Tight contact structures on some bounded Seifert manifolds with minimal convex boundary
aa r X i v : . [ m a t h . G T ] N ov Tight contact structures on some bounded Seifert manifoldswith minimal convex boundary
Fan Ding, Youlin Li and Qiang ZhangAbstract:
We classify positive tight contact structures, up to isotopy fixing the bound-ary, on the manifolds N = M ( D ; r , r ) with minimal convex boundary of slope s and Girouxtorsion 0 along ∂N , where r , r ∈ (0 , ∩ Q , in the following cases:(1) s ∈ ( −∞ , ∪ [2 , + ∞ );(2) s ∈ [0 ,
1) and r , r ∈ [1 / , s ∈ [1 ,
2) and r , r ∈ (0 , );(4) s = ∞ and r = r = .We also classify positive tight contact structures, up to isotopy fixing the boundary, on M ( D ; , ) with minimal convex boundary of arbitrary slope and Giroux torsion greaterthan 0 along the boundary. Keywords: contact structure; bounded Seifert manifolds If M is an oriented 3-manifold, a contact structure on M is a completely non-integrable 2-plane distribution ξ given as the kernel of a global 1-form α such that α ∧ dα = 0 at everypoint of M . Throughout this paper, we assume the contact structures are positive , i.e., givenby one form α satisfying α ∧ dα >
0, and oriented.Classification of tight contact structures on oriented 3-manifolds is a fundamental prob-lem in contact topology. See [1], [7], [8], [9] and [10]. The classification of tight contactstructures on small Seifert manifolds has been the object of intensive study in the last fewyears. See [14], [3], [4] and [2]. In [9], Honda classified tight contact structures on the solidtorus S × D and the thickened torus T × I . In [13], Tanya classified tight contact structureson Σ × I where the boundary condition is specified by a single, nontrivial separating dividingcurve on each boundary component. In this article, we classify tight contact structures onsome bounded Seifert manifolds.Let N be a small bounded Seifert manifold M ( D ; r , r ), where r i ∈ (0 , ∩ Q , i = 1 , N with minimal convex boundary, i.e., thenumber of dividing curves on ∂N is 2. Suppose s ∈ Q . Denote the greatest integer notgreater than s by [ s ]. Let s − [ s ] = ba , where a > b ≥ g.c.d. ( a, b ) = 1.If − ba is not an integer, then write − ba = a − a −··· am − − am , where a j ’s are integers and1 j ≥ j ≥
1. If − ba is an integer, then set a = − ba + 1, a = 1 and m = 2. Let r = a − a −··· am − − am +1 . Theorem 1.1
The number of tight contact structures on N with minimal convex boundaryof slope s and Giroux torsion along ∂N , up to isotopy fixing the boundary, is the numberof tight contact structures, up to isotopy, on the small Seifert manifold M ( − − [ s ]; r , r , r ) in the following cases:1. s ∈ ( −∞ , ;2. s ∈ [0 , and r , r ∈ [ , ;3. s ∈ [1 , and r , r ∈ (0 , ) ;4. s ∈ [2 , + ∞ ) . The idea of the proof of Theorem 1.1 is roughly as follows. Similar to the argumentsin [14], [3], [4], we get an upper bound for the number of tight contact structures, up toisotopy fixing the boundary, on N with given conditions. This upper bound is the same asthe number of tight contact structures, up to isotopy, on M ( − − [ s ]; r , r , r ). For any tightcontact structure η on M ( − − [ s ]; r , r , r ), we can decompose M ( − − [ s ]; r , r , r ) into N and a solid torus V , and isotope η so that ∂N is minimal convex with dividing curves ofslope s . When measured in the coordinates of ∂V , this slope is −
1. Thus, by the uniquenessof tight contact structures, up to isotopy fixing the boundary, on a solid torus with minimalconvex boundary of slope −
1, we conclude that the number of tight contact structures, upto isotopy, on M ( − − [ s ]; r , r , r ) is less than or equal to the number of tight contactstructures, up to isotopy fixing the boundary, on N with given conditions.Using the fact that a double cover of M ( D ; , ) is the thickened torus T × I and theclassification of tight contact structures on T × I , we have the following classification. Theorem 1.2 (1) We can divide the set of tight contact structures on M ( D ; , ) withminimal convex boundary of slope ∞ and Giroux torsion along the boundary, up to isotopyfixing the boundary, into two subsets. The tight contact structures in one subset are in 1-1correspondence with Z . The other subset contains two elements.(2) For any integer t > and any number s ∈ Q ∪ {∞} , there are exactly tight contactstructures on M ( D ; , ) with minimal convex boundary of slope s and Giroux torsion t along the boundary, up to isotopy fixing the boundary. In Section 2, we give some preliminaries. In Section 3, we prove Theorem 1.1 in cases1 and 2, and in Section 4, we prove Theorem 1.1 in cases 3 and 4. In Section 5, we proveTheorem 1.2. The reader is assumed to be familiar with convex surfaces theory (cf. [6], [5])and bypasses (cf. [9]). 2
