Local geometry of symplectic divisors with applications to contact torus bundles
LLocal geometry of symplectic divisors with applications tocontact torus bundles
Tian-Jun Li, Jie MinJanuary 18, 2021
Abstract
In this note we study the contact geometry of symplectic divisors. We show the contactstructure induced on the boundary of a divisor neighborhood is invariant under toric andinterior blow-ups and blow-downs. We also construct an open book decomposition onthe boundary of a concave divisor neighborhood and apply it to the study of universallytight contact structures of contact torus bundles.
Contents
A.1 Contact structure and toric equivalence . . . . . . . . . . . . . . . . . . . . . 15A.2 Half edge and blow-up of a vertex . . . . . . . . . . . . . . . . . . . . . . . . . 19 A topological divisor D refers to a connected configuration of finitely many closed embed-ded oriented smooth surfaces D = C ∪ · · · ∪ C k in a smooth oriented 4-manifold X (possiblywith boundary or non-compact). In this paper, a topological divisor D is required to sat-isfy the following additional properties: D doesn’t intersect the boundary of X , no three C (cid:48) i s intersect at the same point, and any intersection between two surfaces is positive andtransversal. In a symplectic 4-manifold ( X, ω ) (possibly with boundary or non-compact), a symplectic divisor is a topological divisor D embedded in X , with each component beinga symplectic surface and having the positive orientation with respect to ω . Since we are1 a r X i v : . [ m a t h . S G ] J a n nterested in the germ of a symplectic divisor, X is sometimes omitted in the writing and( D, ω ), or simply D , is used to denote a symplectic divisor.Given a divisor D = ∪ ki =1 C i in ( X, ω ), the intersection matrix of D is a k × k matrix Q D =([ C i ] · [ C j ]), where · is used for any of the pairings H ( X ; K ) × H ( X ; K ), H ( X ; K ) × H ( X ; K )and H ( X ; K ) × H ( X, ∂X ; K ). Here K could be either Z or R depending on the situation.We also denote by b + ( D ) the number of positive eigenvalues of Q D .Let ( D = ∪ ki =1 , ω ) be a symplectic divisor. A symplectic divisor is called ω -orthogonal isany two components intersect ω -orthogonally. A closed regular neighborhood of D is calleda plumbing of D . A plumbing N D of D is called a concave plumbing (resp. convexplumbing ) if it is a strong symplectic cap (resp. filling) of its boundary ( − Y D , ξ D ) (resp.( Y D , ξ D )). A concave plumbing is also called a divisor cap of its boundary. Let Q D be theintersection matrix of D and a = ( C i · [ ω ]) ∈ ( R + ) k be the area vector of D . A symplecticdivisor D is said be concave (resp. convex ), if it satisfies positive (resp. negative) GScriterion , i.e. there exists z ∈ ( R + ) k (resp. ( R ≤ ) k ) such that Q D z = a . A topologicaldivisor D = ∪ C i is called non-negative if D · C i ≥ i and D · C j (cid:54) = 0 for some j .Similarly we can define a topological divisor to be non-positive , positive and negative .For an ω -symplectic divisor D , Gay-Stipsicz constructed in [9] a convex plumbing for D satisfying the negative GS criterion. This construction was extended to symplectic divisorssatisfying the postive GS criterion in [16], where a concave plumbing is constructed for eachsuch divisor. To summarize, we have the following theorem. Theorem 1.1 ([9], [16]) . Let D ⊂ ( W, ω ) be an ω -orthogonal symplectic divisor. Then D has a concave (resp. convex) plumbing if ( D, ω ) satisfies the positive (resp. negative) GScriterion. We call this construction the GS construction and review it in Section 2.1. Note that adifferent construction was presented in [20], which works in higher dimensions and does notrequire ω -orthogonality.The convex or concave plumbing depends on the symplectic divisor ( D, ω ) and other pa-rameters, but the contact structure induced on the boundary depends only on the topologicaldivisor D ([20], [16]). This motivates the notion of convex and concave topological divisors(see Section 2.2). A much stronger uniqueness holds for convex divisor D , where the contactstructure is called Milnor fillable. The Milnor fillable contact structure depends only on theoriented diffeomorphism type of Y D instead of the divisor D ([1]). In Section 2.2 we formulatea suitable version of the following natural question. Question 1.2.
Is there a similar unique contact sturcture on − Y D for concave D ? As a first step towards the question, we prove the contact structure ( − Y D , ξ D ) is invariantunder toric equivalence and interior blow-up/down of D (Proposition 2.7) in the Appendix.Such invariance also plays an important role in the study of symplectic fillings of contacttorus bundles in [17].Furthermore, an open book decomposition was constructed on the contact boundary ofthe convex plumbing in [9] for non-positive symplectic divisors. Later a Lefschetz fibrationwas constructed on the convex plumbing of a non-positive symplectic divisor in [8]. In Section2.3, we extend the construction of [9] to non-negative divisors.2 roposition 1.3. Let D be a non-negative symplectic divisor and ( N D , ω ) the concave plumb-ing constructed from the GS construction. Then there is an open book decomposition sup-porting the boundary ( − Y D , ξ D ) of ( N D , ω ) . The page and monodromy of the open bookdecomposition can be read off directly from D . As an application of the open book decomposition, we investigate the universal tightnessof some contact torus bundles. Let X be a smooth rational surface and D ⊂ X an effectivereduced anti-canonical divisor. Such pair ( X, D ) is called an anti-canonical pair and is relatedto Looijenga’s conjecture on dual cusp singularities ([19]), which was proved in [13] and laterin [3]. Symplectic log Calabi-Yau pairs were then introduced in [15] as a symplectic analogueof anti-canonical pairs. Enumerative aspects of symplectic log Calabi-Yau pairs and relationsto toric actions were also studied in [18].We call a topological divisor D consisting of a cycle of spheres a circular sphericaldivisor . A symplectic circular spherical divisor can be seen as a local version of a symplecticlog Calabi-Yau pair as it doesn’t require a closed ambient symplectic manifold. It is well-known that the boundary of a plumbing of a cycle of spheres is a topological torus bundle([21]). By Proposition 5.10 of [17], when b + ( D ) ≥ D admits a concave plumbing and itsboundary ( − Y D , ξ D ) is a contact torus bundle.Golla and Lisca investigated a large family of such contact torus bundles in [11], deter-mined their Stein fillability and studied the topology of Stein fillings. Then for all circularspherical divisors with b + ( D ) ≥
1, the Stein fillability/non-fillability was determined and allminimal symplectic fillings were shown to have a unique rational homology type, by Mak andthe authors ([17]).With their understanding of Stein fillings, Golla and Lisca showed in [11] that a subfamilyof the contact torus bundles they considered are universally tight. This led them to formulatethe following conjecture.
