to multiplictaion with a smoothpositive function. The sign of the divergence of such structures is well-defined in singular points, and as proved by Giroux these divergences arealways nonzero. A more precise definition of open book foliations woulddefine them as a pair ( E, e π ) and the corresponding vector field away fromthe elliptic points is the dual of d e π . But in this case the divergence wouldbe zero everywhere. Thus one cannot even hope that the two singular foli-ations agree. That is why the above statement only states topological con-jugancy. As Giroux proved [8], nonzero divergence on the singularitiesimplies that the set of singular points and the leaves determine the equiva-lence class of the singular foliations, thus for characteristic foliations we donot loose information by using this vague definition.The proof of Proposition 2.3 uses local models for ( B, π ) near the singularand regular points of F ob , and alters the Thurston and Winkelnkemper [11]construction just slightly near B ∩ F .Let ( B, π ) and ( B ′ , π ′ ) be open books for the 3-manifolds Y and Y ′ re-spectively. Assume that for the embeddings ι : F ֒ → Y and ι ′ : F ֒ → Y ′ theopen book foliations F ob ( ι ) and F ob ( ι ′ ) agree and have no circle leaves. Ifthe surfaces ι ( F ) and ι ′ ( F ) are separating then, as we prove in [7], one can DDITIVITY OF SUPPORT NORM OF TIGHT CONTACT STRUCTURES 7 obtain new contact 3-manifolds with open books by open book surgery asfollows. Cut open the 3-manifolds Y and Y ′ along ι ( F ) and ι ′ ( F ) to obtain3-manifolds with boundaries. Denote the parts of Y by Y with ∂Y ∼ = F and Y with ∂Y ∼ = − F . Similarly denote the parts of Y ′ by Y ′ with ∂Y ′ ∼ = F and Y ′ with ∂Y ′ ∼ = − F . Then using the maps ι − ◦ ι ′ between the bound-aries one can form the new objects: Z = Y ∪ Y ′ and Z ′ = Y ′ ∪ Y ζ = ξ | Y ∪ ξ ′ | Y ′ and ζ ′ = ξ ′ | Y ′ ∪ ξ | Y C = ( B ∩ Y ) ∪ ( B ′ ∩ Y ′ ) and C = ( B ′ ∩ Y ′ ) ∪ ( B ∩ Y ) ̺ = π | Y \ B ∪ π ′ | Y ′ \ B ′ : Z \ C → S and ̺ ′ = π ′ | Y ′ \ B ′ ∪ π | Y \ B : Z ′ \ C ′ → S Then it is proved in [7]:
Theorem 2.5 ([7] open book surgery) . With the above notations and conditionsthe contact structures ( Z, ζ ) and ( Z ′ , ζ ′ ) are supported by the open books ( C, ̺ ) and ( C ′ , ̺ ′ ) . (cid:3)
3. P
ROOF OF THE MAIN T HEOREM
The inequality sn ( ξ ) + sn ( ξ ) + 1 ≥ sn ( ξ ) is obvious, as one can formconnected sum of open books for ( Y , ξ ) and ( Y , ξ ) in the neighbourhoodof a point on their bindings, which induces boundary connected sum onthe pages.Let ( Y , ξ ) and ( Y , ξ ) be tight contact 3-manifolds and let ( B, π ) be asupporting open book with minimal norm for their connected sum ( Y, ξ ) =( Y , ξ ) Y , ξ ) . In the following two lemmas and using open book surgerywe will construct open books ( B , π ) and ( B , π ) for ( Y , ξ ) and ( Y , ξ ) with norms that add up to χ ( B, π ) + 1 .Since Y = Y Y there is a separating embedded sphere ι : S ֒ → Y along which the connected sum is formed. By a possible isotopy of theembedding we can make sure that the sphere admits an open book foliation F = F ob with no circles. Lemma 3.1 ( G ++ is a tree) . Let F be an open book foliation with no circles ona sphere, induced by an open book ( B, π ) that supports a tight contact structure.Then the graph G ++ ( F ) is a tree.Proof. By Proposition 2.3 and the uniqueness of contact structures supportedby a given open book, ξ can be isotoped so that F ξ and F are topologicallyconjugates of each other. In particular the graphs G ++ ( F ) and G ++ ( F ξ ) arehomomorphic. The dividing curve Γ on S is given by ∂N ( G ++ ( F ξ )) . As ξ For being able to do open book surgery on the nose one needs that the actual functions π ◦ ι and π ′ ◦ ι ′ agree on F , which is very restrictive. Instead we relax this condition, andonly require that the singular foliations agree, but then for the gluing we need to build upa general theory, that is worked out in [7]. VERA V ´ERTESI is tight Γ has only one component. Thus G ++ ( F ξ ) is a tree, and then G ++ ( F ) is a tree as well. (cid:3) Proposition 3.2 (realisation of a foliation on S as open book foliation in ( B , π ) ) . Let F be an open book foliation on a sphere with no circles. If G ++ ( F ) is a tree, then F can be realised as an open book foliation induced by the standardopen book ( B , π ) for some embedding ι ′ : S ֒ → S .Proof. Let e ± denote the number of positive/negative elliptic points and h ± be the number of positive/negative hyperbolic points in F . Since thenegative elliptic points are paired to the positive elliptic points in Y by thearcs B ∩ Y we have e + = e − . As G ++ is a tree we get e + − h + , andby the Poincar´e–Hopf Theorem we have ( e + + e − ) − ( h + + h − ) = 2 . Thus h − = e + − . From now on let k = e + and we will use induction on k .If k = 1 then there are no hyperbolic points, and the standard embeddingof S with image in the pure imaginary part of S ⊂ C works. Otherwiseassume that k > and take a degree one vertex p of the graph G ++ . Con-sider the star D p of p . As shown on the first picture of Figure 5, D p is adisc whose boundary consists of n hyperbolic points with their unstableseparatrices connecting them to n negative elliptic points. Note that sinceF IGURE