Preliminaries
For r , r , r ∈ Q \ Z , the Seifert manifolds M ( D ; r , r ) and M ( r , r , r ) are described asfollows. Let Σ be an oriented pair of pants, and identify each connected component of − ( ∂ Σ × S ) = T ∪ T ∪ T with R / Z , so that ! gives the direction of − ∂ (Σ × { } ) and ! gives the directionof the S factor. For i = 1 , ,
3, let V i = D × S , and identify ∂V i with R / Z so that ! gives the direction of the meridian ∂ ( D × { } ) and ! gives the direction of the S factor. Then M ( D ; r , r ) (respectively, M ( r , r , r )) is obtained from Σ × S by gluing V i to T i , i = 1 , i = 1 , , ϕ i : ∂V i → T i defined by the matrix ϕ i = p i u i − q i − v i ! , where q i p i = r i , u i q i − p i v i = 1, and 0 < u i < p i .Note that if r = n + a − a −··· am − − am +1 , where m ≥ n and a j ’s are integers, a j ≥ ≤ j < m and a m ≥
1, then ϕ = p u − q − v ! = − n ! a − ! · · · a m − − ! a m + 1 1 − ! and a − ! · · · a m − − ! a m + 1 1 − ! − ! = − aa − b ! , where a > b ≥ g.c.d. ( a, b ) = 1 and ba = 1 − a − a −··· am − − am . Thus wehave Proposition 2.1
In the notations above, ϕ − ! = − ana + a − b ! . (cid:3) Let n , n be integers. The Seifert manifolds M ( D ; r , r ) and M ( D ; r + n , r + n ) areorientation-preserving diffeomorphic. This can be seen as follows. Let f : Σ × S → Σ × S bean orientation-preserving diffeomorphism such that f sends each T i to itself and on each T i , f is given by the matrix f i = − n i ! , where n = − n − n . ( f can be constructed by usinga smooth function g : Σ → SO (2) such that for x ∈ Σ, z ∈ S , f ( x, z ) = ( x, g ( x ) z ).) f can beextended to an orientation-preseving diffeomorphism, still denoted by f , from M ( D ; r , r )to M ( D ; r + n , r + n ). Since f = n + n ! , we have3 roposition 2.2 Under f , a simple closed curve of slope s in T of M ( D ; r , r ) changesto a simple closed curve of slope s + n + n in T of M ( D ; r + n , r + n ) . (cid:3) Similarly, the Seifert manifolds M ( r , r , r ) and M ( r + n , r + n , r − n − n ) areorientation-preserving diffeomorphic. They are also denoted by M ( e ; r − [ r ] , r − [ r ] , r − [ r ]), where e = [ r ] + [ r ] + [ r ].On T × [0 , ∼ = R / Z × [0 ,
1] with coordinates (( x, y ) , t ), consider ξ n = ker(sin( πnt ) dx +cos( πnt ) dy ), with the boundary adjusted so it becomes convex with two dividing curves oneach component, where n ∈ Z + . Let ( M, ξ ) be a contact 3-manifold and T ⊂ M an embeddedtorus. The Giroux torsion along T is the supremum, over n ∈ Z + , for which there exists acontact embedding φ : ( T × [0 , , ξ n ) ֒ → ( M, ξ ), where φ ( T × { t } ) is isotopic to T . (Weset the Giroux torsion to be 0 if there is no such embedding). One can consult [11] for thisdefinition.The main invariant in the classification of tight contact structures on Seifert manifoldsis the maximal twisting number. One can consult [2] for the definition.For the rest of the paper, r , r ∈ (0 , ∩ Q , s ∈ Q , a, b, a j ( j = 1 , . . . , m ) and r aredefined as in the Introduction. For i = 1 ,
2, suppose − r i = a i − a i − ai −··· aili − − aili , where a ij ’s are integers and a ij ≤ − j ≥
0. When we consider the number of tight contactstructures up to isotopy or up to isotopy fixing the boundary, we usually omit the phrase “upto isotopy” or “up to isotopy fixing the boundary”.
For i = 1 ,
2, let ϕ i = p i u i − q i − v i ! , where q i p i = r i , u i q i − p i v i = 1 and 0 < u i < p i .Let ξ be a tight contact structure on N = M ( D ; r , r ) with minimal convex boundaryof slope s ( T ) and Giroux torsion 0 along ∂N . We first isotope ξ to make each V i ( i = 1 ,
2) astandard neighborhood of a Legendrian circle isotopic to the i th singular fiber with twistingnumber t i <
0, i.e., ∂V i is convex with two dividing curves each of which has slope t i whenmeasured in the coordinates of ∂V i . Then, when measured in the coordinates of T i , the slope s i = − q i p i + p i ( t i p i + u i ) < − q i p i . The proof of the following lemma is similar to the proof of [14, Lemma 2.2].