Conjecture 1.4 ([11]) . Suppose D is a circular spherical divisor with b + ( D ) = 1 and ( − Y D , ξ D ) symplectic fillable, then ( − Y D , ξ D ) is universally tight. This was confirmed for divisors with nonsingular intersection matrices by Ding-Li ([2]).Both the results of Golla-Lisca and Ding-Li come from an extrinsic point of view and relieson understanding the symplectic fillings of virtually overtwisted contact torus bundles.We approach this conjecture from an intrinsic angle, based on the Giroux correspondencebetween contact structures and open book decompositions. Via the open book decompositionconstructed in Proposition 1.3, we combine the results of Honda ([14]) and Van-Horn-Morris([23]) to prove the following result in the direction of the above conjecture.
Theorem 1.5.
Let D be a circular spherical divisor toric equivalent to a non-negative one,then ( − Y D , ξ D ) is universally tight, except possibly when − Y D is a parabolic torus bundle withmonodromy (cid:18) n (cid:19) , n > . Because our approach is purely 3-dimensional in nature, our result is stronger than Con-jecture 1.4 in the sense that we don’t require D to be embedded in a rational surface. Infact, most contact structures we considered are not symplectic fillable, and thus cannot be3tudied by extrinsic methods. Also the circular spherical divisor D we considered can have b + ( D ) ≥
2, compared to b + ( D ) = 1 in Conjecture 1.4. Acknowledgments : The authors are grateful to Cheuk Yu Mak for useful discussions.Both authors are supported by NSF grant 1611680.
We briefly review the construction of divisor neighborhood in [9] and [16], i.e. the proof ofTheorem 1.1. The invariance of contact structure under blow-ups (Proposition 2.7) and theconstruction of open book decompositions (Proposition 1.3) are based on this construction.For each topological divisor D , we can associate a decorated graph Γ = ( V, E, g = ( g i ) , s =( s i )) with each vertex v i representing the embedded symplectic surface C i and each edgeconnecting v i , v j corresponds to an intersection between C i and C j . Each vertex v i is weightedby the genus g i = g ( C i ) and self-intersection s i = [ C i ] . If ( D, ω ) is a symplectic divisor, wecan associate an augmented graph (Γ , a ) by adding the area vector a = ([ ω ] · [ C i ]) ki =1 .For an augmented graph (Γ , a ) and a vector z such that Q Γ z = a . Let z (cid:48) = − π z and fixa small (cid:15) >
0. For each vertex v i and each edge e connecting to v i , we choose an integer s i,e such that (cid:80) e ∈E ( v i ) s i,e = s i , where E ( v i ) denotes the set of edges e connecting to v i . Also,set x i,e = − s i,e z (cid:48) i − z (cid:48) j , where v j is the other vertex connected by e .Consider the first quadrant P = [0 , ∞ ) ⊂ R and for some fixed γ and δ , let g : P → [0 , ∞ ) be a smooth function with level sets like in the following figure. So g ( x, y ) = x when y − x > γ , g ( x, y ) = y when y − x < − γ and g is symmetric with respect to the line y = x .Figure 1: Contour of function g ( x, y )The constants γ and δ are chosen to be small enough so that for each vertex v i and eachedge e incident to v i , the line passing through (0 , (cid:15) ) with tangent vector (1 , − s i,e ) shouldintersect g − ( δ ) in the region y − x > γ . By symmetry, we also have the line passing through( (cid:15),
0) with tangent vector ( − s i,e ,
1) intersects g − ( δ ) in the region y − x < − γ .For edge e connecting vertices v i and v j , we can construct a local model ( X e , C e , ω e , V e , f e )as follows. Let µ : S × S → [ z (cid:48) i , z (cid:48) i + 1] × [ z (cid:48) j , z (cid:48) j + 1] be the moment map of S × S onto its4mage. We set p , p be the coordinates for [ z (cid:48) i , z (cid:48) i + 1], [ z (cid:48) j , z (cid:48) j + 1] and set q , q ∈ R / π Z tobe the corresponding fiber coordinates. Then ω = dp ∧ dq + dp ∧ dq is the symplectic formon the preimage of the interior of the moment image. Let g e ( x, y ) = g ( x − z (cid:48) i , y − z (cid:48) j ) and let R e be the open subset of g − e [0 , δ ) between the line passing through ( z (cid:48) i , z (cid:48) j + 2 (cid:15) ) with tangentvector (1 , − s i,e ) and the line passing through ( z (cid:48) j , z (cid:48) i + 2 (cid:15) ) with tangent vector ( − s j,e , X e , ω e ) be the symplectic manifold given as the toric preimage µ − ( R e ). Let C e = µ − e ( ∂R e ), f e = g e ◦ µ e and V e be the Liouville vector field obtained by lifting the radial vector field p ∂ p + p ∂ p in R . y xR i,e R j,e z (cid:48) j z (cid:48) i z (cid:48) j + (cid:15)z (cid:48) j + 2 (cid:15) z (cid:48) i + (cid:15) z (cid:48) i + 2 (cid:15) Figure 2: Region R e Then for each vertex v i with valence d i , we may associate a 5-tuple ( X i , C i , ω i , V i , f i ) asfollows. Let g i be the genus of v i and Σ i be a compact Riemann surface with genus g i and d i boundary components ∂ e Σ i corresponding to each edge e connected to v i . We can finda symplectic form β i and a Liouville vector field W i on Σ i such that there exists a collarneighborhood of ∂ e Σ i parametrized as ( x i,e − (cid:15), x i,e − (cid:15) ] × S on which β i = dt ∧ dα and W i = t∂ t . Then we define X i = Σ i × D ( √ δ ) and ω i = β i + rdr ∧ dθ , where D ( ρ ) is thedisk of radius ρ and ( r, θ ) is the standard polar coordinate on the disk. We define f i = r V i = W i + ( r z (cid:48) i r ) ∂ r and C i = Σ i − ∂ Σ i .Finally, the symplectic neighborhood ( X, C, ω, V, f ) is constructed by gluing the localmodels together appropriately. Let R i,e be the parallelogram in R e cut out by the twolines with tangent vector (1 , − s i,e ) passing through ( z (cid:48) i , z (cid:48) j + (cid:15) ) and ( z (cid:48) i , z (cid:48) j + 2 (cid:15) ) respectively.Similarly R j,e is cut out by the two lines with tangent vector ( − s j,e ,
1) passing through( z (cid:48) j , z (cid:48) i + (cid:15) ) and ( z (cid:48) j , z (cid:48) i + 2 (cid:15) ) respectively. X i can be glued to X e by identifying µ − e ( R i,e ) with( x i,e − (cid:15), x i,e − (cid:15) ) × S × D ( √ δ ). It’s easy to check that symplectic forms, functions andLiouville vector fields all match accordingly.It’s easy to see that when ( D, ω ) satisfies negative GS criterion, i.e. z ∈ ( R − ) k , the Liou-ville vector field V points outward along the boundary. So the glued 5-tuple ( X, C, ω, V, f )gives the desired convex neighborhood. And when (
