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Tight contact structures on the solid torus

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Tight contact structures on the solid torus

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Content Tight Contact Structures on the Solid Torus by David Jin-kyum Kim A Thesis Presented to the FACULTY OF THE USC DORNSIFE COLLEGE OF LETTERS, ARTS, AND SCIENCES UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree MASTER OF SCIENCE (MATHEMATICS) May 2022 Copyright 2022 David Jin-kyum Kim TableofContents ListofFigures iii Abstract iv Chapter1: Introduction 1 1.1 Basic Denitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Local Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Tight and Overtwisted Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Chapter2: ToolsfortheClassicationofTightContactStructuresonManifolds 6 2.1 Legendrian Knots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Convex Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Bypasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.1 Finding Bypasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Chapter3: TheSolidTorus 19 Bibliography 30 ii ListofFigures 1.1 A visual representation of the intersection of (R 3 ; 3 ) with theRplane. . . . . . . . . . 4 2.1 The sign of the crossings is determined by handedness. Right-handed crossings (left) are counted positively. Left-handed crossings (right) are counted negatively . . . . . . . . . . 8 2.2 The standard foliation on a Bypass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Visualizing a Bypass attachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 The way dividing curves of a surface are changed by attaching a bypass. . . . . . . . . . . 12 2.5 A representation of how the mystery move aects dividing curves. . . . . . . . . . . . . . 13 2.6 The possible outcomes of CaseII. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.7 The Farey Tesselation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 iii Abstract In this thesis, we focus on establishing well-known results of tight contact structures on some manifolds, focusing in particular on low dimensional (3dimensional) manifolds. We introduce the basic dichotomy betweentight andovertwisted structures, noting the dierences in classifying both structures. In particular, we focus on the classication of tight contact structures. In order to do so, we introduce and make use of some knot invariants so that we may develop the notion of the bypass, a tool developed by Ko Honda. Finally, making use of bypasses, we state many of the basic classication results of the eld and end with a classication of the tight contact structures on the Solid Torus,T =S 1 D 2 . iv Chapter1 Introduction 1.1 BasicDenitionsandExamples In the contact geometric world, we concern ourselves with a smooth distribution of planes associated to points on our manifold. Generally, the distribution of these planes can be either integrable or non- integrable. In the contact geometric world, we work specically with the non-integrable case. The follow- ing denitions reect these general notions. Denition: A plane eld is a subbundle ofTM (M a manifold) such that for allp2M, p =T p M\ is a 2-dimensional subspace ofTM. Denition: A contact structure is a pair (M;) whereM is a manifold and is a plane eld dened by a 1-form such that = ker(). In addition, it must satisfy the ’non-integrability’ condition mentioned previously, given by^d6= 0. Remark: The denition of contact manifold written here is specialized for 3-dimensions. For higher dimensions, simply replace plane elds with hyperplane elds. The conditions on a contact structure are equivalently a maximal non-integrability condition (given by Frobenius’ Theorem).[7] Though it will not be touched on, this naturally gives rise to mechanics/dynamics related formulations of many of the ideas presented here. 1 Someexamplesofcontactstructures: 1. TakeR 3 with standard coordinates (x;y;z), let 1 =dz +xdy, and let 1 =ker ( 1 ) 2. TakeR 3 with cylindrical coordinates (r;;z), let 2 =dz +r 2 d, and let 2 =ker ( 2 ) 3. TakeR 3 with cylindrical coordinates (r;;z), let 3 = cos(r)dz +r sin(r)d, and let 3 =ker ( 3 ) Remark: Example (1) is generally referred to as the standard contact structure onR 3 . Denition: Two contact structures 0 and 1 on a manifoldM are called contactomorphic if there exists a dieomorphismf :M!M sending 0 to 1 , i.e.f ( 0 ) = 1 . Remark: Notice that examples (1) and (2) are contactomorphic. 