Lemma 3.1 On M ( D ; r , r ) , if s ( T ) ≤ max { r , r } , or if < s ( T ) < and r i ≥ ( i = 1 , , then any tight contact structure with minimal convex boundary of slope s ( T ) admits a vertical Legendrian circle L with twisting number . (cid:3) Now suppose s ( T ) <
0. Using the vertical Legendrian circle L , we can thicken V i ( i = 1 ,
2) to V ′ i such that V ′ i ’s are pairwise disjoint, and T ′ i = ∂V ′ i is a minimal convex torus4ith vertical dividing curves when measured in coordinates of T i . Also, we can thicken T to L = T × [0 ,
1] such that T × { } = T and T × { } = T ′ is a minimal convex torus withvertical dividing curves when measured in the coordinates of T . Choose t i ≪ − −∞ < a i +1 < s i for i = 1 ,
2. By [9, Proposition 4.16], for i = 1 ,
2, there exists a minimalconvex torus T ′′ i in the interior of V ′ i \ V i isotopic to T i that has dividing curves of slope a i +1 .Let V ′′ i be the solid torus bounded by T ′′ i , and Σ ′′ × S = N \ ( V ′′ ∪ V ′′ ).First we consider V ′′ and V ′′ . Since ϕ − i a i + 11 ! = − ( a i + 1) v i − u i ( a i + 1) q i + p i ! (here u i , v i correspond respectively to − u i , − v i in the proof of [14, Theorem 1.6]), the dividingcurves of T ′′ i ( i = 1 ,
2) have slope − ( a i +1) q i + p i ( a i +1) v i + u i when measured in the coordinates of ∂V i . Bya similar argument as in the proof of [14, Theorem 1.6], there are exactly Q l i j =1 | a ij + 1 | tightcontact structures on V ′′ i that satisfy the given boundary condition.Then we consider N \ ( V ′′ ∪ V ′′ ) = Σ ′′ × S . Let L i ( i = 1 ,
2) be the thickened toruswhich is bounded by T ′ i and T ′′ i , then L i has boundary slopes ∞ and a i +1 . By [9, Theorem2.2], there are exactly | a i | minimally twisting tight contact structures on L i that satisfy thegiven boundary condition. The two boundary slopes of the thickened torus L are ∞ and s ( T ) respectively. Case 1. s ∈ ( −∞ , Case 1(a). s ∈ ( −∞ , − . Let s ( T ) = s . We decompose L into m continued fraction blocks (some blocks may beinvariant neighborhoods of convex tori). The first continued fraction block has two boundaryslopes ∞ and [ s ] + 1 − a − , the second continued fraction block has two boundary slopes[ s ] + 1 − a − and [ s ] + 1 − a − a − , . . . , the m th continued fraction block has two boundaryslopes [ s ] + 1 − a − a −··· am − − and s ( T ) = s = [ s ] + 1 − a − a −··· am − − am . By shuffling,there are at most a ( a − . . . ( a m − − a m minimally twisting tight contact structures on L . By a similar argument as in the first paragraph of [14, page 241], the upper bound ofthe number of tight contact structures on N with minimal convex boundary of slope s andGiroux torsion 0 along ∂N is | a a Q i =1 Q l i j =1 ( a ij + 1) | a ( a − . . . ( a m − − a m .Consider the closed Seifert manifold M ( r , r , − − [ s ] + r ). Since − − [ s ] >
0, by [14,Theorem 1.6], it admits | a a Q i =1 Q l i j =1 ( a ij + 1) | a ( a − . . . ( a m − − a m tight contactstructures. By [14, Theorem 1.3], for any tight contact structure η on M ( r , r , − − [ s ] + r ),there is a vertical Legendrian circle with twisting number 0. We isotope η so that there is avertical Legendrian circle L with twist number 0 in the interior of Σ × S , and V is a standardneighborhood of a Legendrian circle isotopic to the 3rd singular fiber with twisting number t <
0, i.e., ∂V is convex with two dividing curves each of which has slope t when measured inthe coordinates of ∂V . Let ϕ = p u − q − v ! , where q p = − − [ s ]+ r , u q − p v = 1 and5 < u < p . Then, when measured in the coordinates of T , the slope s = − q p + p ( t p + u ) .Using L , we can thicken V to V ′ , such that T ′ = ∂V ′ is a minimal convex torus with verticaldividing curves when measured in the coordinates of T . Since 1 − ba = a − a −··· am − − am > a − a −··· am − − am +1 = r , we have −∞ < s = [ s ] + ba < [ s ] + 1 − r . Thus −∞ < s < s forsufficiently small t . By [9, Proposition 4.16], there exists a minimal convex torus T ′′ in theinterior of V ′ \ V isotopic to T that has dividing curves of slope s . Thus we can isotopy η sothat T is minimal convex with dividing curves of slope s when measured in the coordinatesof T . Note that M ( r , r , − − [ s ] + r ) has a decomposition N ∪ ϕ V . By Proposition 2.1, ϕ − − a − [ s ] a − b ! = − ! . Thus the slope of the dividing curves on ∂V is − ∂V . There is exactly one tight contact structure on V withminimal convex boundary of slope −