D, ω ) satisfies positive GS criterion,we have z ∈ ( R + ) k . Then we can choose t small enough such that V is inward pointing5long the boundary of f − ([0 , t ]), which gives a concave neighborhood. We would call thisneighborhood the convex or concave plumbing of D and denote it by ( N D , ω ).In summary, given a symplectic divisor ( D, ω ) (or equivalently an augmented graph(Γ , a )), a vector z satisfying positive/negative GS criterion and choices of parameters (cid:15), δ, t ∈ R + , { s v,e ∈ Z | (cid:80) e ∈E ( v ) s v,e = s v } , g : [0 , ∞ ) → [0 , ∞ ), the above construction gives asymplectic plumbing ( N D , ω ) with Liouville vector field V along the boundary.Although the statement of Theorem 1.1 concerns an ambient symplectic manifold ( W, ω ),it actually only depends on the combinatorial data (Γ , a ). Suppose D is only a topolog-ical divisor with intersection matrix Q D such that there exists z, a satisfying the positive(resp. negative) GS criterion Q D z = a . Then the GS construction actually constructs acompact concave (resp. convex) symplectic manifold ( N D , ω z ) such that D is ω z − orthogonalsymplectic divisor in N D and a is the ω z − area vector of D . Let ( N D , ω ) be a symplectic plumbing of D and Y D = ∂N D be the oriented boundary 3-manifold of the plumbing N D . The Liouville vector field V constructed above induces acontact structure ξ D = ker( α ) on this boundary, where α = ι V ω . Note that when N D isconvex (resp. concave), ξ D is a positive contact structure (i.e. α ∧ dα >
0) on the orientedmanifold Y D (resp. − Y D ).The following uniqueness result implies that the symplectic structure ω may vary but theinduced contact structure on the boundary only depends on the topological divisor D . Proposition 2.1 ([16], cf. [20]) . Suppose D is an ω − orthogonal symplectic divisor which sat-isfies the positive/negative GS criterion. Then the contact structures induced on the boundaryare contactomorphic, independent of choices made in the construction and independent of thesymplectic structure ω , as long as ( D, ω ) satisfies positive/negative GS criterion.Moreover, if D arises from resolving an isolated normal surface singularity, then thecontact structure induced by the negative GS criterion is contactomorphic to the contactstructure induced by the complex structure. This motivates us to consider the notion of convexity for topological divisors. A topolog-ical divisor D is called concave (resp. convex ) if there exists z ∈ ( R + ) r (resp. z ∈ ( R ≤ ) r )such that a = Q D z ∈ ( R + ) r . Then there is a contact manifold ( − Y D , ξ D ) (resp. ( Y D , ξ D ))and, for each choice of such z , a symplectic cap (resp. filling) ( N D , ω z ) containing D as asymplectic divisor. One can check by simple linear algebra that being concave (resp. convex)is preserved by toric blow-up (see for example Lemma 3.8 of [16]). Remark 2.2.
The notions of convex and concave for topological divisors are less restrictivethan that for symplectic divisors, as we do not fix the symplectic area a . When D is convex, ( Y D , ξ D ) is contactomorphic to the contact boundary of some isolatedsurface singularity ([12]) and is called a Milnor fillable contact structure. A closed 3-manifold Y is called Milnor fillable if it carries a Milnor fillable contact structure. For every Milnorfillable Y , there is a unique Milnor fillable contact structure ([1]), i.e. the contact structure ξ D only depends on the oriented homeomorphism type of Y D instead of D when D is convex.6n light of this uniqueness result, it is natural to ask if similar results hold when D isconcave. The answer is no and the following counterexample is given in [16]. Example 2.3 (Example 2.21 of [16]) . Let D be a single sphere with self-intersection and D be two spheres with self-intersections and intersecting at one point as follows. Both divisors have a concave neighborhood. By [21] we can see that − Y D and − Y D areboth orientation preserving homeomorphic to S . However, ξ D is the unique tight contactstructure on S while ξ D is overtwisted. So far all the counterexamples we can construct consist of divisors with different b + andalso only one of them is fillable. So we refine our question to the following: Question 2.4.
Suppose D and D are concave divisors with − Y D ∼ = − Y D . Suppose either b + ( Q D ) = b + ( Q D ) or ξ D , ξ D both symplectically fillable, then is ( − Y D , ξ D ) contactomor-phic to ( − Y D , ξ D ) ? We first introduce two operations on topological divisors.
Definition 2.5.
For a topological divisor D = ∪ C i , a toric blow-up is the operation ofadding a sphere component E with self-intersection − between an adjacent pair of component C i and C j , and changing the self-intersection of C i and C j by − . Toric blow-down is thereverse operation. D and D are toric equivalent if they are connected by toric blow-ups and toric blow-downs. D is said to be toric minimal if no component is an exceptional sphere (i.e. acomponent of self-intersection − ). Definition 2.6.
For a topological divisor D = ∪ C i , an interior blow-up is the operationof adding a sphere component E with self-intersection − intersecting some component C i atone point, and changing the self-intersection of C i by − . The reverse operation is called an interior blow-down . Since blow-ups and blow-downs can be performed in the symplectic category, these op-erations have symplectic analogues by adding an extra parameter of symplectic area. Theywill be described for augmented graphs in Section A.1 and A.2.Note that two divisors give the same oriented plumbed 3-manifold if and only if they arerelated by Neumann’s plumbing moves ([21]), including toric blow-ups/blow-downs and inte-rior blow-ups/blow-downs introduced above. To solve Question 2.4, it suffices to understandhow the induced contact structure changes when we perform Neumann’s plumbing moves.As a first step towards this goal, we have the following proposition, whose proof is technicaland thus deferred to the appendix.