5. The star D p of p for n = 4 . The elliptic pointsare marked with signs in hollow circles, while the hyper-bolic points are black dots. In the second picture the disc D from Proposition 2.2 is encircled with dashed green line.The third picture shows the foliation after the change.each of the hyperbolic points are connected to p they all happen in differenttimes (i.e. they have different π -value), thus the boundary of D p is embed-ded, and even a sufficiently small neighbourhood of D p is embedded in S .Also, since p has degree one in the graph G ++ all but one of the hyperbolicpoints of D p are negative.If n = 2 , then F can be produced from a foliation F ′ where one ex-changes the neighbourhood of D p with the neighbourhood of a single neg-ative elliptic point. The new G ++ ( F ′ ) is still a tree, thus by induction F ′ is induced by an embedding of S into ( B , π ) . Now performing a fingermove, as in Figure 6 pushing the neighbourhood of the negative elliptic DDITIVITY OF SUPPORT NORM OF TIGHT CONTACT STRUCTURES 9 point, that we replaced D p with, through the z -axis gives a new embed-ding of S into ( B , π ) with open book foliation F on it. While doing thisfinger move we can achieve any identification of the boundary of D p and D − D p , thus we can ensure that we get back the same leaves as we startedwith.F IGURE
6. Changing the open book foliation with a finger move.If n > , then since all but one of the hypebolic points of D p are negativewe can perform a local change of the foliation on two consecutive hyper-bolic points as on the last two pictures of Figure 5 and get a new open bookfoliation with smaller n . Thus by a sequence of these local change moveswe can construct a foliation F ′ with n = 2 , and then as before F ′ can be re-alised by an embedding of S , and by Proposition 2.2 we can do the reverseof the local change moves, now by changing the π -values of the hyperbolicpoints in question. This change gives us the required embedding. (cid:3) End of the proof for Theorem 1.1.
We continue the argument from the begin-ing of the section of constructing open books for ( Y , ξ ) and ( Y , ξ ) froman open book ( B, π ) for ( Y, ξ ) . Let F be an open book foliation with no cir-cles induced on the separating sphere ι : S ֒ → Y = Y Y , using Lemma3.1 and Proposition 3.2 we can construct an embedding ι ′ : S ֒ → S suchthat the standard open book ( B , π ) induces the same open book foliation F on S . Then by Theorem 2.5 we can do open book surgery on the twoopen books ( B, π ) and ( B , π ) along F to obtain open books ( B , π ) and ( B , π ) for the contact structures ( Y , ξ ) and ( Y , ξ ) . Since the gluing hap-pened along the same foliation the sum of the Euler characteristics for theold and new open books are the same: sn ( ξ ) + sn ( ξ ) ≤ χ ( B , π ) + χ ( B , π ) = χ ( B , π ) + χ ( B, π ) = sn ( ξ ) − . This inequality together with the obvious inequality in the begining of thesection proves the statement. (cid:3)
4. F
UTURE DIRECTIONS AND U PCOMING PAPERS
However simple this cut and paste technique for open books is, it hasnot yet been fully explored. By understanding the embeddings of S into ( B , π ) that induce a given open book foliation the results of the papercould possibly be extended to prove the additivity of genus for open booksof tight contact structures. Another direction is to generalise this embed-ding result to some foliations on tori, and get an understanding of openbooks in terms of JSJ decompositions of contact 3-manifolds. In a paper inpreparation with Licata [7] we describe the full machinery that is neededfor cutting and pasting open books by defining the notion of open booksthat have been cut. These open books are called foliated open books , and sim-ilarly to ordinary or partial open books they satisfy all desirable propertiesof existence and uniqueness. R EFERENCES [1] Daniel Bennequin. Entrelacements et ´equations de Pfaff. In
Third Schnepfenried geometryconference, Vol. 1 (Schnepfenried, 1982) , volume 107 of
Ast´erisque , pages 87–161. Soc.Math. France, Paris, 1983.[2] Y. Eliashberg. Classification of overtwisted contact structures on -manifolds. Invent.Math. , 98(3):623–637, 1989.[3] John B. Etnyre. Planar open book decompositions and contact structures.
Int. Math.Res. Not. , (79):4255–4267, 2004.[4] Emmanuel Giroux. Convexit´e en topologie de contact.
Comment. Math. Helv. , 66(4):637–677, 1991.[5] Tetsuya Ito and Keiko Kawamuro. Open book foliation.
Geom. Topol. , 18(3):1581–1634,2014.[6] Tetsuya Ito and Keiko Kawamuro. Operations on open book foliations.
Algebr. Geom.Topol. , 14(5):2983–3020, 2014.[7] J. Licata and V. V´ertesi. Foliated open books. 2018.[8] Patrick Massot. Topological methods in 3-dimensional contact geometry. In
Contactand symplectic topology , volume 26 of
Bolyai Soc. Math. Stud. , pages 27–83. J´anos BolyaiMath. Soc., Budapest, 2014.[9] Burak Ozbagci. On the Heegaard genus of contact 3-manifolds.
Cent. Eur. J. Math. ,9(4):752–756, 2011.[10] Peter Ozsv´ath, Andr´as Stipsicz, and Zolt´an Szab´o. Planar open books and Floer ho-mology.
Int. Math. Res. Not. , (54):3385–3401, 2005.[11] W. P. Thurston and H. E. Winkelnkemper. On the existence of contact forms.
Proc. Amer.Math. Soc. , 52:345–347, 1975.IRMA, U
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