Reeb foliations on S^5 and contact 5-manifolds violating the Thurston-Bennequin inequality
RREEB FOLIATIONS ON S AND CONTACT -MANIFOLDSVIOLATING THE THURSTON-BENNEQUIN INEQUALITY ATSUHIDE MORI
Abstract.
This article describes the following results; i) convergence of highdimensional contact structure to codimension one foliation with Reeb compo-nent, ii) relation between
Nil -type and
Sol -type contact submanifolds of S ,iii) definition of convex Thurston-Bennequin inequality and iv) generalizationof Lutz twist via convex hypersurface theory. We perform Lutz twists along Sol -type submanifolds of S to obtain exotic contact structures violating theinequality. We also describe the corresponding modification of foliation. Introduction
The author obtained the following results and distributed a preprint on themin 2009. Since some of the results has been used and brushed-up in literatures,especially in [17] and [18], he would like to belatedly describe them in this article.We omit some details according to the current necessities.i) In §
2, we construct a family of contact structures of a closed manifold ofdimension greater than three which converges to a codimension one folia-tion. The family is presented by a quadratic curve in the space of 1-formscontrastingly to the linear deformation of 3-dimensional contact structureto a foliation in [22]. In certain cases, this result has been generalized to adeformation of contact structure to leafwise symplectic foliation in [23].ii) In §
3, we briefly explain a contact geometrical relation between
N il -typeand
Sol -type submanifolds of the standard S , which has been completedby Naohiko Kasuya via complex singularity theory in [17].iii) In §
4, we describe convex hypersurface theory due to Giroux and add to ita relative version. Then we define convex Thurston-Bennequin inequality.It is likely that any convex hypersurface (with boundary) in the standard S n +1 will satisfy the inequality, while there exists a non-convex hypersur-face in S n +1 which violates the inequality. (We describe it in Appendix,which has been another unpublished manuscript of this author.)iv) In §
5, we describe a generalization of Lutz twist via convex hypersurfacetheory. This was generalized and improved to a satisfactory form by Massot,Niederkr¨uger and Wendl in [18].We notice that the above results relate to each other. Indeed, in §
6, we perform Lutztwists along certain
Sol -type submanifolds of S and obtain exotic (non-fillable)contact structures which violate the convex Thurston-Bennequin inequality. Wealso describe the corresponding modifications of foliations. This provides manyexamples of foliations of S with (generalized) Reeb components. Mathematics Subject Classification.
Primary 57R17, Secondary 57R30, 57R20.
Key words and phrases.
Contact structure; foliation ; Milnor fibration; open-book. a r X i v : . [ m a t h . G T ] M a y A. MORI Convergence of contact structure to foliation
We recall the definition of a supporting open-book structure in Giroux [12].
Definition 2.1.
Let α be a contact form on a closed manifold M n +1 , and N n − a codimension two contact submanifold. Suppose that a tubular neighborhood of N n − is the product N n − × D , and the pull-back of the angular coordinate of D extends to a fiber bundle projection θ : M n +1 \ N n − → S . Then we saythat the open-book structure defined by θ supports the contact structure ker α ifthere exists a function h : M n +1 → R such that dθ ∧ ( d ( e h α )) n > N n − . The contact form α = e h α is called an adapted contact form.A cooriented contact structure is defined by a contact form α . If a supportingopen-book structure is specified, we usually assume that α is adapted to it. Notethat we can take the above h so that the restriction h | N n − is arbitrary. Let ρ denote the square of the radial coordinate of D . Modifying ρ and θ if necessary,we have the axisymmetric expression α = f ( ρ ) β + g ( ρ ) dθ on N n − × D , where β is (the pull-back of) the restriction α | N n − , f ( ρ ) a positive decreasing function of ρ with f (0) = 1, and g ( ρ ) a non-negative non-decreasing function of ρ smoothlytangent to ρ and 1 respectively at ρ = 0 and ρ = 1. Our result of this section is Theorem 2.2.
In the above setting for n > , suppose that the contact submanifold N n − admits a non-zero closed -form ν with ν ∧ ( dβ ) n − = 0 . Then there existsa family of contact forms { α t } ≤ t< on M n +1 which starts with α = α andconverges to a non-zero -form α with α ∧ dα = 0 . That is, the contact structure kerα converges to a foliation defined by α at the end of an isotopic deformation.Proof. Take smooth functions f ( ρ ), g ( ρ ), h ( ρ ) and e ( ρ ) of ρ ∈ [0 ,
1] such thati) f = 1 near ρ = 0, f = 0 on [1 / , f (cid:48) ≤ , g = 1 near ρ = 1, g = 0 on [0 , / g (cid:48) ≥ , h = 1 on [0 , / h = 0 near ρ = 1,iv) e is supported near ρ = 1 /
2, and e (1 / (cid:54) = 0.Put f t ( ρ ) = (1 − t ) f ( ρ ) + tf ( ρ ), g t ( ρ ) = (1 − t ) g ( ρ ) + tg ( ρ ) and α t | ( N × D ) = f t ( ρ ) { (1 − t ) β + th ( ρ ) ν } + g t ( ρ ) dθ + te ( ρ ) dρ, where ν also denotes its pull-back. We extend α t by putting α t | ( M \ ( N × D )) = τ α + (1 − τ ) dθ where τ = (1 − t ) . We see from dν = 0 and ν ∧ ( dβ ) n − = 0 that α t ∧ ( dα t ) n can be written as nf n − t (1 − t ) n ( g (cid:48) t f t − f (cid:48) t g t ) β ∧ ( dβ ) n − ∧ dρ ∧ dθ on N × D and τ n +1 α ∧ ( dα ) n + τ n (1 − τ ) dθ ∧ ( dα ) n on M \ ( N × D ). Therefore we have α t ∧ ( dα t ) n > ≤ t < , α ∧ dα = 0 and α (cid:54) = 0 . This completes the proof of Theorem 2.2. (cid:3)
If a cooriented foliation by oriented leaves has a minimal region bordered byclosed leaves with totally coherent or totally anti-coherent orientation, we call ita dead-end component. Let F be the foliation defined by α . Then F consists oftwo dead-end components, the Reeb component { ρ ≤ / } ≈ N n − × D and itscomplement. Here the orientation of the border leaf depends on the sign of e (1 / EEB FOLIATIONS ON S AND CONTACT 5-MANIFOLDS 3
Remark.
A similar result in the case where n = 1 is contained in the author’spaper [22]. It provides a foliation with Reeb component at the end of an isotopicdeformation of a given cooriented contact structure on a closed 3-manifold. In thiscase the deformation can be presented by a straight line segment in the space of1-forms such that the Reeb field is transverse to the limit leaf along the core of theReeb component for 0 ≤ t <
1. The same result is also contained in Etnyre [6].Note that, contrastingly to this 3-dimensional case, the above family { α t } is not astraight line but a quadratic function of t , and the Reeb field keeps tangent to thelimit leaf along the core { ρ = 0 } of the Reeb component for 0 ≤ t < Codimension two submanifolds
Let T A denote the mapping torus T × R / ( (cid:0) xy (cid:1) , z +1) ∼ ( A (cid:0) xy (cid:1) , z ) of A ∈ SL ( Z ).In the case where trA ≥
2, we have a natural contact form on T A . Namely, Proposition 3.1.
1) In the case of A m, = (cid:18) m (cid:19) ∈ SL ( Z ) for a positiveinteger m , T A m, is a N il -manifold admitting the contact form β = dy + mzdx .2) (Geiges [9] , Ghys [10] , Mitsumatsu [20] ) Let v ± be eigenvectors of A with Av ± = a ± v ± , where a > and dx ∧ dy ( v + , v − ) = 1 . Then the -forms β ± = ± a ∓ z dx ∧ dy ( v ± , · ) define the Anosov foliations for thesuspension flow. Using these forms we can construct the symplectic cylinder ([ − , × T A , d ( β + + sβ − )) ( s ∈ [ − , with contact-type boundary ( − T A ) (cid:116) T A . Note that T A and − T A = T A − are Sol -manifolds and the Anosov foliations are the Lie algebraic ones.