Proposition 2.7.
The contact structure induced by the GS construction is invariant under1. toric blow-ups/blow-downs, . and interior blow-ups/blow-downs. In light of Proposition 2.7, we see that toric equivalence is a natural equivalence ondivisors. For the study of contact structures and symplectic fillings, it suffices to considertoric minimal divisors. In particular it is used in the proof of Theorem 1.5.Note that all Milnor fillable contact structures are Stein fillable. Then we raise anotherquestion related to the fillability of divisor contact structures when D is concave. Question 2.8.
Is there a graph 3-manifold Y such that ( − Y, ξ ) is symplectically fillable forsome contact structure ξ , but − Y has no fillable divisor contact structure, i.e. for any concave D with Y D = Y , ( − Y, ξ D ) is not fillable? This subsection is devoted to the proof of Proposition 1.3. We first recall some generalitieson open book decompositions and refer the readers to [5] and [22] for further details. Anopen book decomposition of a 3-manifold Y is a pair ( B, π ) where B is an oriented link in Y such that π : Y \ B → S is a fiber bundle where the fiber π − ( θ ) is the interior of acompact surface Σ θ with boundary B , for all θ ∈ S . For each θ ∈ S , Σ θ is called a pagewhile B is called the binding of the open book. An open book decomposition can also bedescribed as (Σ , h ) where Σ is an oriented compact surface with boundary and h : Σ → Σ isa diffeomorphism such that h is identity in a neighborhood of ∂ Σ. The map h is called themonodromy.An open book decomposition ( B, π ) of a 3-manifold Y supports a contact structure ξ on Y if ξ has a contact form α such that α ( B ) > dα (Σ) >
0. Suppose we have an openbook decomposition with page Σ and monodromy h . Attach a 1-handle to the surface Σalong the boundary ∂ Σ to obtain a new surface Σ (cid:48) . Let γ be a closed curve in Σ (cid:48) transverselyintersecting the cocore of this 1-handle exactly once. Define a new open book decompositionwith page Σ (cid:48) and monodromy h (cid:48) = h ◦ τ γ , where τ γ denotes the right Dehn twist along γ . Theresulting open book decomposition is called the a positive stabilization of the original one.The inverse of this process is called a positive destabilization. In [10] Giroux established theone-to-one correspondence between oriented contact structures on Y up to isotopy and openbook decompositions of Y up to positive stabilization. This correspondence is of fundamentalimportance and enables us to study contact structures through open book decompositions.For the construction of open book decompositions, it is convenient to introduce the fol-lowing notions. Let D = ∪ C i be a topological divisor such that D · C i (cid:54) = 0 for some i . Then D is called non-negative if D · C i ≥ i . Equivalently its associated decorated graphΓ would satisfy s i + d i ≥ i and s j + d j (cid:54) = 0 for some j , where d i is the valence ofvertex v i . Similarly we can define a topological divisor to be non-positive , positive , and negative in the obvious way. These notions were first introduced in [4] and are intimatelyrelated to open book decompositions. Here is an easy observation. Lemma 2.9.
A topological divisor being non-negative is preserved by toric blow-down.
It was shown in [8] that all non-positive divisors are actually negative definite and thusconvex. For non-negative divisors, we can show the following:8 emma 2.10.
All non-negative divisors are concave.Proof.
Let D be a non-negative divisor with r components and denote by Q D = ( Q ij ) theintersection matrix. We will find a pair of vectors z and a through an iterated perturbationprocess.Start with z = (1 , . . . , T and a = Q D z . Since D is non-negative, we have a j ≥ j and a i > i . So the index set I = { i | a i > } is nonempty.Suppose a l = 0 and Q il > i ∈ I . Let z (cid:48) be a new vector such that z (cid:48) i = z i + (cid:15) for some small positive (cid:15) and z (cid:48) j = z j for all other j . Then we let a (cid:48) = Q D z (cid:48) such that a (cid:48) j = a j + (cid:15)Q ji for all j . Since Q ji ≥ j (cid:54) = i , we have a (cid:48) j ≥ a j for all j (cid:54) = i . Inparticular, a (cid:48) l = a l + (cid:15)Q li = (cid:15)Q li > Q li >
0. For (cid:15) small enough, we can also require that a (cid:48) i = a i + (cid:15)Q ii >
0. So we have I (cid:48) = { i | a (cid:48) i > } ⊃ I ∪ { l } .We could repeat the process using I (cid:48) , z (cid:48) , a (cid:48) as the new I, z, a . Since the divisor D is finite,this process stops at some finite time and produces a pair of vectors z, a ∈ ( R + ) r such that Q D z = a .Based on their construction of convex divisor neighborhoods, Gay and Stipsicz constructedan open book decomposition supporting the induced contact structure on the boundary whenthe divisor is non-positive ([9]). We first recall their construction and then extend it to thecase of non-negative divisors. Lemma 2.11 (cf. [9]) . Let M ± = ± [0 , × S × S with coordinates t ∈ [0 , , α ∈ S and θ ∈ S . Given a nonnegative integer m there exists an open book decomposition ob ± m = ( B, π ± ) on M ± such that the following conditions hold:1. π ± | { }× S × S = θ π ± | { }× S × S = θ ± mα B has m components B , . . . , B m , which we take to be B i = { } × { πim } × S