1. Note that η , when restricted to N , has Giroux torsion0 along ∂N . Hence the number of tight contact structures on N with given conditions is atleast | a a Q i =1 Q l i j =1 ( a ij + 1) | a ( a − . . . ( a m − − a m .Therefore, there are exactly | a a Q i =1 Q l i j =1 ( a ij + 1) | a ( a − . . . ( a m − − a m tightcontact structures on N with minimal convex boundary of slope s and Giroux torsion 0 along ∂N . Case 1(b). s ∈ [ − , s ( T ) = s . Note that the outermost continued fraction block of L has two boundaryslopes ∞ and − a − , and hence contains a − a a a − ( a + 1)( a + 1)( a − a − . . . ( a m − − a m Q i =1 Q l i j =1 | a ij + 1 | tight contact structures on N with minimal convexboundary of slope s and Giroux torsion 0 along ∂N .Consider the small Seifert manifold M ( r , r , r ). By [3, Theorem 1.1], it admits exactly[ a a a − ( a + 1)( a + 1)( a − a − . . . ( a m − − a m Q i =1 Q l i j =1 | a ij + 1 | tight contactstructures. Let ϕ = p u − q − v ! , where q p = r , u q − p v = 1 and 0 < u < p . Notethat M ( r , r , r ) has a decomposition N ∪ ϕ V . By a similar argument as in Case 1(a), forany tight contact structure η on M ( r , r , r ), we can isotopy η so that T is minimal convexwith dividing curves of slope s when measured in the coordinates of T . By Proposition2.1, ϕ − − aa − b ! = − ! . Thus the slope of the dividing curves on ∂V is − ∂V . Similar to Case 1(a), we conclude that the number oftight contact structures on N with given conditions is at least [ a a a − ( a + 1)( a + 1)( a − a − . . . ( a m − − a m Q i =1 Q l i j =1 | a ij + 1 | .Therefore, there are exactly [ a a a − ( a + 1)( a + 1)( a − a − . . . ( a m − − a m Q i =1 Q l i j =1 | a ij + 1 | tight contact structures on N with the given boundary condition andGiroux torsion 0 along ∂N . Case 2. s ∈ [0 ,
1) and r , r ∈ [ , M ( D ; r , r ) with minimal convex bound-ary of slope s contains a Legendrian vertical circle with twisting number 0.By Proposition 2.2, a tight contact structure on M ( D ; r , r ) with minimal convexboundary of slope s corresponds to a tight contact structure on M ( D ; − r , r ) withminimal convex boundary of slope s −
1. We consider tight contact structures on M ( D ; − r , r ) with minimal convex boundary of slope s −
1. Without loss of generality, assume that r ≥ r .Suppose ξ is a tight contact structure on N = M ( D ; − r , r ) with minimal convexboundary of slope s − ∂N . Using a vertical Legendrian circlewith twisting number 0, we can thicken standard neighborhoods of two Legendrian singularfibers to U and U such that the slopes of the dividing curves on ∂U and ∂U are ∞ whenmeasured in the coordinates of T and T , respectively. We can thicken T to a thickenedtorus L so that the slope of the other boundary component of L is ∞ .Consider the closed Seifert manifold M ( − r , r , r ). Let ϕ = p u − q − v ! , where q p = r , u q − p v = 1 and 0 < u < p . M ( r , r , r ) has a decomposition N ∪ ϕ V . ByProposition 2.1, ϕ − − aa − b ! = − ! . Thus we can find layers N ij of M ( − r , r , r )in [4, page 1432] in ( N , ξ ). Note that L corresponds to U \ V ′ in [4, page 1432]. We canobtain similar results as in [4, Propositions 6.1 and 6.3] for ( N , ξ ). Since r is not an integer, L contains at least two layers and we can obtain a similar result as in [4, Proposition 6.4] for( N , ξ ) (we only encounter Case 2 in the proof of [4, Proposition 6.4]). Therefore we obtainan upper bound of the number of tight contact structures on N with given conditions, whichis the same as the number of tight contact structures on M ( − r , r , r ).By [4, Proposition 5.1], for any tight contact structure η on M ( − r , r , r ), there is aLegendrian vertical circle with twisting number 0. Then similar to Case 1(a), we can isotopy η so that T = ∂V is minimal convex with dividing curves of slope − ∂V . Then we conclude that the number of tight contact structures on M ( − r , r , r ) is less than or equal to the number of tight contact structures on N withminimal convex boundary of slope s − ∂N .Therefore, the number of tight contact structures on N with given conditions is exactlythe number of tight contact structures on M ( − r , r , r ). By Proposition 2.2, a tight contact structure on M ( D ; r , r ) with minimal convex boundaryof slope s corresponds to a tight contact structure on M ( D ; r − , r −
1) with minimalconvex boundary of slope s − M = M ( D ; r − , r −
1) and let s ( T ) = s −
2. For i = 1 , i − − − a i − ai −··· aili − − aili . Suppose r i − − q i p i , where p i , q i are integers, 0 < q i < p i and g.c.d. ( p i , q i ) = 1. Let ϕ i = p i u i q i v i ! , where p i > u i > p i v i − q i u i = 1.Let ξ be a tight contact structure on M with minimal convex boundary of slope s ( T )and Giroux torsion 0 along ∂M . The proof of the following lemma is similar to the proof of[14, Theorem 1.4]. Lemma 4.1 On M = M ( D ; − q p , − q p ) , if s ( T ) ≥ − , then any tight contact structurewith minimal convex boundary of slope s ( T ) and Giroux torsion along ∂M does not admitLegendrian vertical circles with twisting number . (cid:3) The proof of the following lemma is similar to the proof of the corresponding resultcontained in the proof of [14, Theorem 1.7].