Honda [14], in a part of his classification of tight contact structures of T -bundles,showed that, for any A ∈ SL ( Z ) with tr A ≥
2, there exist a positive integer m and a m -tuple k = ( k , . . . , k m ) of non-negative integers such that the product A m,k = (cid:18) (cid:19) (cid:18) k (cid:19) . . . (cid:18) (cid:19) (cid:18) k m (cid:19) is conjugate to A in SL ( Z ). Then Van Horn-Morris [26] showed that the abovecontact structure ker β or ker( β − + β + ) of T A m,k is supported by the open-bookstructure determined up to equivalence by the following data: Page:
The page P , i.e., the fiber of θ in the previous section is the m timespunctured torus which is divided into m copies P i ( i = 1 , . . . , m ) of thethree times punctured sphere along disjoint loops γ i between P i and P i +1 . Monodromy:
The monodromy is the composition τ ( ∂P ) ◦ (cid:81) mi =1 { τ ( γ i ) } k i ,where τ ( γ ) denotes the right-handed Dehn twist along a loop γ on P , and τ ( ∂P ) the simultaneous right-handed Dehn twist along ∂ -parallel loops.Using this description, the author proved the following theorem. Theorem 3.2.
In the case where m ≤ , the above T A m,k is contactomorphic tothe link of the singularity (0 , , of the complex surface f m,k ( ξ, η, ζ ) = 0 , where f , ( k ) = ξ + ( η − ζ )( η + 2 ζ η + ζ − ζ k ) and f , ( k ,k ) = ξ + { ( ζ + η ) − ζ k }{ ( ζ − η ) + ζ k } . A. MORI
Since we may consider the singularity links as contact submanifolds of S , thistheorem provides a contact submanifold with N il - or
Sol -geometry according to k =0 or not. We call it a N il -type or
Sol -type contact submanifold. The author askedwhether there exists a similar result for m = 3 since ker β is still contactomorphicto a Brieskorn N il -manifold for ( m, k ) = (3 , Theorem 3.3 (Kasuya [17]) . The above contact manifold T A m,k is contactomorphicto a link of isolated surface singularity in C if and only if m ≤ , where thesingularity can be taken as (0 , , of the surface ξ a + η b + ζ c + ξηζ = 0 where ( a, b, c ) stands for (2 , , k ) in the case where m = 1 , for (2 , k , k ) inthe case where m = 2 , and for (3 + k , k , k ) in the case where m = 3 . We would like to sketch the proof of Theorem 3.2. First we prepare an easyproposition which enabled the author to find the singularities.
Proposition 3.4.
The complex curve C : ξ = − ( η − p ) · · · ( η − p m +2 ) on the ξη -plane is topologically a once or twice punctured, properly embedded, oriented surfacewith Euler characteristic ( − m ) if the points p i are mutually distinct. These pointsare the critical values of the hyperelliptic double covering π η | C where π η denotesthe projection to the η -axis. Let B : p = p ( θ ) , . . . , p m +2 = p m +2 ( θ ) be a closedbraid on C × S ( θ ∈ S ) . Then the above curve C = C θ traces a surface bundleover S . Fix a proper embedding l of R into the η -axis such that l (1) = p (0) , . . . , l ( m + 2) = p m +2 (0) . Suppose that the closed braid B is isotopic to the geometric realization of a word J (cid:89) j =1 { σ i ( j ) } q ( j ) ( q ( j ) ∈ Z , i ( j ) ∈ { , . . . , m + 1 } ) , of the right-handed exchanges σ i : C → C of strands p i and p i +1 along the arc l ([ i, i + 1]) ( i = 1 , . . . , m + 1) . Then the monodromy of the surface bundle C θ is J (cid:89) j =1 { τ ( (cid:96) i ( j ) ) } q ( j ) where (cid:96) i = ( π η | C ) − ( l ([ i, i + 1])) . Sketch of the proof of
Theorem 3.2 . Regard ζ as a parameter and take the brancheddouble covering π η | C ζ of the curve C ζ : f m,k = 0 on the ξη -plane. Then the criticalvalues of π η | C ζ are p , p = − ζ { − ( ζ k ) / } and p = 2 ζ if m = 1(resp. p , p = − ζ { − ( ζ k ) / } , p , p = ζ { − ( − ζ k ) / } if m = 2) . As ζ rotates along a small circle | ζ | = ε once counter-clockwise, the points p ( ζ ),. . . , p m +2 ( ζ ) traces a closed braid, which is clearly a geometric realization of( σ ◦ σ ) ( σ ) k (resp. ( σ ◦ σ ◦ σ ) ( σ ) k ( σ ) k ) . From Proposition 3.4 and the relation τ ( ∂C ζ ) (cid:39) ( τ ( (cid:96) ) ◦ τ ( (cid:96) )) (resp. τ ( ∂C ζ ) (cid:39) ( τ ( (cid:96) ) ◦ τ ( (cid:96) ) ◦ τ ( (cid:96) )) ) , we see that the singularity link of { f m,k = 0 } admits the above mentioned open-book structure. To be more precise, we have to consider everything in a small ball,take ε much smaller, and use the result of Caubel-Nemethi-Popescu-Pampu [2]. (cid:3) EEB FOLIATIONS ON S AND CONTACT 5-MANIFOLDS 5 The Thurston-Bennequin inequality for convex hypersurfaces
Preliminaries on the Thurston-Bennequin inequality.
Let Σ be a com-pact connected oriented hypersurface embedded in a contact (2 n + 1)-manifold( M n +1 , ker α ), and S + (Σ) (resp. S − (Σ)) the set of the positive (resp. nega-tive) tangent points of Σ to ker α . Here the sign of the tangency coincides withthe sign of ( dα | Σ) n at each tangent point. We assume that the union S (Σ) = S + (Σ) ∪ S − (Σ) is a discrete subset in int Σ. Considering on the symplectic hyper-plane (ker α, dα | ker α ) at each point on Σ, we see that the symplectic orthogonalof the intersection T Σ ∩ ker α forms an oriented line field L on Σ. Then the set ofsingular points of L coincides with the set S (Σ) of tangent points. Definition 4.1.
The singular foliation F Σ defined by L is called the characteristicfoliation of Σ with respect to the contact structure ker α .Put λ = α | Σ and take any volume form dvol on Σ. Then we see that the vectorfield X on Σ defined by ι X dvol = λ ∧ ( dλ ) n − is a positive section of L . Moreover ι X { λ ∧ ( dλ ) n − } = − λ ∧ ι X ( dλ ) n − = 0 ( λ ∧ ( dλ ) n − (cid:54) = 0) . This implies that λ ∧ ι X dλ = 0, and therefore the flow of X preserves the conformalclass of λ . Since dvol is arbitrary, we may take X as any positive section of L .Therefore λ defines a holonomy invariant transverse contact structure of F Σ . Notethat, even for another volume form dvol (cid:48) on Σ, the sign of div X = ( L X dvol (cid:48) ) /dvol (cid:48) at each singular point p ∈ S (Σ) coincides with the sign of the tangency at p . Thus F Σ itself contains the information about the sign of the tangency.On the other hand, by using a positive section X of L , we can define the indexInd p of a singular point p ∈ S (Σ) as the vector field index Ind X p . Definition 4.2.
Suppose that the boundary ∂ Σ of the above hypersurface Σ is non-empty, and the characteristic foliation F Σ is positively (i.e., outwards) transverseto ∂ Σ. Then we say that Σ is a hypersurface with contact-type boundary. Thecontact-type boundary ∂ Σ inherits the contact form λ | ∂ Σ = α | ∂ Σ. Remark.