4. The binding and pages can be oriented so that ± ∂ θ is positively tangent to B i andpositively transverse to pages.Proof. This lemma was proved in [9] for M + only, where ob + m is constructed by stacking m copies of building blocks with page shown on the left of Figure 3. It’s easy to see the sameproof works for M − by stacking copies of building blocks with page shown on the right ofFigure 3.The building blocks on the right are P = − [0 , × [0 , × S with coordinates ( x, y, θ ).It is equipped with the open book decomposition ( B P , π P ) satisfying B P = { } × { } × S , π | { }× [0 , × S = θ , π | [0 , ×{ }× S = θ , π | [0 , ×{ }× S = θ and π | { }× [0 , × S = θ − πy . Notethat when pages are oriented so that − ∂ θ is positively transverse, then B P is oriented (asboundary of the page) so that − ∂ θ is positively tangent.9 P x yθ Figure 3: Building blocks for the open bookRecall that the boundary Y D = ∂N D is constructed by gluing f − i ( l ) and f − e ( l ) togetherif the edge e connects to vertex v i . For each vertex v i , we set the open book decomposition θ : f − i ( l ) = C i × S √ l → S to be the projection to second factor. For each edge e , f − e ( l )is a submanifold with toric coordinates ( p , q , p , q ). We set the open book decomposition f − e ( l ) → S to be q + q . Recall that each gluing region can be parametrized as ( x i,e − (cid:15), x i,e − (cid:15) ) × S × S √ l with coordinates ( t, α, θ ) and q + q transforms into ( − s i,e − α + θ .So if − s i − d i ≥
0, we can choose s i,e so that p i,e = − s i,e − x i,e − (cid:15), x i,e − (cid:15) ) × S × S √ l to be ob + p i,e andinterpolate from q + q to θ .Now we extend the construction to the concave case. The main difference from theconvex case is that the open book decomposition supports the positive contact structureon the negative boundary − Y D of the concave neighborhood N D instead of the positiveboundary. So this open book is constructed by gluing θ : − f − i ( l ) = − Σ i × S ρ → S ρ and q + q : − f − e ( l ) → S together. Along the gluing region − ( x i,e − (cid:15), x i,e − (cid:15) ) × S × S ρ withcoordinate ( t, α, θ ), q + q transforms to the function − ( s i,e + 1) α + θ . We can modify theopen book using the building block ob − q i,e if q i,e = s i,e + 1 ≥
0. And such a choice of { s i,e } exists if s i + d i ≥ − f − i ( l ) the Reeb vector field is a negativemultiple of ∂ θ and on − f − e ( l ) the Reeb vector field is a negative multiple of b ∂ q + b ∂ q forsome b , b >
0. They are both positively transverse to the pages and positively tangent tothe bindings.For each vertex v i , let S i be a compact surface with genus g i and s i + d i boundarycomponents. It’s easy to see that the page S of the above open book decomposition is givenby connect-summing the surfaces S i according to Γ. Let { γ , . . . , γ l } be the collection of simpleclosed curves on S consisting of one circle around each connect-sum neck and { δ , . . . , δ q } bethe collection of simple closed curves in S parallel to each boundary component. Here l = | E | q = (cid:80) ki =1 ( s i + d i ) = (cid:80) ki =1 s i + 2 l . For any simple closed curve c in S , let τ c denote theright Dehn twist along c . Then the monodromy is given by ( τ γ . . . τ γ l ) − ( τ δ . . . τ δ q ). Thisfinishes the proof of Proposition 1.3. Example 2.12.
The open book on the right of Figure 4 corresponds to the divisor on theleft. Here each vertex is decorated by ( s i , g i ) where s i is the self-intersection number and g i is the genus. Red curves are labeled with + or − to indicate that the monodromy consists ofa positive or negative Dehn twist along the curve. (1 ,
1) ( − ,
0) (2 , D (left) and open book decomposition for ( Y D , ξ D ) Remark 2.13.
The open book decomposition we constructed in the concave case matches theone constructed by Gay in [7] and [6]. The construction of Gay makes use of handlebodytheory and only works for positive divisors. Our construction is stronger as it works moregenerally for non-negative divisors.The open books constructed in both convex and concave cases match the ones constructedby Etgu and Ozbagci in [4], where the construction is purely topological and is not required tobe compatible with a certain contact structure.
Honda has classified tight contact structures on torus bundles in [14], which are mostlydistinguished by their S − twisting β S . In his thesis ([23]), Van-Horn-Morris described acorrespondence between open book decompositions of tight contact torus bundles and worddecompositions of their monodromies. Combining their results, we can determine that thecontact torus bundles ( − Y D , ξ D ) are universally tight for a large family of circular sphericaldivisors D .Given a convex torus Σ = R / Z inside a tight contact manifold, its slope is the slope of aclosed linear curve on Σ that is parallel to a dividing curve. In this case, the dividing curvesare parallel and homologically essential, so the slope is well-defined. To any slope s of a line in R we can associate its angle ¯ α ( s ) ∈ RP = R /π Z . For ¯ α , ¯ α ∈ RP , let [ ¯ α , ¯ α ] be the imageof the interval [ α , α ] ⊂ R , where α i ∈ R are representatives of ¯ α i and α ≤ α < α + π . Aslope s is said to be between s and s if ¯ α ( s ) ∈ [ ¯ α ( s ) , ¯ α ( s )].11et ξ be a contact structure on T × I with convex boundary and has boundary slopes s i for T × { i } , i = 0 , ξ is minimally twisting if every convex torus T × t has dividingset with slope between s and s . For a minimal twisting ξ , the I − twisting of ξ is given by β I = α − α . For general ξ , cut ( T × I, ξ ) into minimally twisting segments T k ∼ = T × I k , k = 1 , . . . , l and its I − twisting is the sum of each: β I = β I + · · · + β I l . Then for a tightcontact torus bundle M , we define the S − twisting β S to be the supremum of the I − twisting (cid:98) β I (cid:99) over all splittings of M into T × I along a convex torus isotopic to a fiber, where (cid:98) β I (cid:99) isdefined to be nπ if nπ ≤ β I < ( n +1) π . ( M, ξ ) is called minimally twisting in the S − directionif β S < π .Now we are ready to state Honda’s result in the non-minimal twisting case. Proposition 3.1 (Proposition 2.3 of [14]) . For a torus bundle with monodromy A , thereexist infinitely many tight contact structures with non-minimal twisting. The universally tightcontact structures are distinguished by the