Lemma 4.2 On M = M ( D ; − q p , − q p ) , if s ( T ) ≥ , or if − ≤ s ( T ) < and q i p i > ( i = 1 , , then the maximal twisting number of Legendrian vertical circles in ( M, ξ ) is − . Now assume that s ( T ) ≥
0, or − ≤ s ( T ) < q i p i > ( i = 1 , ξ , we can find a Legendrian vertical circle L in the interior of Σ × S with twisting number −
1. Then we make each V i ( i = 1 ,
2) a standard neighborhood of aLegendrian circle which is isotopic to the i th singular fiber with twisting number t i < − ∂V i is convex with two dividing curves each of which has slope t i when measured in thecoordinates of ∂V i . Let s i be the slope of the dividing curves of T i = ϕ i ( ∂V i ) measured in thecoordinates of T i . Then we have s i = q i t i + v i p i t i + u i = q i p i + p i ( p i t i + u i ) . The fact that t i < − < s i < q i p i . In particular, if q i p i > ( i = 1 , < s i < q i p i .We can assume that T i = ϕ i ( ∂V i ) ( i = 1 ,
2) and T have Legendrian rulings of slope ∞ when measured in the coordinates of T i and T , respectively. Using L , we can thicken V i to V ′ i ( i = 1 ,
2) and T to L to get a decomposition, M = (Σ ′ × S ) ∪ ( V ′ ∪ V ′ ∪ L ),such that T ′ i = ∂V ′ i ( i = 1 ,
2) has two dividing curves of slope [ s i ] = 0 when measured in thecoordinates of T i , and the thickened torus L has two boundary slopes [ s ( T )] and s ( T ) (cf.the proof of [14, Theorem 1.7]).By [10, Lemma 5.1], there are exactly 2 + [ s ( T )] tight contact structures on Σ ′ × S satisfying the boundary condition and admitting no Legendrian vertical circles with twistingnumber 0.The slope of the dividing curves on ∂V ′ i is − q i v i when measured in the coordinates of ∂V i .So by a similar argument as in the proof of [14, Theorem 1.7], there are exactly Q l i j =0 | a ij + 1 | tight contact structures on V ′ i satisfying such boundary condition.We consider Case 4 first. Case 4. s ∈ [2 , + ∞ ). 8e decompose L into m continued fraction blocks (some blocks may be invariant neigh-borhoods of convex tori). The first one has boundary slopes [ s ] − s ] − − a − , thesecond one has boundary slopes [ s ] − − a − and [ s ] − − a − a − , . . . , the last one has bound-ary slopes [ s ] − − a − a −··· am − − and [ s ] − − a − a −··· am − − am = s ( T ). By shuffling, thereare at most ( a − a − . . . ( a m − − a m minimally twisting tight contact structures on L (except s = [ s ]). So there are at most [ s ] Q i =1 Q l i j =0 | a ij + 1 | ( a − a − . . . ( a m − − a m tight contact structures on M with minimal convex boundary of slope s − ∂M .Consider the small closed Seifert manifold M ( r − , r − , − [ s ]+1+ r ) = M ( − q p , − q p , − [ s ]+1+ r ). Since [ s ] ≥
2, applying [14, Theorem 1.7], there are exactly [ s ] Q i =1 Q l i j =0 | a ij + 1 | ( a − a − . . . ( a m − − a m tight contact structures on M ( − q p , − q p , − [ s ]+1+ r ). According tothe proof of [14, Theorem 1.7], for any tight contact structure η on M ( − q p , − q p , − [ s ]+1+ r ),the maximal twisting number of a Legendrian vertical circle is −
1. After an isotopy of η , wecan find a vertical Legendrian circle L ′ with twist number − × S andmake V a standard neighborhood of a Legendrian circle isotopic to the 3rd singular fiber withtwisting number t <
0, i.e., ∂V is convex with two dividing curves each of which has slope t when measured in the coordinates of ∂V . Let ϕ = p u q v ! , where q p = [ s ] − − r , p v − u q = 1 and 0 < u < p . Then, when measured in the coordinates of T , the slope s = q p + p ( t p + u ) .Using L ′ , we can thicken V to V ′ such that T ′ = ∂V ′ has two dividing curves ofslope [ s ] −
2. Since [ s ] − ≤ s − < [ s ] − − r , [ s ] − ≤ s − < s for sufficientlysmall t . By [9, Proposition 4.16], there is a convex torus T ′′ in the interior of V ′ \ V which is parallel to T and has two dividing curves of slope s ( T ) = s −
2. Thus we canisotopy η so that T is minimal convex with dividing curves of slope s ( T ) when measuredin the coordinates of T . M ( r − , r − , − [ s ] + 1 + r ) has a decomposition M ∪ ϕ V .By Proposition 2.1, ϕ − − a − ([ s ] − a − b ! = − ! . Thus the slope of the dividingcurves on ∂V is − ∂V . Similar to the argument inCase 1(a), the number of tight contact structures on M with given conditions is at least[ s ] Q i =1 Q l i j =0 | a ij + 1 | ( a − a − . . . ( a m − − a m .Therefore, there are exactly [ s ] Q i =1 Q l i j =0 | a ij + 1 | ( a − a − . . . ( a m − − a m tightcontact structures on M with minimal convex boundary of slope s − ∂M . Case 3. s ∈ [1 ,