1.2 LocalStructure Theorem 1.2.1: (Darboux’s Theorem): Given (M;) an arbitrary contact 3-manifold andp a point inM, there exist neighborhoodsN such thatp2NM,U such that (0; 0; 0)2UR 3 , and a contactomor- phismf : (N;j N )! (U; 1 j U ) (where 1 refers to the standard contact structure onR 3 ). Essentially,Darboux’sTheorem states that all contact structures "look" similar near a point. The impor- tant consequences of this theorem are twofold: 1) The local structure of a contact structure is uninteresting and therefore 2) the classication of contact structures is entirely related to their global structure. Proof. Takep2M. We may assume there exists a local coordinate chart that mapsp2M to 02R 3 and that the contact structure at 0 is thexy-plane (i.e. (0) = ker(dz)). Then in a neighborhood of 0, we can take to be given by the 1-form =dz +fdx +gdy. Now notice that on thexz-plane, restricts to a vector eldv = @ x f@ z . Noticev is transverse to thez-axis. Then using the fundamental theorem of ODE’s, we can integrate alongv starting at thez-axis. 2 We let (x;z) be the timex ow, which starts at (0; 0;z). This gives us new coordinates wherev = @ x . Under these coordinates, we have normalized on thexz-plane. Now consider under the restriction tox =K (K a constant). This restriction yields the vector eld v =@ y g@ z . We then have a vector eldv dened in a neighborhood of 02R 3 such thatv is transverse to they-axis. Integrating as before, we obtain new coordinates so that we can write our contact form as follows: = dz +f(x;y;z)dx withf(x;y; 0) = 0 and@ y f < 0 by the contact condition. Now simply change coordinates (x;y;z)7! (x;f(x;y;z);z). Remark: Though we refer to it here as Darboux’s Theorem, it may also be referenced as Pfa’s The- orem. Darboux’s is more commonly used with regards to symplectic geometry. Some tools of symplectic geometry are also used in contact geometry, as we will see in section 1.3. Denition: A foliation of codimensionnq partitions annmanifoldM into submanifolds of di- mensionq (called leaves), such that locally,M is a productR q R nq . In our specic case, our foliations will be of codimension 1 (and so our leaves will be of dimension n 1). Denition: Let be an embedded oriented surface in a contact manifold (M;). For all x2 , considerl x = x \T x . We dene singular points to be pointsx at whichl x =T x . Now there exists a singular foliationF of that is tangent tol x at eachx. The leaves of this foli- ation are the disjoint union of 1-manifolds which contain no singularities (i.e. the leaves are the com- plement of the singularities), and the leaf through x must be tangent to l x . This foliation is called the characteristic foliation of and is denoted . Example: Let be the disk of radius in theR-plane of the contact structure (R 3 ; 3 ). One can "measure" what the contact structure looks like on by taking the intersection between the plane elds of (R 3 ; 3 ) to yield a structure as depicted below. Moving the interior of up or down slightly will lead to 3 a dierent lines of intersection, but the tangency points at the center and the boundary will remain. See Figure 1.1. Figure 1.1: A visual representation of the intersection of (R 3 ; 3 ) with theRplane. Remark: The example above is known as the overtwisted disk. It is an essential part of the study of contact manifolds. 1.3 TightandOvertwistedManifolds Denition: A contact structure on a manifold M is called overtwisted if its characteristic foliation admits an overtwisted disk. If it does not, it is called tight. Example: 3 is often regarded as the standard tight contact structure onR 3 The following theorem is the rst "big" result in the classication of contact structures. Theorem1.3.1: (Eliashberg’s Overtwisted Classication): Given a compact, closed 3Manifold, letH be the set of homotopy classes of plane elds onM andC 0 the set of isotopy classes of overtwisted contact structures onM. The natural inclusion mapC 0 intoH induces a homotopy equivalence.