S -twisting β S which take values in { mπ | m ∈ Z + } .There exists virtually overtwisted contact structure only when A = (cid:18) n (cid:19) , n > . So to decide whether a tight contact torus bundle is universally tight, it suffices to showit is non-minimal twisting, except the positive parabolic cases mentioned above. In order tocalculate S -twisting from the divisor, we utilize the explicit open book decomposition forcontact torus bundles described by Van-Horn-Morris. Let Word denote the set of words in { a, a − , b, b − } . To a and b − we associate corresponding relative open book decompositionswith pages and monodromies as in Figure 5. The relative open books for a − and b are thesame as that with sign reversed.Figure 5: Relative open book for a (left) and b − (right)To any word w ∈ Word , we can then associate an open book decomposition ob w =(Σ w , φ w ) with torus pages Σ w by stringing together the annular regions associated to eachletter in w and identifying the remaining pair of circle boundaries to form a many-puncturedtorus as in Figure 6. The monodromy φ w is given by Dehn twists along the all the signedcurves. Denote the corresponding contact manifold by ( Y w , ξ w ). Lemma 3.2 ([23]) . Suppose words w, v are related by a sequence of braid relations b − a − b − = a − b − a − . Then the associated open book decompositions ob w and ob v are stably equivalent and thustheir supported contact structures are isotopic. aa − or a − a leaves the pageunchanged and only adds canceling pairs of Dehn twists τ c τ − c or τ − c τ c to the monodromy,which does not change the open book decomposition.There is a natural mapΦ : Word → Aut + ( T ) ∼ = (cid:104) a − , b − | b − a − b − = a − b − a − , ( ab ) = Id (cid:105) ∼ = SL (2 , Z )defined by Φ( a ) = (cid:18) (cid:19) , Φ( b ) = (cid:18) − (cid:19) , Φ( a − ) = Φ( a ) − and Φ( b − ) = Φ( b ) − . Here Aut + ( T ) is identified with SL (2 , Z ) by identifying T with R / Z . In the rest of this section,we will not distinguish between a word w and its image Φ( w ) in SL (2 , Z ) when we work withmatrix multiplications. The 3-manifold supported by the open book decomposition ob w isdetermined by the conjugacy class of Φ( w ) in SL (2 , Z ), but the contact structure varies whenwe take different words. Lemma 3.3 ([23]) . Let Y be the ambient manifold of the open book ob w . Then Y is home-omorphic to the torus bundle T A with monodromy is A = Φ( w ) . Using the open book decomposition constructed in Proposition 1.3, we can associatedto any non-negative circular spherical divisor D a word w ( D ) that is solely composed of a − , b − as follows. Recall in our construction of open book decompositions, each vertex v i with self-intersection s i contributes ( s i + d i ) boundary-parallel positive Dehn twists in themonodromy and each edge contributes a negative Dehn twist along the connect-sum neck.Thus each vertex corresponds to the word b − − s i and each edge corresponds to the word a − .The word w ( D ) is obtained by taking product of all these words in the clockwise order. Thenthe divisor D = ( s , . . . , s l ) corresponds to the word w ( D ) = b − − s a − . . . b − − s l a − . Theopen book decomposition ob w ( D ) associated to this word is exactly the one constructed inProposition 1.3 for ( − Y D , ξ D ). Proposition 3.4 ([23]) . Any word in { a, a − , b − } gives an open book decomposition com-patible with a weakly fillable contact structure. w ( D ) is a word in { a − , b − } , we have that ( Y D , ξ D ) is weakly fillable, and inparticular, tight. We will usually write w for w ( D ) as our choice of divisor D would be clearfrom the context.From Honda’s classification, we know that if β S ≥ π and the monodromy is not conjugateto a n , n >
1, then the contact structure is universally tight. We can compute the S − twistingof a contact structure from the word associated to its compatible open book decompositionas in [23]. For a word w = a k b − . . . a k l b − , which we read from left to right as we movefrom t = 0 to t = 1. To compute the change of angles, we end with V l = (1 , T and workbackwards to t = 0. Then V l − = a k l b − (1 , T , so on and so forth. Let c w denote the totalangle change, then β S of ( Y w , ξ w ) is at least (cid:98) c w (cid:99) . Note that when calculating β S , we arefree to change the word by braid relation, cyclic permutation and adding canceling pairs of a and a − . Example 3.5.
Consider parabolic bundle ( − Y D , ξ D ) given by the concave divisor in thefollowing graph, with − ≤ n so that the graph is non-negative. Its monodromy is A = A ( − n, − = − (cid:18) − n (cid:19) . n The word associated to this divisor is a − ( b − ) n +2 a − ( b − ) . Through cyclic permutationit becomes w = b − a − ( b − ) n +2 a − b − . We can check that b − a − ( b − ) n +2 a − b − (cid:18) (cid:19) = (cid:18) − (cid:19) . The rotation is c w = π . So β S ≥ π and the contact structure is universally tight.Proof of Theorem 1.5. We start by noticing that a word corresponding to a concave divisor D of length l takes the form a − b − − s . . . a − b − − s l . By Proposition 2.7, we may assume D is toric minimal or D = ( − , p ). Then D is still non-positive by Lemma 2.9 and thus concave,we must have b + ( Q D ) ≥ s i ≥ i , or D = ( − , −
2) or ( − , − s i ≥ i . By cyclic permuting, we may assume s l ≥ w = b − − s l . . . a − b − − s l − a − b − with − − s l ≤ −
1. If − − s l − = 0, then w canbe written as the product w = w (cid:48) b − a − n b − with n ≥ w (cid:48) in { a − , b − } . If − − s l − ≤ −
1, then w = w (cid:48) b − a − l b − m a − b − with l ≥ m ≥ b − a − n b − and b − a − l b − m a − b − ro-tates the vector (1 , T by at least π : b − a − n b − (cid:18) (cid:19) = (cid:18) − n − n (cid:19) , n ≥ b − a − l b − m a − b − (cid:18) (cid:19) = (cid:18) − l − l (cid:19) , l ≥ , m ≥ .