2) and r , r ∈ (0 , ).Since r i ∈ (0 , ) ( i = 1 , q i p i = 1 − r i > ( i = 1 , Q i =1 Q l i j =0 | a ij + 1 | ( a − a − . . . ( a m − − a m tight contact structures on M withgiven conditions.Consider the small closed Seifert manifold M ( r − , r − , r ) = M ( − q p , − q p , r ). Weclaim that this small closed Seifert manifold is an L -space (see [12] for the definition). Note9hat since r + r + r − = 0, M ( r − , r − , r ) is a rational homology sphere. By [12,Theorem 1.1], it suffices to show that − M ( − q p , − q p , r ) = M ( − q p , q p , − r ) carries nopositive, transverse contact structures. Suppose otherwise, by [2, Theorem 4.5], there areintegers h , h , h and k >
0, such that (1) h k < − q p , h k < − q p , h k < r −
1, and (2) h + h + h k = − − k . Let n be a positive integer such that 1 − r ≥ n . Combining (1) and(2), we have − − k < − q p − q p + r − < − − n . So 1 ≤ k ≤ n −
1. If k is even, then h ≤ − k − h ≤ − k − h ≤ −
1. Thus we have − − k = h + h + h k ≤ − k − k = − − k .This is absurd. If k is odd, then h ≤ [ − k ] = − k − , h ≤ [ − k ] = − k − and h ≤ −
1. Thuswe have − − k = h + h + h k ≤ − k − ) − k = − − k . This is also absurd.By [2, Theorem 1.3], there are exactly Q i =1 Q l i j =0 | a ij + 1 | ( a − a − . . . ( a m − − a m tight contact structures on M ( r − , r − , r ). Moreover, in each of these tight contactstructures, there is a Legendrian vertical circle with twisting number −
1. By a similarargument as in Case 4, there are at least Q i =1 Q l i j =0 | a ij + 1 | ( a − a − . . . ( a m − − a m tight contact structures on M with given conditions. So, in this case, there are exactly Q i =1 Q l i j =0 | a ij +1 | ( a − a − . . . ( a m − − a m tight contact structures on M with minimalconvex boundary of slope s − ∂M . Consider the thickened torus S × S × I , where I = [0 , x, y, z ) the coordinatesof S × S × I , where x ∈ R / π Z , y ∈ R / π Z and z ∈ [0 , x ∈ [ − π, π ], y ∈ [ − π, π ], and − π is identified with π . Let N be the quotient space of S × S × I byidentifying ( x, y, z ) with ( x + π, − y, − z ). Let p : S × S × I −→ N be the covering projection.The covering transformation ( x, y, z ) ( x + π, − y, − z ) is denoted by τ . N can be identifiedwith M ( D ; − , ) (the images under p of S × { } × { } and S × { π } × { } correspond tothe singular fibers). On the boundary T = − ∂N = S × S × { } , S × { pt } × { } gives thefiber direction (i.e., corresponds to ! in the notation of Section 2) and { pt } × S × { } corresponds to − ! in the notation of Section 2. By Proposition 2.2, a simple closedcurve of slope s in T of N = M ( D ; − , ) corresponds to a simple closed curve of slope s + 1 in T of M ( D ; , ).We regard N as the quotient space of a thickened cylinder [0 , π ] × S × I by identifying(0 , y, z ) with ( π, − y, − z ). See Figure 1. The coordinates of the four points P , Q , R and S at the left end are (0 , − π , , − π , , π ,
1) and (0 , π ,
0) respectively. The coordinatesof the two points X and Y at the left end are (0 , , ) and (0 , π, ) respectively.10igure 1.Let ξ be a tight contact structure on N with minimal convex boundary of slope s ( T ) = ∞ . This means that the dividing curves of T consist of two simple closed curves parallel to S × { pt } × { } . Note that the image of { pt } × S × I under p is an essential annulus in N andthe metric closure of its complement in N is a solid torus. Assume that T is a convex torusin standard form with dividing curves S × { } × { } and S × { π } × { } , and { pt } × S × { } , pt ∈ S , are the Legendrian rulings. See Figure 1. The upper bold line and the upper dashedline form a dividing curve, and the lower bold line and the lower dashed line form the otherdividing curve. The plus sign “ + ” in Figure 1 denotes the region p ( S × [0 , π ] × { } ) in T bounded by the two dividing curves.Let A denote the annulus which is the image of { }× S × I under p . After perturbation, A is convex with Legendrian boundary. Also assume that ♯ Γ A , the number of connectedcomponents of the dividing set Γ A of A , is minimal among all convex annuli in its isotopyclass relative to the boundary. Γ A contains exactly two properly embedded arcs. Withoutloss of generality, we assume that the endpoints of these two dividing arcs are P , Q , R and S . Case 5.1.