We usually fix a primitive 1-form λ on an exact symplectic manifold(Σ , dλ ). This is equivalent to fix a vector field X with ι X dλ = λ . If X is positivelytransverse to the boundary ∂ Σ, then ( ∂ Σ , λ | ∂ Σ) is called the contact-type boundary.The above definition is a natural shift of this notion into our setting.Let D be an embedded disk with contact-type boundary in a contact 3-manifold.We say that D is overtwisted if the singularity S ( D ) consists of a single sinkpoint. Note that a sink point is a negative singular point since it has negativedivergence. If a contact 3-manifold contains overtwisted disks, we say that it isovertwisted, else it is tight. We can show that the existence of an overtwisted diskwith contact-type boundary is equivalent to the existence of an overtwisted diskwith Legendrian boundary, which is an embedded disk D (cid:48) similar to the above D except that the characteristic foliation F D (cid:48) is tangent to the boundary ∂D (cid:48) , where ∂D (cid:48) (or − ∂D (cid:48) ) is a closed leaf of F D (cid:48) . Indeed perturbing the above D if necessary,we can find D (cid:48) in D . Conversely, the characteristic foliation of the boundary ofan usual neighborhood of D contains a sink point N (north pole), a source point S (south pole) and a pair of closed orbits which bounds a tubular neighborhoodof the equator. Thus, removing a narrow tubular neighborhood of the closed orbitwith negative divergence, we obtain the above D as the disk containing N . A. MORI
Let Σ be any surface with contact-type (i.e., positive transverse) boundary em-bedded in the standard S . Then Bennequin [1] proved the following inequalitywhich implies the absence of overtwisted disks in S , i.e., the tightness of S . Thurston-Bennequin inequality (I) . (cid:88) p ∈ S − (Σ) Ind p ≤ n +1) dimensionalanalogue of an overtwisted disk, i.e., a plastikstufe (or an overtwisted family) whichis the product D × L n − of an overtwisted disk D with a closed isotropic subman-ifold L n − (see § n -dimensionalanalogue of overtwisted disk. Here we should notice that a folklore says that theabove inequality does not hold in higher dimension as a merely algebraic inequalitydescribed in the following remark (see also Appendix). In order to bring out thegeometric flavor of the inequality, we put a strong limit on the test hypersurfaces.It is the “convexity” in the next subsection. Remark.
The Thurston-Bennequin inequality can also be written in terms of rel-ative Euler number as follows. The above vector field X ∈ T Σ ∩ ker α is a sectionof ker α | Σ which is canonical (i.e. exhausting) near the boundary ∂ Σ. Thus undera certain coherent boundary condition we have (cid:104) e (ker α ) , [Σ , ∂ Σ] (cid:105) = (cid:88) p ∈ S + (Σ) Ind p − (cid:88) p ∈ S − (Σ) Ind p. Then the Thurston-Bennequin inequality can be expressed as −(cid:104) e (ker α ) , [Σ , ∂ Σ] (cid:105) ≤ − χ (Σ) . There is also an absolute version of the inequality for a closed hypersurface Σ with χ (Σ) ≤ |(cid:104) e (ker α ) , [Σ] (cid:105)| ≤ − χ (Σ), or equivalently as (cid:88) p ∈ S − (Σ) Ind p ≤ (cid:88) p ∈ S + (Σ) Ind p ≤ . The absolute version trivially holds if the Euler class e (ker α ) ∈ H n ( M n +1 ; Z )is a torsion. Note that the inequality and its absolute version can be defined forany oriented plane field on an oriented 3-manifold M (see Eliashberg-Thurston[5]). They are originally proved for a foliation on M without Reeb componentsby Thurston (see [25]). On the other hand, we can deform even a tight contactstructure to a foliation with Reeb component ([22]; see also [21]). Thus manyfoliations with Reeb components also satisfy the inequality.4.2. Convex hypersurfaces.
In this subsection, we review the convex hypersur-face theory doe to Giroux [11], and add to it a relative version.A vector field Z on a contact manifold ( M n +1 , ker α ) is called a contact vec-tor field if it satisfies α ∧ L Z α = 0, i.e., preserves ker α . Let V ker α denote thespace of contact vector fields for ker α . It is fundamental that the contact form α determines the linear isomorphism α ( · ) : V ker α → C ∞ ( M n +1 ). Indeed we canidentify V ker α to the space of Hamiltonian vector fields on the symplectization (cid:0) R ( (cid:51) s ) × M n +1 , d ( e s α ) (cid:1) for functions which split as e s H ( H ∈ C ∞ ( M n +1 )). EEB FOLIATIONS ON S AND CONTACT 5-MANIFOLDS 7
Definition 4.3.
1) For a contact vector field Z , the function H = α ( Z ) is called thecontact Hamiltonian function. Conversely for a function H on M n +1 , the contactvector field Z uniquely determined by α ( Z ) = H is called the contact Hamiltonianvector field. The contact Hamiltonian vector field of the constant function 1 lies inthe degenerate direction of dα and is called the Reeb field.2) A closed oriented hypersurface Σ embedded in a contact manifold is said tobe convex if there exists a contact vector field transverse to Σ.Let Z be a contact Hamiltonian vector field of a function H which is positivelytransverse to a closed convex hypersurface Σ. Perturbing H if necessary, we mayassume that the level set { H = 0 } is a regular hypersurface transverse to Σ. LetΣ × ( − ε, ε ) denote a neighborhood of Σ on which Z can be expressed as ∂/∂z ( z ∈ ( − ε, ε )). In the rest of this subsection, we restrict everything to Σ × ( − ε, ε ).Further we assume that α is z -invariant by multiplying it by a positive function ifnecessary. Then the restrictions ± Z |{± H > } are the Reeb fields of ± α/H and Z is tangent to their partition { H = 0 } . Indeed we can write α = λ + Hdz where λ is (the pull-back of) the restriction α | Σ and H is a z -invariant function. Definition 4.4.
The above Γ is called the dividing set on Σ with respect to Z . Γdivides Σ into the positive region Σ + = { ( H | Σ) ≥ } ⊂ Σ and the negative region − Σ − = { ( H | Σ) ≤ } ⊂ Σ. We orient Γ as Γ = ∂ Σ + or equivalently as Γ = ∂ Σ − .We see that the 2 n -form Ω = ( dλ ) n − ∧ ( Hdλ + nλdH ) satisfies Ω ∧ dz = α ∧ ( dα ) n .Thus the characteristic foliation F Σ is positively transverse to the dividing setΓ. Since λ defines the holonomy invariant transverse contact structure of F Σ ,Γ is a contact submanifold. Let β be any contact form presenting the contactstructure. Then, changing α and H with keeping ker α and Γ if necessary, we have α = e − H β + Hdz on a small neighborhood {− ε (cid:48) < H < ε (cid:48) } of Γ × ( − ε, ε ).Let dλ ± be exact symplectic forms on int Σ ± such that λ ± ∧ λ = 0 and λ + | Γ = λ − | Γ = β . Then λ ± ± dz is a z -invariant contact form on int Σ ± × ( − ε, ε ). Definition 4.5.
1) Let (int Σ + , dλ + ) be an exact symplectic manifold. Then thepair (int Σ + × R , λ + + dz ) ( z ∈ R ) is called its contactization with respect to λ + .2) Let (Σ + , dλ + ) be the compactification of an exact symplectic manifold withcontact-type end. Precisely, assume that the primitive 1-form λ + can be expressedas e − s β on a collar neighborhood ( − ε (cid:48) , × ∂ Σ + of the manifold Σ + , where s is thecoordinate of ( − ε (cid:48) , β is the pull-back of λ + | ∂ Σ. Take a function g + on Σ + such that g + is a decreasing function of s on ( − ε (cid:48) , × ∂ Σ + , g + = − s holds near ∂ Σ + , and g + = 1 holds except on the collar. Then α + = λ + + g + dz is a contactform on the cylinder Σ + × R . We call (Σ + × R , ker α + ) the modified contactizationof the exact symplectic manifold (int Σ + , dλ + ) with respect to λ + .We can identify the boundaries of the modified contactizations Σ + × R ( z ∈ R )w.r.t. λ + and Σ − × ( − R ) w.r.t. λ − as is depicted in Figure 1. Then we obtain acontact form (cid:101) α = α ± on (Σ + ∪ ( − Σ − )) × R which is expressed as (cid:101) α = e − s β + gdz on ( − ε (cid:48) , ε (cid:48) ) × ∂ Σ + × R , where g = ± g ± on Σ is the smoothing of ± ± Σ ± . Definition 4.6.