[1] Remark: Notice that this theorem essentially states that the classication of overtwisted contact struc- tures on 3-manifolds is the same as the classication of homotopy classes of plane elds. In addition, this implies that all 3-manifolds admit overtwisted contact structures. 4 Theorem1.3.2: Etnyre-Honda: There exist closed, compact 3-manifolds that do not support any tight contact structures.[6] Though this theorem states that there exist manifolds that do not admit tight contact structures, "most" do. In most cases, the simplest method for constructing tight contact structures is to ll it with a compact symplectic manifold (using symplectic lling) and make use of the following theorem: Theorem1.3.3: Eliashberg,Gromov: If a contact structure can be lled by a compact symplectic man- ifold, then it is tight.[3][9] The nal important result comes from the following theorem. Theorem1.3.4: Eliashberg: IfF is a singular foliation onS 2 induced by a tight contact structure, there exists a unique (up to isotopy xing boundary) tight contact structure onB 3 such that (@B 3 ) =F.[2] This theorem is useful in classifying tight contact structures on a manifold by the general procedure as follows: start by removing parts of the manifold on which the local contact structure is known (i.e. the local characteristic foliation is known) until all that remains is a collection of 3-balls. Then use the above theorem to determine that we know what the structure looks like (by uniqueness). 5 Chapter2 ToolsfortheClassicationofTightContactStructuresonManifolds 2.1 LegendrianKnots Legendrian Knots are classic topological objects whose properties turn out to have important consequences in contact geometry. In this section we dene Legendrian Knots and their invariants. Denition: Let (M;) be a contact manifold. A curve : S 1 ! M is called Legendrian if (S 1 ) is always tangent to, i.e. for allx2S 1 ,d(T x (S 1 )) is contained in (x) . Remark: An exactly similar denition to above can be made for Transverse Knots (replace tangent with transverse). Denition: LetK be a knot. A framing ofK is a choice of homotopy class of trivializations of the normal bundle, vK, toK. This naturally gives an identication of a tubular neighborhoodN(K) with S 1 D 2 . We will denote a framing byF. We now introduce the twisting number followed by two classical knot invariants. Denition: The Relative Thurston-Bennequin Invariant (more commonly referred to as the twisting- number) of a knotK is denonted byt(K;F). It is dened as follows: Pick an orientation onK. Note thatK has a natural choice of framing, called the normal framing, induced by by takingv p 2 p so the (v p ; _ L(p)) forms an oriented basis for p . Thent(K;F) is equal to 6 the integer (signed) dierence in number of twists betweenF and the normal framing. By convention, left twists are negative. Now take a seifert surface such that@ =K. This induces a framingF and gives rise to the rst classical invariant of Knot Theory. Denition: The Thurston-Bennequin Invariant of a knot K is denoted by tb(K). It is dened as follows: tb(K) =t(K;F ). Here remember thatF is the framing adapted to the characteristic foliation . Note thattb(K) is independent of the choice of the seifert surface . Now we dene the second classical invariant of Knot Theory. Denition: Take an oriented Legendrian knot K inR 3 . The rotation number of K is denoted by r(K). It is dened as follows: choose a trivialization of onR 3 , thenr(K) is the winding number of _ K alongK with respect to the trivialization. Though visualizing knots and their properties may be dicult in three dimensions, it turns out all knots can be approximated by their front projections. Denition: The front projection of a Legendrian Knot K to a plane is a projection such that the following hold: 1. all crossings of strands of project the strand with the smaller slope in front of the strand with the larger slope, 2. all vertical tangencies are rather denoted using cusps. Given a front projection, the Thurston-Bennequin Invariant and the rotation number can be computed as follows: 7 Figure 2.1: The sign of the crossings is determined by handedness. Right-handed crossings (left) are counted positively. Left-handed crossings (right) are counted negatively tb(K) = 1 2 (#cusps) + #positive crossings#negative crossings r(K) = 1 2 (#downward cusps#upward cusps) Notice that it is easy to decreaset(K;F) simply by adding zigzags along the front projection. Increas- ing it however, is not so trivial. As a nal aside before moving on, note the following consequence of Pfa’s Theorem (Theorem 1.2.1): Lemma2.1.1: Any closed curve in a contact manifold can beC 0 -approximated by a Legendrian curve. 