Both of the two dividing arcs in Γ A connect the two different components of ∂A . The two dividing arcs in Γ A must connect the points P, Q and
R, S respectively. Asshown in Figures 2 and 3, when we cut N along the convex annulus A and round the edges,we obtain a solid torus with two dividing curves on the boundary. Moreover, each of thesetwo dividing curves intersects a meridian of this solid torus exactly once. There exists aunique tight contact structure on this solid torus by [9, Proposition 4.3]. This implies thatin this case, for every choice of Γ A , there exists at most one tight contact structure.11igure 2.Figure 3.Similar to the proof of [9, Proposition 4.9], we define the holonomy k ( A ) by passingto the cover { } × R × I ⊂ S × R × I and letting k ( A ) be the integer such that there is adividing curve which connects from (0 , π ,
0) to (0 , k ( A ) π + π , − α = cos ydx + sin ydz on S × S × I . Then ξ = ker α is the I -invariant neighbor-hood of a convex S × S with dividing curves S ×{ } and S ×{ π } . If we take ξ and isotope S × S × { } via ( x, y ) ( x, y − kπ ) and isotope S × S × { } via ( x, y ) ( x, y + kπ ),while fixing S × S × { } , namely, we take a self-diffeomorphism of S × S × I by sending( x, y, z ) to ( x, y + 2 kπ ( z − ) , z ), then we obtain a tight contact structure ξ k on S × S × I with holonomy k (in the sense of [9, Proposition 4.9]), and the corresponding contact form α k = cos( y + 2 kπ ( − z )) dx + sin( y + 2 kπ ( − z )) dz .Since τ ∗ ( α k ) = α k , each nonrotative tight contact structure ξ k on S × S × I is τ -invariant. So ξ k induces a tight contact structure on N with holonomy k ( A ) = k . By [9,Proposition 4.9], the nonrotative tight contact structures ξ k , k ∈ Z , on S × S × I arenon-isotopic, so they induce non-isotopic tight contact structures on N . All these tightcontact structures on N have Giroux torsion 0 along ∂N since each ξ k has Giroux torsion 0along the boundary. These tight contact structures on N form the subset in Theorem 1.2(1)whose elements are in 1-1 correspondence with Z . Note also that for all these tight contactstructures, a convex torus parallel to ∂N must have slope ∞ since each ξ k is nonrotative. Case 5.2.
The two endpoints of each dividing arc in Γ A belong to the same componentof ∂A .If Γ A contains an odd number of closed dividing curves, see Figure 4, then, when we cut N along A and perform edge-rounding, we find two dividing curves which bound disks. Thiscontradicts Giroux’s criterion (see [9, Theorem 3.5]). So Γ A must contain an even number ofclosed dividing curves. Figure 4.Suppose Γ A contains 2 t closed dividing curves, where t ≥
0. As shown in Figure 5, whenwe cut N along the convex annulus A and round the edges, we obtain a solid torus S × D with 4 t + 2 vertical dividing curves. 13igure 5.Next cut S × D along a meridional disk D after modifying the boundary to be standardwith horizontal rulings. Since ♯ Γ A is minimal, by a similar argument as in the proof of [9,Lemma 5.2], the dividing set of the convex meridional disk D has a unique configuration asfollows. Let γ and γ be the two dividing curves on ∂ ( S × D ) which intersect ∂N . Thenall γ ∈ Γ D must separate D ∩ γ from D ∩ γ (hence the dividing curves of D are parallelsegments, with only two boundary-parallel components, each containing one D ∩ γ i in theinterior); otherwise there would exist a bypass which allows for a reduction in the number ofdividing curves on A .Therefore, the tight contact structure ξ on N depends only on Γ A , which in turn isdetermined by the sign of the boundary-parallel components of A along ∂N , together with t + 2 = ♯ Γ A . So in this case, for each t ≥
0, there exist at most two tight contact structureson N .For each t ∈ { } ∪ Z + , let η t be the contact structure on S × S × I given by 1-form β t = sin((2 t + 1) πz ) dx + cos((2 t + 1) πz ) dy , with the boundary adjusted so it becomes convexwith two dividing curves on each component. Let η ′ t denote the contact structure given by − β t . By [9, Lemma 5.3], any two of these tight contact structures on S × S × I are distinct.For each t ∈ { } ∪ Z + , since τ ∗ ( β t ) = β t , both η t and η ′ t are τ -invariant, and hence inducecontact structures ζ t and ζ ′ t on N respectively. Since these two induced contact structureson N lift to distinct tight contact structures on S × S × I , they are tight and distinct.Moreover, both ζ t and ζ ′ t have minimal convex boundary of slope ∞ and Giroux torsion t along ∂N by the explicit formula of β t and [11, Proposition 3.4].Similar to [9, Lemma 5.2] and [11, Proposition 3.2], if Γ A contains 2 t closed curves,then ξ is ζ t or ζ ′ t . ζ and ζ ′ form the subset in Theorem 1.2(1) which contains two elements.14his completes the proof of Theorem 1.2(1). For t ≥