The contact manifold ((Σ + ∪ ( − Σ − )) × R , ker (cid:101) α ) is called theunified contactization of Σ = Σ + ∪ ( − Σ − ).Since the convex hypersurface Σ ⊂ ( M n +1 , α ) is compact, the original neigh-borhood Σ × ( − ε, ε ) is contactomorphic to a neighborhood of (Σ + ∪ ( − Σ − )) × { } A. MORI
Figure 1.
The modified contactizations: The arrows present theReeb fields of α ± and the shaded parts are identified.in the unified contactization. This is the convex hypersurface theory due to Giroux.We add to it a relative version. Definition 4.7.
Let Σ be a compact connected oriented hypersurface with non-empty contact-type boundary embedded in a contact manifold ( M n +1 , α ). ThenΣ is said to be convex if there exists a contact vector field Z which is positivelytransverse to Σ and satisfies the boundary condition α ( Z ) | ∂ Σ > H = α ( Z ) after suitably perturbing the contact vector field Z . Thenthe dividing set Γ = { H = 0 } ∪ Σ divides Σ into the positive region Σ + and the(possibly empty) negative region − Σ − in the same way as above. Then we have ∂ Σ = ∂ Σ + \ ∂ Σ − (cid:54) = ∅ (see Figure 2). Figure 2.
A convex hypersurface with contact-type boundaryWe can also construct the modified contactization and use it as the model of aneighborhood of the convex hypersurface as long as we do not modify the contacti-zation of Σ + near ∂ Σ × R ⊂ ∂ Σ + × R . We explain the treatment of the boundary ∂ Σ in the following subsection.4.3.
Open-book structure with convex pages.
We can construct a contactstructure from a given open-book structure with convex pages. Then the open-book structure can be considered as a generalization of “quasi-compatible” open-book structure introduced by Etnyre and Van Horn-Morris in [8]. In fact Etnyrecommented that their result on Giroux torsion might be generalized to the settingof this article. His idea motivated the work of Massot, Niederkr¨uger and Wendl in[18] as is mentioned there. The idea of placing a Lutz tube along the binding of anopen-book structure is also found in Ishikawa’s work [16].
EEB FOLIATIONS ON S AND CONTACT 5-MANIFOLDS 9
Proposition 4.8 (Construction of open-book structure with convex pages) . Let (Σ ± , dλ ± ) be two compact exact symplectic manifolds with contact-type boundary.Suppose that there exists an inclusion ι : ∂ Σ − → ∂ Σ + preserving the contact form,i.e., ι ∗ ( λ + | ∂ Σ + ) = λ − | ∂ Σ − . Similarly to the construction of unified contactization,we modify the exact symplectic structure so that they match up to define an exact -form dλ on the union Σ = Σ + ∪ ι ( − Σ − ) . Precisely, the modification is supportedin a small neighborhood ( − ε (cid:48) , ε (cid:48) ) × Γ of the dividing set Γ = ∂ ( − Σ − ) ⊂ Σ , andthe primitive -form λ is locally expressed as λ = e − s β for − ε (cid:48) < s < ε (cid:48) , where s denotes the coordinate of the interval ( − ε (cid:48) , ε (cid:48) ) , and β (the pull-back of ) the above ι -invariant contact form on Γ . Let ϕ : Σ → Σ be a diffeomorphism such that i) the support of ϕ does not intersect with (( − ε (cid:48) , ε (cid:48) ) × Γ) ∪ ∂ Σ , and ii) ϕ ∗ λ − λ = ± dh (according to the sign of the region) holds for a suitablepositive function h on Σ which is equal to on (( − ε (cid:48) , ε (cid:48) ) × Γ) ∪ ∂ Σ .Then the contact form (cid:101) α of the unified contactization Σ × R determines a contactform on the mapping torus Σ × R / ( x, z + h ) ∼ ( ϕ ( x ) , z ) . We cap-off the boundary ∂ Σ × ( R / Z ) by the tube (cid:18) ∂ Σ × D , ker (cid:18) f ( ρ ) f (1) λ | ∂ Σ + hg ( ρ )2 π dθ (cid:19)(cid:19) , where f ( ρ ) and g ( ρ ) are the functions in the setting of Theorem 2.2. This pro-vides a closed contact manifold ( M n +1 , ker α ) on which the family of the convexhypersurfaces { θ = 2 πz/h = const } defines an open-book structure. The hypersurface { θ ≡ π } is clearly convex. In other word, the pageΣ (cid:48) = { θ = 0 } is an extension of Σ such that ∂ Σ (cid:48) is contained in the dividing setof Σ (cid:48) . We can define the unified contactization of Σ (cid:48) just by partially glueing themodified contactizations in Definition 4.5 as is described in Definition 4.6. Clearly,the unified contactization of Σ is a contact submanifold of the extended unifiedcontactization of Σ (cid:48) . The construction in Proposition 4.8 can be considered asfollows. First we extend the mapping torus of Σ to that of Σ (cid:48) . Then we shrink itsboundary ∂ Σ (cid:48) × R / Z into ∂ Σ (cid:48) to obtain the open-book structure. Definition 4.9.
We say that one unbinds the open-book structure to the mappingtorus and one rebinds the latter to the former. We use the same terminologies evenwhen the open-book structure is defined only near the binding.
Remark.
Giroux proved that any symplectomorphism supported in int Σ + is iso-topic through such symplectomorphisms to ϕ with ϕ ∗ λ + − λ + = dh + ( ∃ h + > h | ∂ Σ + = 1. It is remarkable that he can prove the existence of sup-porting open-book decomposition. Namely, we can obtain a supporting open-bookstructure of a given closed contact manifold by using the result of Ibort, Mart´ınez-Torres and Presas in [15] on applicability of the Donaldson-Auroux approximatelyholomorphic method to complex valued functions on contact manifolds (see [12]).Trivially, this also implies the existence of the above open-book structure.4.4. The inequality for convex hypersurfaces.
For a convex hypersurface Σwith contact-type boundary, the inequality in § Thurston-Bennequin inequality (II) . χ (Σ − ) ≤ − = ∅ ). Suppose that there exists a convex disk Σ with contact-type boundary in a contact3-manifold which is the union Σ + ∪ ( − Σ − ) of a negative disk region Σ − and apositive annular region Σ + . Then, since χ (Σ − ) = 1 >
0, the convex disk Σ violatesthe Thurston-Bennequin inequality. We call the convex disk Σ a convex overtwisteddisk. The Giroux approximation in [11], implies that any overtwisted disk withcontact-type boundary is approximated by a convex overtwisted disk. Moreover,it also implies that the inequality for any convex surface holds if and only if thecontact 3-manifold is tight. This leads us to the following definition.
Definition 4.10.
A convex overtwisted hypersurface is a connected convex hy-persurface Σ + ∪ ( − Σ − ) with non-empty contact-type boundary which satisfies χ (Σ − ) >
0. We say that a contact structure is convex-overtwisted (resp. convex-tight) if it contains some (resp. no) overtwisted convex hypersurface.Allegorically, a convex hypersurface percepts ‘extra parts’ in a contact manifold atthe sensor Σ − through the counter χ (Σ − ) where each extra part must stick out a‘tight’ wear (i.e. tight contact structure). In other words, we see that the manifoldis wearing a ‘loose’ (i.e. overtwisted) contact structure if the number χ (Σ − ) ispositive. Indeed the original Thurston-Bennequin inequality expresses the tightnessof a contact 3-manifold as the absence of ‘extra parts’ (i.e. Lutz tubes). Remark.