2.2 ConvexSurfaces Convex surfaces embedded in contact manifolds become essential to the study of tight contact structures. Many of the ideas on Convex Surfaces can be attributed to the work of Emmanuel Giroux.[8] Denition: Given a contact manifold (M;), we say a vector eld is contact if its ow preserves the contact vector eld. Then a surface is called convex if there exists a contact vector eldv transverse to it. Now, let be the set of points on where the contact vector eld is tangent to. Generically, is a multi-curve, i.e. a collection of curves, such that the following holds: 1. is a properly embedded 1-manifold, possibly disconnected, and possibly with boundary, 2. is transverse to , 8 3. The isotopy class of does not depend onv, 4. n = + F (here + denotes the set of points where the orientation given byv(x) agrees with, respectively disagrees with, the orientation given by x ), 5. is nonempty. IfF is a singular foliation on , then is dened to divide the foliation if the above conditions hold when is replaced byF. Then the curves on are called the dividing curves (or dividing set) ofF. We will denote by # the number of components of . The following theorem increases the usefulness of convex surfaces by making them easy to nd. Theorem2.2.1: Any closed surface embedded in a contact structure isC 1 close to a a convex surface. Alternatively, anyC 1 -generic surface is convex. Theorem2.2.2: Giroux Flexibility: Suppose divides bothF and . Then for any neighborhoodN of , and fort2 [0; 1], there exists an isotopy t : !N of such that: 1. 0 is the natural inclusion of intoN, 2. t () is convex for allt2 [0; 1], 3. t does not change , 4. ( 1 ()) = 1 (F). Essentially, this theorem says that the characteristic foliation can be taken to be anything we wish it to be as long as it respects the dividing curves. In essence, the dividing set encodes all important information about our contact structure in a neighborhood of the convex surface.[8] Proof. Firstly notice that dividingF implies there must be another contact structure 0 on such that 0 =F. Now on R, and 0 are respectively given given by =fdt + and 0 =f 0 dt + 0 where 9 they have common dividing setf = f 0 = 0. We may isotope both of our contact structures near so that they and their foliations agree there. Away from , we can normalize byf and assumef =1 (depending on our orientation as dened in the denition of our dividing curves). Sayf = 1 (i.e. we are on + ). Then and 0 are area forms on + and so we can interpolate between them by letting s = (1s) +s 0 . Let s =dt + s . We can then use the Moser technique to solve for a vector eldX =g@ t +Y in the equation: L X s = ds ds . i X d s +d( s (X)) = 0 . i Y d s +d(g + s (Y )) = 0 . We can split the equations to give:i Y d s = 0 andg = s (Y ). Once we solve forY in the rst equation,g is automatically determined. In any case,X is independent oft, hence the corresponding isotopy s () must be transverse to@ t for alls. An important consequence of the theorem is the following: Theorem2.2.3: (LegendrianRealizationPrinciple): Let be a convex surface with dividing set , and C a multi-curve on . AssumeC and intersect transversely such that each connected component of nC intersects nontrivially with (the dividing set). Then there is an isotopy ofC such that 1 (C) is Legendrian. This isotopy is as in Giroux’s Flexibility Theorem. 2.3 Bypasses In the contact geometric world, the dividing set of a convex surface embedded in a tight contact structure is aected in discrete units. These discrete units are given by bypasses, where one bypass attachment is the fundamental unit of isotopy. The notion of the bypass was developed by Ko Honda.