1, there are exactly two tight contactstructures, namely, ζ t and ζ ′ t , on N with minimal convex boundary of slope ∞ and Girouxtorsion t along ∂N . This proves Theorem 1.2(2) when s = ∞ .Now let ξ be a tight contact structure on N with minimal convex boundary of slope s ( T ) = s ∈ Q and Giroux torsion t ≥ ∂N . There is a minimal convex torus T ′ inthe interior of N which is parallel to T and has slope s , such that the restriction of ξ on thethickened torus U ′ bounded by T ′ and T has Giroux torsion t . According to [9, Lemma 5.2],( U ′ , ξ | U ′ ) is universally tight.There is a minimal convex torus T in the interior of U ′ which is parallel to T and hasslope ∞ . We assume that the restriction of ξ on the thickened torus U bounded by T and T is minimally twisting. Note that U ⊂ U ′ .The contact submanifold ( N \ U, ξ | N \ U ) belongs to Case 5.2. If the contact submanifold( N \ U, ξ | N \ U ) belongs to Case 5.1, then each convex torus in N \ U which is parallel to T hasslope ∞ , contradicting the fact that T ′ has slope s . Note that for the contact structure ζ on N , the slope of a convex torus parallel to T is greater than or equal to 0. Thus if s ≥ N \ U, ξ | N \ U ) is t −
1, and if s <
0, then the Giroux torsion of( N \ U, ξ | N \ U ) is t . So there are at most two possibilities of ( N \ U, ξ | N \ U ).Since U ⊂ U ′ and ( U ′ , ξ | U ′ ) is universally tight, ( U, ξ | U ) is universally tight. By [9,Proposition 5.1], there are at most two possibilities of ( U, ξ | U ). Moreover, these two possibil-ities are distinguished by their relative Euler classes. If ( U, ξ | U ) is given, then at most onepossibility of ( N \ U, ξ | N \ U ) can make ( N, ξ ) tight by [9, Lemma 5.2]. Hence there are atmost two tight contact structures on N with the given conditions.For a given t ′ ∈ Z + ∪{ } , let 0 < w < t ′ +1 satisfy that − s = cot((2 t ′ +1) πw ). Let η t ′ (weuse the same notation as in Case 5.2) be the tight contact structure on S × S × [ − w, w ]given by 1-form β t ′ = sin((2 t ′ + 1) πz ) dx + cos((2 t ′ + 1) πz ) dy , with the boundary adjusted soit becomes convex with two dividing curves on each component. η ′ t ′ is given by the 1-form − β t ′ . Think of N as the quotient space of S × S × [ − w, w ] by identifying ( x, y, z ) with( x + π, − y, − z ). The transformation ( x, y, z ) ( x + π, − y, − z ) on S × S × [ − w, w ]is still denoted by τ . Since β s is τ -invariant, η t ′ and η ′ t ′ induce tight contact structures ζ t ′ and ζ ′ t ′ on N with minimal convex boundary of slope s ( T ) = s . ζ t ′ and ζ ′ t ′ are distinct since η t ′ and η ′ t ′ are distinct.Note that the restriction of the contact structure η t ′ on S × S × [ − w,
0] is minimallytwisting. We conclude that if s <
0, then the Giroux torsion along ∂N of ζ t ′ and ζ ′ t ′ is t ′ andif s ≥
0, then the Giroux torsion along ∂N of ζ t ′ and ζ ′ t ′ is t ′ + 1.Therefore for each t ∈ Z + , there are exactly two tight contact structures on N withminimal convex boundary of slope s and Giroux torsion t along ∂N . This finishes the proofof Theorem 1.2(2). Acknowledgements.
The first author is partially supported by grant no. 10631060 ofthe National Natural Science Foundation of China. The second author is partially supportedby grant no. 11001171 of the National Natural Science Foundation of China.15 eferences [1] Y. Eliashberg, Contact 3-manifolds twenty years since J. Martinet’s work, Ann. Inst.Fourier 42 (1992), 165–192.[2] P. Ghiggini, On tight contact structures with negative maximal twisting number on smallSeifert manifolds, Algebr. Geom. Topol. 8 (2008), no. 1, 381–396.[3] P. Ghiggini, P. Lisca, A. Stipsicz, Classification of tight contact structures on smallSeifert 3-manifolds with e ≥
0, Proc. Amer. Math. Soc. 134 (2006), no. 3, 909–916(electronic).[4] P. Ghiggini, P. Lisca, A. Stipsicz, Tight contact structures on some small Seifert fibered3-manifolds, Amer. J. Math. 129 (2007), no. 5, 1403–1447.[5] H. Geiges, An introduction to contact topology, Cambridge Studies in Advanced Math-ematics, vol. 109, Cambridge University Press, Cambridge, 2008.[6] E. Giroux, Convexit´e en topologie de contact, Comment. Math. Helv. 66 (1991), 637–677.[7] E. Giroux, Structures de contact en dimension trois et bifurcations des feuilletages desurfaces, Invent. Math. 141 (2000), 615–689.[8] E. Giroux, Structures de contact sur les vari´et´es fibr´ees en cercles audessus d’une surface,Comment. Math. Helv. 76 (2001), 218–262.[9] K. Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000),309–368 (electronic).[10] K. Honda, On the classification of tight contact structures II, J. Differential Geom. 55(2000), no. 1, 83–143.[11] K. Honda, W. Kazez, G. Matic, Convex decomposition theory, Int. Math. Res. Not.2002, no. 2, 55–88.[12] P. Lisca, A. Stipsicz, Ozsv´ath-Szab´o invariants and tight contact 3-manifolds, III, J.Symplectic Geom. 5 (2007), no. 4, 357–384.[13] C. Tanya, A class of tight contact structures on Σ × I , Algebr. Geom. Topol. 4 (2004),961–1011 (electronic).[14] H. Wu, Legendrian vertical circles in small Seifert spaces, Commun. Contemp. Math. 8(2006), no. 2, 219–246.School of Mathematical Sciences, Peking University, Beijing 100871, China E-Mail address : [email protected] of Mathematics, Shanghai Jiaotong University, Shanghai 200240, China16 -mail address : [email protected] of Science, Xi’an Jiaotong University, Xi’an 710049, China