Any convex overtwisted hypersurface Σ must contain a connected com-ponent S + of the positive region Σ + with ∂S + ∩ ∂ Σ (cid:54) = ∅ and ∂S + ∩ ∂ Σ − (cid:54) = ∅ . Thus S + is a connected symplectic manifold with disconnected contact-type boundary.The existence problem of such a symplectic manifold is called Calabi’s question,and McDuff [19] found the first example. Then a simpler example appeared inthe literatures referred in Proposition 3.1 2). It is a cylinder with 3-manifold basepresenting a gradual exchange between positive and negative contact structures.Mitsumatsu [20] further studied on coexistence situation of positive and negativecontact structures by means of a generalization of Anosov flow (i.e. projectivelyAnosov flow). On the other hand a Lutz twist can be considered as a local ex-change of the direction of a closed orbit K of a Reeb flow. Here the direction isdetermined by the sign of the restricted contact form on K . These facts motivatedthis author to generalize Lutz twist. Let N be a codimension two oriented sub-manifold of a contact 5-manifold ( M , ker α ) with trivial normal bundle. Supposethat N is tangent to the Reeb flow of α , and the restriction α | N is a positivecontact form. Further suppose that an Anosov flow on N presents an exchange of α | N with a negative contact form. Then, as is described in the next section, wecan make a modification of the contact structure supported near N which realizesthat exchange. Note that the Anosov flow on the submanifold tangent to both ofthe positive and negative contact structures. In the original 3-dimensional case, thecontact structures are the oriented zero sections of T K , and we may consider thatthe 0-dimensional flow is tangent to them. In other word, the notion of Anosovflow on 3-manifold generalizes the identity on the circle.5.
A generalization of Lutz twist
Definitions and examples.
We will perform a generalization of Lutz twistalong a twistable contact knot defined as follows.
Definition 5.1.
A codimension two closed contact submanifold ( K n − , ker β ) ofa contact manifold ( M n +1 , ker α ) is called a contact knot if it is connected and its EEB FOLIATIONS ON S AND CONTACT 5-MANIFOLDS 11 normal disk bundle is trivial. Suppose that there exists a cylinder I × K n − whichis the compactification of an exact symplectic manifold with contact-type end. Herewe assume that the end { } × K n − is contactomorphic to ( K n − , ker β ). Thenwe say that the contact knot K n − ⊂ M n +1 is twistable.The contact submanifold N n − in Definition 2.1 (or ∂ Σ in Proposition 4.8)is a contact knot if it is connected. Now we assume that K n − = N n − istwistable. Then we firstly unbind the contact manifold M n +1 along K n − as isdescribed in Definition 4.9; secondly insert the quotient I × K n − × ( R / Z ) of themodified contactization of the cylinder I × K n − with turning it upside down,i.e., as ( − I × K n − ) × ( − R / Z ); and lastly rebind the new boundary ( − K n − ) × ( − R / Z ) = K n − × ( R / Z ) to regain a closed contact manifold diffeomorphic to M n +1 . This modification is topologically the insertion of the tube K n − × D along K n − . From open-book point of view, it is the addition of exact symplecticcollar I × K n − to the page with reversed orientation. Note that the reversion ofthe orientation of the convex hypersurface with non-empty contact-type boundarymakes the sign of the region next to the boundary negative, and therefore breaksthe boundary condition in Definition 4.7. Thus the addition of the positive collaris mandatory. Since it does not change the manifold M n +1 , it is considered as amodification of the contact structure. Moreover, since the contact structure aroundany contact knot ( K n − , ker β ) can be written asker( e − ρ β + ρdθ ) (( √ ρ, θ ) ∈ D , ρ (cid:28) , we can also unbind the contact manifold ( M n +1 , ker α ) along K n − to obtain aportion of modified contactization. (This is just a local construction. Indeed, if( K n − , β ) is not fillable, we can not construct a whole modified contactization.)Thus we can perform a similar insertion of tube along any twistable contact knot. Proposition 5.2 (Definition of a generalization of Lutz twist) . We call the abovetube K n − × D an abstract Lutz tube. To show its existence, we describe explicitlythe cylinder I × K n − and the other end −{ } × K n − , which become the pageand the binding of the open-book structure of the Lutz tube. Suppose that a contactknot K n − with contact form β also admits a -form µ such that i) µ ∧ ( dµ ) n − = − β ∧ ( dβ ) n − ( < and ii) the other (2 n − -forms presented by products of β, dβ, µ and dµ vanish.Then we see that the -form α (cid:48) = sin πρ β + cos πρ µ − sin( πρ ) dθ defines the contact structure of the tube ( − K n − ) × ( − D ) whose core − K n − inherits the contact form µ . where ( √ ρ, θ ) is the polar coordinates of D . We callit the (half ) Lutz tube with core ( − K n − , ker µ ) and the longitude ( K n − , ker β ) .The core and the longitude span the page of the open-book structure { θ = const } of the Lutz tube. Thus we can insert it along any contact knot contactomorphic to ( K n − , ker β ) . We call this insertion a (half ) Lutz twist. Remark.
1) Proposition 5.2 has been generalized to a better form in [18].2) The twice iteration of half Lutz twist can be considered as a full Lutz twist.We notice that this is (perhaps essentially) different from the other generalizationof full Lutz twist found by Etnyre and Pancholi in [7]. Their full twist is much more elaborated and sophisticated than the original full Lutz twist even thoughphenomenally the former shrinks to the latter in 3-dimensional case.
Example 5.3.
In the case where n = 1, each connected component of K is thecircle S oriented by a non-zero 1-form β >
0. Then µ = − β satisfies the aboveconditions. The contact form α (cid:48) = − cos( πρ ) β − sin( πρ ) dθ defines the original halfLutz tube { ρ ≤ } = S × D . (The full Lutz tube is formally { ρ ≤ } .) Example 5.4.
Suppose that
T K ( n = 2) admits a frame ( e , e , e ) with[ e , e ] = e , [ e , e ] = e and [ e , e ] = 0 , that is, K is a Sol -manifold. Then the dual coframe ( β, µ, ψ ) satisfies dβ = µ ∧ ψ, dµ = β ∧ ψ and dψ = 0Then we see that β and µ satisfies the above conditions. For the Sol -type contactsubmanifolds in Theorem 3.2 (or 3.3), we may put e = 12 ( a z v − + a − z v + ) , e = 12 ( a z v − − a − z v + ) and e = 1log a ∂∂z , in the setting of Example 3.1. Then we have β = β + + β − and µ = β + − β − . Example 5.5.
Suppose that
T K ( n = 2) admits a frame ( e , e , e ) with[ e , e ] = e , [ e , e ] = e and [ e , e ] = e , that is, K is a (cid:102) SL ( R )-manifold. Since the dual coframe ( β, µ, ψ ) satisfies dβ = µ ∧ ψ, dµ = β ∧ ψ and dψ = µ ∧ β, we can see that β and µ satisfies the above conditions. We also have examples of (cid:102) SL ( R )-type submanifolds in S . However, since they lack relation with codimen-sion one foliations of S , we omit them in this article. Example 5.6.
As is shown in Geiges[9], there is also a (2 n +1)-dimensional solvableLie group for any n which admits a pair of left invariant 1-forms β and µ satisfyingthe above conditions. Moreover he found an explicit T -bundle over the circle whichadmits such β and µ . That is, we have an example of a seven dimensional Lutztube. The author suspects that seven dimensional Lutz twists enable us to changenot only the contact structure but also the homotopy class of the almost contactstructure of a given contact 7-manifold. See Question 5.5 in Etnyre-Pancholi [7].(See [17] snd [18] for subsequent developments.) Remark.