[12] 10 Figure 2.2: The standard foliation on a Bypass Remark: The fundamental existence of bypasses depends on the following Edge-Rounding-Lemma. Though the lemma itself is not so complicated, it does require some setup. Take a compact convex surface embedded in a contact structure (M;). For such a to exist, we must have thatt(L;F ) 0 for each connected componentL of@. For now, assumet(L;F ) < 0. Using a normal form theorem onL, we can write the neighborhoodN(L) =S 1 D 2 =R=Zfx 2 +y 2 g (note we are using coordinates (x;y;z)). Then the contact 1-form becomes = sin(2nz)dx + cos(2nz)dy, where we haven =t(L;F ) andL is the linex = y = 0. After aC 0 small pertur- bation, we can let [N(L) =fy = 0;x 0g. We will generally assume such a surface to have a collared legendrian boundary, which consists of parallel legendrian ruling curves. Such an assumption allows us to make use of several properties in cancelling singularities not discussed here. Once we have those properties, we can take the following. Lemma 2.3.0.1: (Edge-Rounding Lemma): Let 1 ; 2 be two convex surfaces with collared legen- drian boundary, which intersect transversely along a common boundary curveL such thatt(L;F 1 ) = t(L;F 2 ) =n. WithN(L) as above, we can take our above properties and proceed withedge-rounding. = (( 1 [ 2 )nfx 2 +y 2 2 g)[ (f(x) 2 + (y) 2 = 2 g[fx 2 +y 2 2 g), where< is a convex surface with dividing curvesz = k 2n on 1 connects to the dividing curvez = k 2n 1 4n on 2 wherek = 0;:::; 2n 1. Our new surface is essentially a gluing of our two old surfaces along a common edge in a way that respects dividing curves. 11 Denition(*): Let be a convex surface and a Legendrian arc in that intersects the dividing set at three pointsp 1 ;p 2 ;p 3 (letp 1 andp 3 denote the endpoints of). A bypassD for along is a convex diskD with Legendrian boundary such that the following hold: 1. D\ = 2. tb(@D) =1 3. @D =[ 4. \ =p 1 ;p 3 are corners ofD and elliptic singularities ofD . Figure 2.3: Visualizing a Bypass attachment Lemma 2.3.0.2: (Bypass Attachment Lemma): LetD, be as in denition (*). If is isotoped along D, then we obtain a new convex surface whose dividing curves are derived from as in Figure 2.4. Figure 2.4: The way dividing curves of a surface are changed by attaching a bypass. Theorem2.3.0.3: Let be a convex surface and 0 be the surface obtained from by bypass attach- ment in a tight contact manifold. Then one of the following holds: 1. 0 = (Trivial Bypass Attachment) 12 2. # 0 = # + 2 3. # 0 = # 2 4. 0 is obtained from by a positive Dehn twist about some curve in 5. 0 is obtained by a "mystery move" as in Figure 2.5. Figure 2.5: A representation of how the mystery move aects dividing curves. Proof. Let denote the curve along which the bypass is attached andp 1 ;p 2 ;p 3 denote the points of inter- section of and the dividing curves of . Letp 1 ;p 3 denote the endpoints of. Now suppose i denotes the connected component of the dividing set on whichp i lies. We have the following four cases: CaseI: All three i ’s are distinct. In this case, by the Bypass Attachment Lemma, after our attachment, 1 is joined to 2 , and 2 is joined to 3 , so we have reduced the number of connected components of our dividing set by 2. This is result (3). CaseII: All three i ’s are the same. See Figure 2.6. These are the only possible congurations in such a case, and result in (1), (2), and (4). CaseIII: 1 = 3 , but 2 6= 1 ; 3 . The result is either a Dehn Twist (4) or a Mystery Move (5). CaseIV: 1 6= 3 , but 2 = 1 or 3 . We can analyze similarly toCaseII and conclude (1), (4), or (5) holds. 13 Figure 2.6: The possible outcomes of CaseII. Example: Let us look atT 2 . The previous theorem is simplied to the following cases: 1. The trivial bypass attachment (bypass only intersects one dividing curve and does not wind around any topology). 2. Increasing the number of dividing curves (bypass only intersects one dividing curve and does wind around some topology). 3. Decreasing the number of dividing curves (bypass only intersects three dierent dividing curves). 4. Right Dehn Twist (bypass intersects two dividing curves. This can only happen when # = 2). 