14y adding canceling pairs of a and a − , w (cid:48) can always be written as product of ( aba ) − , a and a − . Note that a , a − preserve the half space a vector sits in and ( aba ) − rotates a vectorby π w (cid:48) does not rotate the vector back to theupper half space. So c w ≥ π and β S ≥ π for the contact structure induced on the boundary.By Proposition 3.1, the contact structure is universally tight except when Y is torus bundlewith monodromy A = a n = (cid:18) n (cid:19) , n >
1. The remaining case of ( − , −
1) and ( − , − A Invariance of contact structure
A.1 Contact structure and toric equivalence
In this section we prove the first statement of Proposition 2.7. We want to show that toricblow-up on the divisor doesn’t change the induced contact structure on boundary of plumbing.The construction in this section will be adapted a little to prove the second statement ofProposition 2.7 in the next section.First we introduce the blow-up of an augmented graph, which is the symplectic versionof toric blow-up. Consider the following local picture of an augmented graph (on the left),where each vertex is decorated by its self-intersection number, genus and symplectic area.The blow-up of this augmented graph with weight 2 πa is given on the right, which is thetoric blow-up with areas specified in the graph. We call this an augmented toric blow-upof edge e . Similarly, the reverse operation is called an augmented toric blow-down .( s , g , a ) v ( s , g , a ) v e = ⇒ ( s − , g , a − πa ) v ( − , , πa ) v ( s − , g , a − πa ) v e e Denote the original augmented graph by (Γ (1) , a (1) ) and the blown-up graph (Γ (2) , a (2) ). Notethat Q Γ (2) z (2) = a (2) is still solvable after the augmented toric blow-up. If z (1) = ( z , z , . . . )and a (1) = ( a , a , . . . ) satisfy Q Γ (1) z (1) = z (1) , then after blow-up of area a , z (2) = ( z , z + z − πa , z , . . . ) and a (2) = ( a − πa , πa , a − πa , . . . ) satisfy Q Γ (2) z (2) = z (2) . So wecould apply GS construction to both augmented graphs. In the following, we will denote theconstruction based on (Γ (1) , a (1) ) by GS-1 and denote the construction based on (Γ (2) , a (2) )by GS-2.For the choice of { s v,e } , note that the two graphs differ only near e . We could choose { s v,e } for GS-1 first and then choose the same { s v,e } for all vertices and edges for GS-2,except the ones involved in the toric blow-up. We could choose s v ,e = 0, s v ,e = − s v ,e + s v ,e = s = − s v ,e = s v ,e − s v ,e = s v ,e −
1. Then we have x v ,e = x v ,e − a , x v ,e = x v ,e − a , x v ,e = − z (cid:48) and x v ,e = z (cid:48) + a . The choice ofother parameters will be specified later. Note that the choice of parameters won’t affect theboundary contact structure by Proposition 2.1.In GS-1, the edge e corresponds to the local model ( X (1) e , C (1) e , ω (1) e , V (1) e , f (1) e ) with toric15 xR (1) v ,e R (1) v ,e z (cid:48) z (cid:48) z (cid:48) + (cid:15) (1) z (cid:48) + 2 (cid:15) (1) z (cid:48) + (cid:15) (1) z (cid:48) + 2 (cid:15) (1) z (cid:48) + a z (cid:48) + a (cid:18) − s v ,e (cid:19) (cid:18) − s v ,e (cid:19) Figure 7: Region R (1) e with moment map µ e y x R (2) v ,e R (2) v ,e z (cid:48) + z (cid:48) + a z (cid:48) z (cid:48) + z (cid:48) + a + (cid:15) (2) z (cid:48) + z (cid:48) + a + 2 (cid:15) (2) z (cid:48) + (cid:15) (2) z (cid:48) + 2 (cid:15) (2) (cid:18) − s v ,e (cid:19) = (cid:18) − s v ,e + 1 (cid:19)(cid:18) − s v ,e (cid:19) = (cid:18) (cid:19) (a) R (2) e y x R (2) v ,e R (2) v ,e z (cid:48) z (cid:48) + z (cid:48) + a z (cid:48) + (cid:15) (2) z (cid:48) + 2 (cid:15) (2) z (cid:48) + z (cid:48) + a + (cid:15) (2) z (cid:48) + z (cid:48) + a + 2 (cid:15) (2) (cid:18) − s v ,e (cid:19) = (cid:18) (cid:19) (cid:18) − s v ,e (cid:19) = (cid:18) − s v ,e + 11 (cid:19) (b) R (2) e Figure 8: Toric picture of edges e , e in GS-2image R (1) e in Figure 7. The gluing region R (1) v ,e is characterized by the vector (cid:18) − s v ,e (cid:19) and R (1) v ,e is characterized by (cid:18) − s v ,e (cid:19) .In GS-2, the edge e corresponds to the local model ( X (2) e , C (2) e , ω (2) e , V (2) e , f (2) e ) with toricimage R (2) e as in Figure 8(a) with gluing region R (2) v ,e characterized by vector (cid:18) − s v ,e (cid:19) = (cid:18) − s v ,e + 1 (cid:19) and R (2) v ,e characterized by (cid:18) − s v ,e (cid:19) = (cid:18) (cid:19) . Using the transformation16 x R (1) v ,e R (1) v ,e R (1) v ,e R (1) v ,e R (1) e R (1) e R (1) v z (cid:48) + 2 (cid:15) (1) z (cid:48) + a + (cid:15) (2) z (cid:48) + 2 (cid:15) (1) z (cid:48) + a + (cid:15) (2) z (cid:48) + a z (cid:48) + a (cid:18) − s v ,e (cid:19) (cid:18) − s v ,e (cid:19) Figure 9: Region R ble after blow-up. R (1) e is the region on the upper left, enclosed by black and red solid lines. R (1) e is the region on the lower right, enclosed by black and red solid lines. R (1) v is the rectangular region in the middle, enclosed black dashed and solid lines. R (1) v ,e , R (1) v ,e are the small rectangular regions in the middle bounded by both red and black lines. R (1) v ,e , R (1) v ,e are the small parallelogram regions on the upper left and lower right. (cid:18) − (cid:19) ∈ GL (2 , Z ), we could map R (2) e onto R (1) e in Figure 9. This gives a symplectomor-phism Φ e : ( µ − e ( R (2) e ) , ω (2) e ) → ( µ − e ( R (1) e ) , ω (1) e ) and identifies the Liouville vector field V (2) e with V (1) e . Similarly, the edge e corresponds to the local model ( X (2) e , C (2) e , ω (2) e , V (2) e , f (2) e )with toric image R (2) e as in Figure 8(b) with gluing region R (2) v ,e characterized by (cid:18) − s v ,e (cid:19) = (cid:18) (cid:19) and R (2) v ,e by (cid:18) − s v ,e (cid:19) = (cid:18) − s v ,e + 11 (cid:19) . Using the transformation (cid:18) −