Once we can perform a Lutz twist, we can iterate it any number oftimes by switching the roles of β and µ . We can also generalize Giroux twist to theinsertion of even number of unbinded Lutz tubes I × S × K n − along a family S × K n − of twistable knots. Note that the original Giroux twist is the insertionof the toric annulus [0 , m ] × T (cid:51) ( ρ, θ, ϕ ) with the “propeller” contact structuredefined by ker( − cos( πρ ) ϕ − sin( πρ ) dθ ) along a pre-Lagrangian torus in M . EEB FOLIATIONS ON S AND CONTACT 5-MANIFOLDS 13
Plastikstufes in Lutz tubes.
In this subsection, we show that the Lutztubes of Example 5.4 contain plastikstufes. First we fix the model.
Definition 5.7.
Let ( M n +1 , ker α ) be a contact manifold, L n − a closed manifold,and ι : D × L n − → M n +1 an embedding such that the restriction ι | ( ∂D × L n − )is Legendrian. Then the image ι ( D × L n − ) is called a plastikstufe if ι ∗ ( α ) ∧ { f ( ρ ) dρ − ρdθ } = 0 , lim ρ → f ( ρ ) ρ = 0 and lim ρ → | f ( ρ ) | = ∞ holds for some functions h and f ( ρ ) (( √ ρ, θ ) ∈ D ). See Figure 3. Figure 3.
A plastikstufe D × S in ( M , ker α ).Niederkr¨uger and Chekanov introduced this notion and proved Theorem 5.8 ([24]) . The contact-type boundary of a compact semipositive sym-plectic manifold contains no plastikstufe.
Then we prove
Theorem 5.9.
Each of the Lutz tubes given in the above Example 5.4 contains aplastikstufe. Thus we can perform a Lutz twist along each of the
Sol -type contactsubmanifolds of S realized as the singularity links in Theorem 3.2 (or 3.3) to obtainan exotic contact structure of S . Remark.
This theorem has been generalized and improved to a satisfactory formin [18]. The next proof is now just an explicit example.
Proof.
In the tube T A × D (cid:51) (( p, q, z ) , ( √ ρ, θ )), with the contact form α (cid:48) = β + − cos( πρ ) β − − sin( πρ ) dθ = a − z dq − cos( πρ ) a z dp − sin( πρ ) dθ, we can take the plastikstufe P = { p = εa − z g ( πρ ) , q = − εa z cos( πρ ) g ( πρ ) } ⊂ T A × D where ε > g ( πρ ) a function of πρ such that g ( πρ ) = 1 (cid:112) cos( πρ ) ( πρ (cid:28)
1) and g ( πρ ) = sin( πρ ) ( πρ ≈ π ) . Indeed the restriction of α (cid:48) to P is α (cid:48) | P = επ { sin( πρ ) g ( πρ ) − ε cos( πρ ) g (cid:48) ( πρ ) } dρ − sin( πρ ) dθ = sin( πρ ) ρ { ε ( πρg ( πρ ) − πρ cot( πρ ) g (cid:48) ( πρ )) dρ − ρdθ } Then f ( ρ ) = ε ( πρg ( πρ ) − πρ cot( πρ ) g (cid:48) ( πρ )) satisfies f ( ρ ) ρ = 0 ( ρ (cid:28)
1) and lim ρ → f ( ρ ) = lim ρ → (cid:18) πρ ) − πρ ) (cid:19) = −∞ . (cid:3) Remark. As ε →
0, the above plastikstufe converges to a solid torus S × D foliated by S times the straight rays on D , i.e., the leaves are { θ = const } .Note that this solid torus is the preimage of the closed orbit { p = 0 , q = 0 } ofthe suspension Anosov flow (( x, y ) , z ) (cid:55)→ (( x, y ) , z + t ) on the core T A − under thenatural projection. This is called an overtwisted family in [7].6. Violation of the inequality and generalized Reeb components
Milnor fibrations as supporting open-book structures.
We can see thatthe exotic contact structure of S constructed in Theorem 5.9 is convex-overtwistedfrom the following lemma essentially contained in the note [13] of Giroux. Lemma 6.1 (see [13]) . The Milnor tube of an isolated singularity (0 , . . . , ∈ C n +1 of complex hypersurface determines an isotopy class of contact structures of S n +1 via the exact symplectic open-book structure associated to it. Moreover the isotopyclass is represented by the standard contact structure of S n +1 .Sketch of the proof. On the small ball B ε = {| z | + · · · + | z n +1 | ≤ ε } in C n +2 ,we consider the graph G k of the function z n +1 = kf ( z , . . . , z n ). From the Graystability, the contact structure of G k is isotopic to that of the standard S n +1 = Γ .Writing z n +1 as x n +1 + √− y n +1 , we see from the obvious inequality( − y n +1 dx n +1 + x n +1 dy n +1 )( − y n +1 ∂/∂x i +1 + x n +1 ∂/∂y n +1 ) ≥ , and dz n +1 | Σ ∞ = 0, that arg( f | ∂ Σ k ) defines a supporting open-book structure of ∂G k equivalent to the Milnor tube {{ f = δ } ∩ B ε } | δ | = ε (cid:48) if k is sufficiently largeand ε (cid:48) > ∂G k which is close to the rotation vector field − y n +1 ∂/∂x i +1 + x n +1 ∂/∂y n +1 andtransverse to the contact structure as well as to the pages { arg f | G k = const } . (cid:3) Theorem 6.2.
1) Suppose that the Euler characteristic of the page { θ = const } ofan exact symplectic open-book structure θ : M n +1 \ N n − → S is positive, andeach connected component of the contact submanifold N n − is twistable. Then wecan insert the Lutz tube along N n − to obtain a new contact structure which isconvex-overtwisted. Indeed since we reversed the orientation of pages and addedpositive collars, the pages are convex overtwisted hypersurfaces.2) Especially each exotic contact structure of S constructed in Theorem 5.9 isconvex-overtwisted. (The Milnor fiber is homotopically a bouquet of -spheres.) Generalized Reeb components.
Putting ν = dψ in Example 5.4, we seethat the open-book structures associated to the Milnor fibrations of the singularitiesin Theorem 3.2 (or Theorem 3.3) satisfy the condition of Theorem 2.2. Let F denote the limit foliation of a deformation of the standard contact structure ker α associated to a Sol -type contact submanifold N . Theorem 6.3.
Let ker α (cid:48) be the exotic convex-overtwisted contact structure ob-tained by inserting the Lutz tube with core ( − N , µ ) along the above Sol -type sub-manifold ( N , β ) ⊂ ( S , ker α ) . Then we can deform the exotic contact structure EEB FOLIATIONS ON S AND CONTACT 5-MANIFOLDS 15 ker α via contact structures to a foliation F (cid:48) with two compact leaves which areparallel to each other. The foliation F (cid:48) can be obtained by cutting and turbulizingthe above foliation F along the boundary ∂U of a regular neighborhood U of theReeb component of F . Then ∂U ≈ N × S becomes a compact leaf. We can prove this theorem by combining Theorem 2.2 with the following conver-gence of contact structures to generalized Reeb components.
Theorem 6.4. (Convergence to generalized Reeb components) Let (cid:101) α be the contactform of the unified contactization (Σ + ∪ ( − Σ − )) × R in Definition 4.6. Take adiffeomorphism ϕ and a positive function h which satisfies the conditions describedin Theorem 4.8 except that in this case ∂ Σ = ∅ . Then the mapping torus M n +1 =Σ × R / ( x, z + h ) ∼ ( ϕ ( x ) , z ) possesses the contact form α induced from (cid:101) α . Thenthere exists a family { α t } ≤ t< of contact forms which starts from α and convergesto a defining -form α of a codimension one foliation F . Here F coincides withthe level foliation of h except near the compact leaves Γ × R / Z into which the levelfoliation spiral. Moreover, we may assume that the compact leaves decomposes thefoliation into the union of dead-end components, i.e., generalized Reeb components.Proof. On a neighborhood ( − ε (cid:48) , ε (cid:48) ) × Γ × R / Z ( ⊂ M n +1 ) of the hypersurface Γ × R / Z , the contact form α is expressed as α = e − s β + g ( s ) dz ( s ∈ ( − ε (cid:48) , ε (cid:48) )) , where β is the contact form on Γ and g ( s ) is a decreasing function which coincideswith − s near s = 0 and which is smoothly tangent to ∓ s = ± ε (cid:48) .Put τ = (1 − t ) and take a function e ( s ) supported in ( − ε (cid:48) , ε (cid:48) ) with e (0) > α t = τ α + (1 − τ ) g ( s ) dz + (1 − τ ) e ( s ) ds (0 ≤ t < α which defines the foliation F described in the theorem. (cid:3) Remark.