14 Remark: Not listed is the "mystery move" as it can never occur onT 2 embedded in a tight contact manifold. It is, however, possible in an overtwisted one. Now let us focus specically on the case where # T 2 = 2 (i.e. our torus has two boundary com- ponents) andT 2 is embedded in a tight contact manifold. Returning to our proof of the generally pos- sible bypass attachments, we see that the only non-trivial bypass attachment possible is option (4), the Dehn Twist. This gives us another useful formulation for possible bypass attachments onT 2 using the Farey Tesselation. Denition: LetD be the unit disk inR 2 . Label (1; 0) on@D by 0 = 0=1 and (1; 0) by1 = 1=0. Join them by a geodesic (as we can think of this construction as the disk model of a hyperbolic plane). For any two labelled points p q and p 0 q 0 on@D with non-negativey-coordinate, label their midpoint as p+p 0 q+q 0 and join the midpoint to both endpoints by a geodesic. Continue until all positive rationals are satised. Repeat for negative rationals. This construction is the Farey Tesselation. Theorem 2.3.0.4: LetT 2 be a convex surface such that # T 2 = 2, with dividing slopes and ruling sloper6= s. LetD be a bypass forT 2 attached along a ruling curve. Then, # T 2 0 = 2 and the dividing slopes 0 is determined as follows: Let [r;s] be the arc running counterclockwise fromr tos. Then,s 0 is equal to the point in [r;s] closest tor such thats ands 0 are joined by a geodesic. Proof. First, a lemma. Lemma2.3.0.5: Applying an elementA ofSL 2 (Z) does not change 1) the ordering and 2) the edge- connectedness properties of elements of the Farey Tesselation. Proof. Omitted, but straightforward. Now letD denote the Farey Tesselation. Lets 0 be as described in the theorem, ands 00 be ass 0 but on the arc@Dn[r;s]. By the choice ofs 0 , there is an edge betweens 0 ands in the Farey Tesselation (and same fors 00 ). Then, the corresponding vectors fors 0 ands form an integral basis forZ 2 and so we can take an elementB ofSL 2 (Z) that mapss7! 0 ands 0 7!1. 15 Figure 2.7: The Farey Tesselation. B will sendr ands 00 to some positive numbers. Sinces 00 has an edge tos 0 , it must map to some fraction 1 n . Assumen> 1. By denition ofs 00 ,r must map to betweens 00 and 1. But thenB(r) is closer to 1 than 1, a contradiction to ordering (by Lemma 2.3.0.5). Thusn = 1, i.e. there must exist an edge betweens 0 ands 00 (by Lemma 2.3.0.5, since there is an edge from 1 to1). Now we have thats,s 0 , ands 00 are all connected by edges. Then, we may nd an elementC ofSL 2 (Z) that mapss7! 0,s 0 7!1,s 00 7!1. Then,r must be correspondingly mapped to some number less than 1. Then by an easy application of our bypass attachments on a torus (i.e. looking at the trivial attachment and Dehn twist cases), we see that attaching a bypass along a ruling curve with slope equal toC(r) must produce a torus with dividing slope1. Transforming everything back to our starting tesselation using C 1 , we see that attaching a bypassing along a ruling curve with sloper must produce a dividing slope s 0 . 16 2.3.1 FindingBypasses Denition: Let be an arc properly embedded in a surface with boundary. Then, is said to be boundary pa- rallel if one of the components of the complement of is a disk. If is part of a dividing set , it is said to be outermost boundary parallel if it is boundary parallel and the disk that it separates contains no other components of . Lemma2.3.1.1: Let be a convex surface and 0 be a convex surface with Legendrian boundary such that one component of@ 0 is a subset of . Assume \ 0 @ 0 . If 0 has an outermost boundary parallel dividing component , and # 06= 1, then 0 may be isotoped relative to boundary so that is isotopic in 0 to a Legendrian curve such that the disk that cuts o from 0 is a bypass for along part of@. Lemma2.3.1.2: Suppose and 0 are as above. Assume that 0 is a disk. Iftb(@ 0 )<1, then there is a bypass for . Proof. Iftb(@ 0 ) =n, then 0 will consist ofn arcs. At least one of them must be outermost boundary parallel. Then Lemma 2.3.1.1 nishes the proof. Theorem2.3.1.3: (ImbalancePrinciple): Suppose and 0 are as above. Assume that 0 =S 1 [0; 1] (an annulus) withS 1 f0g , and@ 0 Legendrian. Ift(S 1 f0g; 0 ) < t(S 1 f1g; 0 ) < 0, then there is a bypass for . Proof. Clearly 0 must have more than one component. By denition, the dividing curves 0 will in- tersectS 1 fig according to the twisting number, i.e.2t(S 1 fig; 0 ) times. Thus all arcs starting at S 1 f0g cannot run toS 1 f1g. Thus one must be outermost boundary parallel. Then Lemma 2.3.1.1 nishes the proof. 