10 1 (cid:19) ∈ GL (2 , Z ), we could map R (2) e onto R (1) e in Figure 9. This gives symplectomorphism Φ e :( µ − e ( R (2) e ) , ω (2) e ) → ( µ − e ( R (1) e ) , ω (1) e ), and identifies the Liouville vector field V (2) e with V (1) e .For vertex v , take X (2) v = [ − z (cid:48) − a + (cid:15) (2) , − z (cid:48) − (cid:15) (2) ] × S × D √ δ (2) , ω (2) v = dt ∧ dα + rdr ∧ dθ and V (2) v = t∂ t + ( r z (cid:48) r ) ∂ r . So we see that the local model ( X (2) v , C (2) v , ω (2) v , V (2) v , f (2) v ) isexactly ( µ − e ( R (1) v ) , µ − e ( L ) , ω (1) e , V (1) e , f (1) e ), where L is the line segment from point ( z (cid:48) +17 xz (cid:48) + a z (cid:48) + a R (1) v ,e R (1) v ,e R (1) v z (cid:48) + a − (cid:15) (2) z (cid:48) + 2 (cid:15) (2) z (cid:48) + 2 (cid:15) (2) z (cid:48) + a − (cid:15) (2) z (cid:48) + a − (cid:15) (2) z (cid:48) + (cid:15) (2) z (cid:48) + (cid:15) (2) z (cid:48) + a − (cid:15) (2) Figure 10: Zoomed picture of vertex region R (1) v (cid:15) (2) , z (cid:48) + a − (cid:15) (2) ) to ( z (cid:48) + a − (cid:15) (2) , z (cid:48) + (cid:15) (2) ) in Figure 10. We can check the gluing of µ − e ( R (1) e ) with µ − e ( R (1) v ) along µ − e ( R (1) v ,e ) coincides with the gluing of µ − e ( R (2) e ) with X (2) v along µ − e ( R (2) v ,e ). Similarly, the gluing along µ − e ( R (1) v ,e ) coincides with the gluing along µ − e ( R (2) v ,e ). So the glued local model X (2) e ∪ X (2) v ∪ X (2) e is symplectomorphic to the preimageof the region R (1) e ∪ R (1) v ∪ R (1) e with Liouville vector fields identified.Blow up the intersection point in P ( D (1) ) corresponding to the edge e symplecticallywith area 2 πa to get ( P ( D (1) ) CP , ω bl ). This corresponds to cutting the corner from R (1) e as shown in Figure 2 and the resulting region is called R ble . Since blowing up aninterior point doesn’t change the boundary, we have ( Y D (1) , ξ D (1) ) = ∂ ( P ( D (1) ) , ω (1) ) ∼ = ∂ ( P ( D (1) ) CP , ω bl ).To make the intervals in Figure 9 and Figure 10 well-defined, the following inequalitiesmust be satisfied:2 (cid:15) (2) < a − (cid:15) (2) and z (cid:48) i + a + 2 (cid:15) (2) ≤ z (cid:48) i + 2 (cid:15) (1) , i = 1 , . Also, in order for the embeddings and the blow-up to remain inside the neighborhood X (1) e ,the following restrictions on sizes of these neighborhoods should be satisfied: δ (2) ≤ δ (1) and a < δ (1) . So we could choose δ (1) , δ (2) , (cid:15) (1) , (cid:15) (2) , a so that they satisfy 0 < δ (2) ≤ δ (1) , < (cid:15) (2) < (cid:15) (1) , a = 2 (cid:15) (1) − (cid:15) (2) and 4 (cid:15) (2) < a < δ (1) . Such choice of a ensures that there is enough18rea to blow-up and the interval in X (2) v is well defined. So the region R (1) e ∪ R (1) v ∪ R (1) e isembedded in R (1) e . Since all other local models are the same for GS-1 and GS-2, by shrinkingthe region R (1) e , we get an contact isotopy from ( Y D (1) , ξ D (1) ) ∼ = ∂ ( P ( D (1) ) CP , ω bl ) to( Y D (2) , ξ D (2) ) = ∂ ( P ( D (2) ) , ω (2) ). A.2 Half edge and blow-up of a vertex
The construction outlined in Section 2.1 actually only works for graphs with at least twovertices, but it can be modified to take care of the single vertex case. Now consider theaugmented graph (Γ , a ) where Γ has only one vertex v decorated with genus g and self-intersection s . As long as s (cid:54) = 0, there is always a solution z = as .According to GS construction, the vertex v corresponds to a local model ( X v , C v , ω v , V v , f v ).Here X v = Σ v × D √ δ where Σ v is a genus g surface with one boundary component. To closeup and get a disk bundle over a closed genus g surface with Euler class s , we need to glue X v to a disk bundle over disk and add the suitable twisting. Consider the region R ˜ e in Fig-ure 11, which is similar to the region R e in Figure 2 except we only have one gluing region R v, ˜ e . This region gives a local model ( X ˜ e , C ˜ e , ω ˜ e , V ˜ e , f ˜ e ) in the same way as the ordinary GSconstruction. Note that x ˜ e = µ − e ( R ˜ e ) ∼ = D × D √ δ . Here the gluing region is specified bythe vector (cid:18) − s (cid:19) . By gluing these two local models, we get the desired disk bundle. y xR v, ˜ e z (cid:48) (cid:15) (cid:15) z (cid:48) + δ (cid:18) − s (cid:19) Figure 11: Region R ˜ e corresponding to the half edge ˜ e This region R ˜ e works almost the same as an edge in ordinary GS construction and we19all it a half edge , as shown below.( s, g, a ) v ⇐⇒ ( s, g, a ) v ˜ e For any vertex v in an augmented graph (Γ , a ), we have X v ∼ = Σ v × D . Take any point p ∈ Σ v and a small disk neighborhood D of p . This local neighborhood D × D can beregarded as the local model X ˜ e corresponding to a half edge ˜ e . Here we could choose theparameter s v, ˜ e = 0 so that s v, ˜ e + (cid:80) E ( v ) s v,e = s v .For an augmented graph (Γ , a ), let v be a vertex in Γ. We introduce the symplectic versionof interior blow-up and blow-down. The following is called an augmented interior blow-up of vertex v with weight a , of which the reverse operation is also called an augmentedinterior blow-down .( s, g, a ) v. . .. . . = ⇒ ( s − , g, a − a ) v. . .. . . ( − , , a ) v ˜ e An augmented interior blow-up of vertex v can be regarded as the augmented toric blow-up of a half edge ˜ e stemming from v as shown in the following diagram, where the right arrowindicates an augmented toric blow-up of ˜ e .( s, g, a ) . . .. . . ⇐⇒ ( s, g, a ) . . .. . . ˜ e = ⇒ ( s − , g, a − a ) . . .. . . ( − , , a )˜ e (cid:48) ⇐⇒ ( s − , g, a − a ) . . .. . . ( − , , a )The construction from Section A.1 also works for regions like R ˜ e with a suitable choice of (cid:15) and a . So we have that toric blowing up a half edge ˜ e doesn’t change its boundary contactstructure. Thus we conclude that the boundary contact structure is invariant under interiorblow-up of a vertex. References [1] Cl´ement Caubel, Andr´as N´emethi, and Patrick Popescu-Pampu. Milnor open books andmilnor fillable contact 3-manifolds.
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School of Mathematics, University of Minnesota, Minneapolis, MN, US
E-mail address : [email protected] School of Mathematics, University of Minnesota, Minneapolis, MN, US
E-mail address : [email protected]@umn.edu