We can change simultaneously the orientation of the compact leaves bychanging the sign of the value e (0) totally. However in order to obtain the dead-endcomponents we can not change it partially.6.3. Topology of the pages.
In this subsection, we decide the Euler characteristicof the Milnor fiber of each singularity in Theorem 3.2. Note that this is equal tothat of the corresponding singularity in Theorem 3.3 which is well known (see [17]).Thus the followings calculation has become just an alternative approach.The Milnor fiber is diffeomorphic to F = { f m,k ( ξ, η, ζ ) = δ } ∩ {| ξ | + | η | + | ζ | ≤ ε } , where δ ∈ C and 0 < | δ | (cid:28) ε (cid:28)
1. Note that the Euler characteristic χ ( F ) is equalto µ ( f m,k , (0 , , µ () denotes the Milnor number of the function germ.Let π ξ , π η and π ζ denote the projections to the axes.In the case where m = 1, the critical values of π ζ | F are the solutions of thefollowing system: f , ( k ) − δ = 0 , ∂∂ξ f , ( k ) = 2 ξ = 0 , ∂∂η f , ( k ) = 0 , and | ζ | (cid:28) ε. Therefore, for each critical value ζ of π ζ | F , we have the factorization( η − ζ )( η + 2 ζ η + ζ − ζ k ) − δ = ( η − a ) ( η + 2 a ) of the polynomial of η , where the parameter a ∈ C depends on ζ . By comparisonof the coefficients of the η -terms and the η -terms we have − ζ + ζ − ζ k = a − a − a and − ζ + 2 ζ k − δ = 2 a . Eliminating the parameter a , we obtain the equation4 ζ k (9 − ζ k ) = 108 ζ (1 − ζ k ) δ + 27 δ . Then we see that π ζ | F has 12 + k critical points, which indeed satisfy a (cid:54) = − a ,i.e., the map π ζ | F defines a Lefschetz fibration F → D with 12+ k singular fibers.Thus we have χ ( F ) = 1 − k = 11 + k . In the case where m = 2, we have the factorization { ( ζ + η ) − ζ k }{ ( ζ − η ) + ζ k } − δ = ( η − a ) ( η + a − b )( η + a + b ) . By comparison of the coefficients we have ζ (2 + ζ k − ζ k ) = 2 a + b ζ ( ζ k + ζ k ) = ab ζ (1 − ζ k )(1 + ζ k ) − δ = a ( a − b ) . In order to eliminate a, b , we put a = u + v and ζ (2 + ζ k − ζ k ) = 6 uv . Then wehave uv − u + v ) = b ζ ( ζ k + ζ k ) = − u + v ) ζ (1 − ζ k )(1 + ζ k ) − δ = ( u + v ) + 2( u + v )( u + v ) . Following Cardano’s method, we put p = uv, q = u + : v and r = ( u + v ) + 2( u + v )( u + v ) . Then p, q and r are polynomials of ζ . Eliminating a from q (= q ( p, a )) = 3 pa − a and r (= r ( p, a )) = − pa + 3 a , we obtain (27 q − r ) + 54( prq − p q ) + 18 p r − p r = 0 , which is a polynomial equation of ζ . As δ →
0, the left hand side converges to ζ k + k (cid:26) − ζ k − ζ k ζ k + ζ k ) (cid:27) . Therefore π ζ | F has 12 + k + k critical points, which indeed satisfy 4 a (cid:54) = b and b (cid:54) = 0, i.e., the map π ζ | F defines a Lefschetz fibration F → D with 12 + k + k singular fibers. Thus we have χ ( F ) = 1 − k + k = 10 + k + k . EEB FOLIATIONS ON S AND CONTACT 5-MANIFOLDS 17
Open problems.
At present, the following natural questions are open.
Problem 6.5.
1) Is there any convex-tight contact structure of dimension > S n +1 convex-overtwisted ?4) Is there any relation between our half Lutz twist and the full Lutz twist ofEtnyre and Pancholi ?The next problem can be considered as a variation of Calabi’s question. Problem 6.6.
Does the standard S n +1 ( n >
1) contains a convex hypersurfacewith disconnected contact-type boundary?If there is no such hypersurfaces, the following conjecture trivially holds.
Conjecture 6.7 (As an affirmative answer for the above 1)) . S n +1 is convex-tight. Acknowledgement
The author would like to cordially thank John Etnyre, Naohiko Kasuya andKlaus Niederkr¨uger for encouraging the author who had almost gave up publish-ing the results of this article. He would also like to thank Yoshihiko Mitsumatsufor helpful comments especially for giving an idea on Lutz twists along Brieskorn (cid:102) SL ( R )-type contact submanifold which will be discussed in another place. References [1] D. Bennequin:
Entrelacements et ´equations de Pfaff , Ast´erisque (1983), 83–161.[2] C. Caubel, A. Nemethi and P. Popescu-Pampu:
Milnor open books and Milnor fillable contact3-manifolds , Topology (3) (2006) 673-689[3] Y. Eliashberg: Filling by holomorphic discs and its applications , Geometry of low-dimensionalmanifolds 2, London Math. Soc. Lect. Note Ser. (1990), 45–72.[4] Y. Eliashberg:
Contact 3-manifolds twenty years since J. Martinet’s work , Ann. Inst. Fourier, (1991), 165–192.[5] Y. Eliashberg and W. Thurston: Confoliations , A.M.S. University Lecture Series, (1998).[6] J. Etnyre: Contact structures on 3-manifolds are deformations of foliations , Int. Math. Res.Notices, (2007), 775–779.[7] J. Etnyre and D. Pancholi: On generalizing Lutz twists , J. London Math. Soc., (2011),670–688.[8] J. Etnyre and J. Van Horn-Morris: Fibered transverse knots and the Bennequin bound , Int.Math. Res. Notices, (2011), 1483–1509.[9] H. Geiges:
Symplectic manifolds with disconnected boundary of contact type , Int. Math. Res.Notices, (1994), 23–30.[10] E. Ghys:
D´eformation de flots d’Anosov et de groupes fuchsiens , Ann. Inst. Fourier, (1992), 209–247.[11] E. Giroux: Convexit´e en topologie de contact , Comm. Math. Helv. (1991), 637–677.[12] E. Giroux: G´eom´etrie de contact: de la dimension trois vers les dimensions sup´erieures ,Proc. ICM-Beijing, (2002), 405–414.[13] E. Giroux: Contact structures and symplectic fibrations over the circle , Notes of the summerschool “Holomorphic curves and contact topology”, Berder, 2003.[14] K. Honda:
On the classification of tight contact structures II , J. Diff Geom., (2000),83–143.[15] A. Ibort, D. Mart´ınez-Torres and F. Presas: On the construction of contact submanifoldswith prescribed topology , J. Diff. Geom., (2000), 235–283.[16] M. Ishikawa: Compatible contact structures of fibered Seifert links in homology -spheres ,Tohoku Math. J., (2012), 25–59.[17] N. Kasuya: The canonical contact structure on the link of a cusp singularity , Tokyo J. Math., (2014), 1–20. [18] P. Massot, K. Niederkr¨uger and C. Wendl: Weak and strong fillability of higher dimensionalcontact manifolds , Invent. Math. (2013), 287–373.[19] D. McDuff:
Symplectic manifolds with contact type boundaries , Invent. Math. (1991),651–671.[20] Y. Mitsumatsu:
Anosov flows and non-Stein symplectic manifolds , Ann. Inst. Fourier (1995), 1407–1421.[21] Y. Mitsumatsu and A. Mori: On Bennequin’s isotopy lemma, appendix to Y. Mitsumatsu:Convergence of contact structures to foliations, in Foliations 2005 (ed. P. Walczak et al.),World Scientific, 2006, 365–371.[22] A. Mori: A note on Thurston-Winkelnkemper’s construction of contact forms on -manifolds ,Osaka J. Math. (2002), 1–11.[23] A. Mori: A note on Mitsumatsu’s construction of a leafwise symplectic foliation , preprint(2012), arXiv: 1202.0891.[24] K. Niederkr¨uger:
The plastikstufe — a generalization of the overtwisted disk to higher di-mensions , Algebr. Geom. Topol. (2006), 2473–2508.[25] W. Thurston: Norm on the homology of -manifolds , Memoirs of the A. M. S., (1986),99–130.[26] J. Van Horn-Morris: Constructions of open book decompositions , Thesis (2007), Univ. ofTexas at Austin.