17 2.4 BasicResults This section is mostly concerned with basic classication results often used in more complex classication results. Theorem2.4.1: (Thickened Sphere): LetM =S 2 [0; 1]. Fix a tight contact structure 0 in a neigh- borhood of@M such that # S 2 t = 1 fort = 0; 1. Then there exists a unique tight contact structure onM up to isotopy relative to boundary. Proof Sketch. The proof of this theorem makes use of another theorem proving the equivalence of bifurcations and bypasses. Given this theorem, construct an explicit model ofS 2 [0; 1] with zero and one bifurcations then prove the two models are isotopic relative to boundary. Theorem 2.4.2: (3-Ball): Assume there exists a tight contact structure on a neighborhood of@B 3 which makesB 3 convex with # @B 3 = 1. Then there exists a unique tight contact structure onB 3 up to isotopy relative boundary. This tight contact structure comes from an extension of. Proof. B 3 can be broken up into a standard 3-ball andS 2 [0; 1].S 2 [0; 1] is unique relative to boundary by above theorem, the standard 3-ball is unique by denition. Remark: These unique structures onB 3 form the basic building blocks for understanding tight contact structures. This is essentially a reformulation of Theorem 1.3.4. Theorem2.4.3: There exist unique tight contact structures on each ofR 3 ;S 3 ;S 2 S 1 . Remark: The proof for each of these results is similar. Break up the manifold into small parts and classify the possible contact structures on those parts. 18 Chapter3 TheSolidTorus In this chapter we prove a classication result on the number of tight contact structures on the solid torus T =S 1 D 2 given the following conditions: 1. # T = 2. 2. s 0 =1 ands 1 = p q where1< p q 1, wheres i denotes the slopes of the two components of T (The slope can be normalized as such). 3. The xed characteristic foliationF is adapted to T . Notice that notationally, we will refer toT asS 1 D 2 , andT 2 as the surface@S 1 D 2 (or alternatively S 1 S 1 ). In addition, #Tight(M; ) will denote the number of tight contact structures on a contact manifoldM with boundary conditions. First we deal with the case where p q =1 under a more general case where p q = 1 n . Proposition 3.1: (Kanda, Makar-Limanov): GivenT with conditions as above with the modication that on property (2), slope p q =1, there exists a unique contact structure onT up to isotopy relative to boundary. 19 Proof. LetLT 2 be a curve that bounds the meridianD ofT 2 . The surfaceT 2 is an embedded convex surface in T , which allows us to use the Legendrian Realization Principle on L to transform L into a Legendrian curve withtb(L) =1. Now take the Seifert surfaceD such that@D =L. By theorem 2.2.1, we can transformD to be convex while preserving the Legendrianness ofL. Astb(L) =1, there is only one possibility for the dividing set S . Next, x some characteristic foliationF onD such that S dividesF. Then any two tight contact structures onS 1 D with the boundary conditionF can be isotoped to agree onT 2 [D. Thus these contact structures are contactomorphic. Tn (T 2 [D) =B 3 . Use the theorem for uniqueness of tight contact structures onB 3 . Now assume p q <1. Before continuing, we introduce continued fraction expansion. Denition: Let p q be a fraction with integer entries. Then p q has a continued fraction expansion as follows: p q =r 0 1 r 1 1 r 2 ::: 1 r k Wherer i 2. We then write p q $ (r 0 ;r 1 ;:::;r k ). Example: 14 5 =3 1 5 . Then 14 5 $ (3;5). In addition, we introduce the standard neighborhood of a Legendrian curve. Denition: Take a (closed) Legendrian curveL witht(L;F) =n< 0;n2Z + (i.e. our framingF is chosen in order to make the twisting number negative). A standard neighborhoodS 1 D 2 =R=Z fx 2 +y 2 g with coordinates (z;x;y) of the Legendrian curveL =S 1 f(0; 0)g is given by the 1-form =sin(2nz)dx +cos(2nz)dy and satises the following: 1. T 2 is convex. 2. # T 2 = 2. 