Appendix: On the violation of Thurston-Bennequin inequality for acertain non-convex hypersurface
In this short note, we give an example of arbitrary small hypersurface withcontact-type boundary in a Darboux chart of dimension greater than three whichviolates the Thurston-Bennequin inequality. We also confirm its non-convexity.Let ( r, θ, z ) be the cylindrical coordinates of R , and take the functions λ ( r ) = 2 r − µ ( r ) = r ( r − . Then the contact form β = λ ( r ) dz + µ ( r ) dθ defines a contact structure with the overtwisted disk D r ≤ × { } . Let U denote asmall neighborhood of D r ≤ × { } . Since even a 3-ball in a Darboux chart (shortlya Darboux 3-ball) contains immersed overtwisted disks, it is easy to see that aDarboux 5-ball contains an embedded overtwisted disk. Thus we can embed theproduct U × ( D <ε ) n − equipped with the contact form α = β + n − (cid:88) i =1 ( x i dy i − y i dx i )into a Darboux (2 n + 1)-ball, where D <ε = { x i + y i < ε } are small disks on the x i + √− y i -axes. (Note that f n − (cid:88) i =1 ( x i dy i − y i dx i ) = n − (cid:88) i =1 ( f x i d ( f y i ) − f y i d ( f x i ))holds for any function f .) Now we take the hypersurfaces (cid:101) Σ = (cid:40) r + ε − (cid:32) z + n − (cid:88) i =1 ( x i + y i ) (cid:33) = 1 + ε (cid:41) and Σ = { r − z ≤ } ∩ (cid:101) Σ . EEB FOLIATIONS ON S AND CONTACT 5-MANIFOLDS 19
The characteristic foliation F (cid:101) Σ is presented by the vector field X = ε − r ( r − z∂ r + (1 + 2 ε − ε − z ) ∂ θ + (cid:8) ( r − + (2 r − ε − z − ε ) (cid:9) ∂ z + ε − (2 r − z n − (cid:88) i =1 ( x i ∂ x i + y i ∂ y i )+ ε − (2 r − r + 1) n − (cid:88) i =1 ( − y i ∂ x i + x i ∂ y i ) . Indeed the following calculations shows that the vector field X satisfies X ∈ T (cid:101) Σ,( α | T (cid:101) Σ)( X ) = 0, and L X ( α | T (cid:101) Σ) = 2 ε − (2 r − zα | T (cid:101) Σ. (cid:40) rdr + ε − (cid:32) zdz + 2 n − (cid:88) i =1 ( x i dx i + y i dy i ) (cid:33)(cid:41) ( X )= 2 ε − (2 r − z (cid:40) r + ε − (cid:32) z + n − (cid:88) i =1 ( x i + y i ) (cid:33) − − ε (cid:41) ,α = (2 r − dz + r ( r − dθ + n − (cid:88) i =1 ( x i dy i − y i dx i ) ,α ( X ) = (2 r − (cid:8) ( r − + (2 r − ε − z − ε ) (cid:9) + r ( r − { − ε − z − ε ) } + ε − (2 r − r + 1) n − (cid:88) i =1 ( x i + y i )= (2 r − r + 1) (cid:40) r + ε − (cid:32) z + n − (cid:88) i =1 ( x i + y i ) (cid:33) − − ε (cid:41) ,dα = 4 rdr ∧ dz + 2 r (2 r − dr ∧ dθ + 2 n − (cid:88) i =1 dx i ∧ dy i ,ι X dα = ε − r ( r − z (4 rdz + 2 r (2 r − dθ ) − r (2 r − r + 1) dr +2 ε − (2 r − z n − (cid:88) i =1 ( x i dy i − y i dx i ) − ε − (2 r − r + 1) n − (cid:88) i =1 ( x i dx i + y i dy i )= 2 ε − (2 r − zα − (2 r − r + 1) (cid:40) rdr + ε − (cid:32) zdz + 2 n − (cid:88) i =1 ( x i dx i + y i dy i ) (cid:33)(cid:41) . Further we see that the vector field X satisfies( dr − dz )( X ) | ∂ Σ = ( r − { ε − ( − r + r + 1) − ( r + 1) } + ε (2 r − > z = r −
1. Thus ∂ Σ is a contact-type boundary.The singularity of X | Σ is the union of the one point set S + (Σ) = { ((0 , θ, − ε √ ε ) , ∈ U × B n − <ε } and the other one point set S − (Σ) = { ((0 , θ, + ε √ ε ) , ∈ U × B n − <ε } . They are respectively a source point and a sink point. Since the indices of thesepoints are equal to 1, we see that the hypersurface Σ violates the Thurston-Bennequin inequality. Figure A depicts (the four-fold covering of) the well-definedpush-forward X (cid:48) of X under the natural projection p from (cid:101) Σ to the quarter-sphereΣ (cid:48) = (cid:8) ( z, r, | ( x, y ) | ) | r + ε − ( z + | ( x, y ) | ) = 1 + ε (cid:9) ( r ≥ , | ( x, y ) | ≥ . Fifure A.
The covering of the vector field X (cid:48) The vector field X (cid:48) on the quoter 2-sphere defines the singular foliation F (cid:48) = { ε − z = ( Cr − r −
1) + ε } −∞≤ C ≤ + ∞ . The singularity consists of the following five points; two (quarter-)elliptic points (cid:0) ∓ ε √ ε, , (cid:1) , whose preimages under p are the above singular points; other two(half-)elliptic points ( ± ε √ ε, , P ± of X ;and a single hyperbolic point (cid:18) , (cid:113) ε − (cid:112) ε (1 + ε ) , (cid:113) ε (cid:112) ε (1 + ε ) (cid:19) presentingthe double point of the singular level C = 1 + 2 ε + 2 (cid:112) ε (1 + ε ). Slightly changingthe small positive constant ε if necessary, we may assume that the preimage H ofthis hyperbolic point ( ≈ S × S n − ) is the union of periodic orbits of X .Now we assume that (cid:101) Σ is convex. Then it is divided along a contact submanifoldΓ into the positive region (cid:101) Σ + and the negative region (cid:101) Σ − . Moreover X is transverseto Γ from positive region to negative region, and there the sign of the naturaldivergence of X changes from positive to negative. This implies that S i (Σ) , P i ⊂ (cid:101) Σ ( i = + , − ) and Γ ∩ H = ∅ . (Note that if (cid:101) Σ itself is not convex but becomes convex after a small perturbation,the same properties also hold.) Then we can see that the dividing set Γ mustcontain a spherical component. This contradicts to the famous Eliashberg-Floer-McDuff theorem, which says that S n − (cid:96) (other components) can not be realized EEB FOLIATIONS ON S AND CONTACT 5-MANIFOLDS 21 as the contact-type boundary of a connected symplectic manifold. This proves that (cid:101)
Σ is not convex. Similarly we can prove the non-convexity of Σ.
Osaka city univerity advanced mathematical institute, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585 Japan
E-mail address ::