20 3. if the meridian has slope equal to zero and longitude given by x = y = K a constant, then slope( T 2) = 1 n . The importance of a standard neighborhood is that it allows the use of the Edge-Rounding Lemma in that neighborhood, further allowing the use of Bypass Attachments. Remark: Our p q =1 case actually comes from these properties, which turn out to be identical to the boundary conditions we are working with in that case. Finally we introduce basic slices and one important theorem on them. Denition: Take the contact manifold (N =T 2 I;) (I = [0; 1]). We will say (N;) is a basic slice if the following holds: 1. is tight. 2. T i are convex and # T i = 2 fori = 0; 1. 3. The minimal integral representatives ofZ 2 corresponding tos i form aZ-basis forZ 2 . 4. is minimally twisting, i.e. the slope of the dividing curves on any convex torusT inN parallel to the boundary is betweens 0 ands 1 . When talking about basic slices, we will say i = T 2 fig . Basic slices will be used extensively in our characterization of tight contact structures and have associated with them the following, extremely useful property. Theorem3.2: There are exactly two basic slices withs 0 = 0 ands 1 =1. Their relative Euler classes are given by(0; 1)2H 1 (T 2 ;Z). In other words, there are exactly two contact structures on each basic slice. Now we state the proposition. 21 Proposition3.3: (Kanda,Makar-Limanov): GivenT with conditions as given at the beginning of this section, there exist exactlyj(r 0 + 1)(r 1 + 1)::: (r k1 + 1)r k j tight contact structures onT up to isotopy xing boundary. We will prove the above proposition by way of three separate propositions. The rst is as follows: Proposition3.4: #Tight min (N; 0 [ 1 )j(r 0 + 1)(r 1 + 1)::: (r k1 + 1)r k j (where 0 and 1 are as in the denition of basic slices). Here the #Tight min denotes the number of tight, minimally twisting contact structures. Proof. Recall that we assume the slopes of to bes 0 =1 ands 1 = p q Take the characteristic foliation on@N to be standard with dividing slopes i . LetA be the annulus A =S 1 I whereS 1 is the circle onT 2 with slope equal to zero. We also take@A to be the Legendrian ruling curves on@N. Note that t(S 1 f0g;A) = 1 2 (S 1 f0g\ 0 ) =det 2 6 6 4 1 1 0 1 3 7 7 5 =1 t(S 1 f1g;A) = 1 2 (S 1 f1g\ 1 =det 2 6 6 4 1 q 0 p 3 7 7 5 =p. Thus, the Imbalance Principle allows us to nd a bypass D for T 2 1 (where T 2 i = T 2 fig) along S 1 f0g. Attaching our bypassD gives us a neighborhood ofT 2 1 D that we denoteT 2 [ 1 2 ; 1]. Thus we can writeN =T 2 [0; 1 2 ][T 2 [ 1 2 ; 1]. HereT 2 [ 1 2 ; 1] is itself a basic slice. Claim 3.4.1: The contact structure restricted to T 2 [ 1 2 ;a] is a basic slice with s 1 = p q and s1 2 given by s 1 , where if p q $ (r 0 ;:::;r k ), then s1 2 = p 0 q 0 $ (r 0 ;:::;r k1 ;r k + 1). If r k =2, then (r 0 ;:::;r k + 1) = (r 0 ;:::;r k1 + 1). Proof. First, we shows1 2 = p 0 q 0 wherep 0 qpq 0 = 1. 22 We use Theorem 2.3.0.4 to determine slopes after bypass attachments on a torus. Now attaching a bypass along a torus gives us that the original torus is in the back face of the new construction. However, in our case, our original torus is the front face ofT 2 [ 1 2 ; 1]. We reect theT 2 and the interval direction (i.e. (x;y)7! (x;y) and interval direction) which makes our original torus the back face of the basic slice. Thus our slope becomess 1 = p q > 1. Then according to Theorem 2.3.0.4 a bypass attached along S 1 f0g gives slopes1 2 = p 0 q 0 where p 0 q 0 is a positive rational closest to 0 with an edge to p q . Then we know p 0 qpq 0 =1. To showp 0 qpq 0 = 1, we can take following transformation: A = 2 6 6 4 p 0 q 0 p q 3 7 7 5 Under this mapping, (p;q)7! (1; 0); (1; 0)7! (p 0 ;p). We will prove immediately below thatp > p 0 > 0. Assuming this, by Theorem 2.3.0.4, the boundary slope must be1, and inverting the mapping gives us our desired result. The fact thatp>p 0 > 0 follows from the following argument. Let p 00 q 00 be the point counterclockwise to p q that is closest to 0 with an edge to p q . Similarly to the proof of Theorem 2.3.0.4, there must also exist an edge from p 00 q 00 to p 0 q 0 . Thus p q = p 0 +p 00 q 0 +q 00 . Then asp 00 > 0 andq 00 0, we getp>p 0 > 0 andqq 0 > 0. All that remains is to verify the continued fraction expansion of p 0 q 0 . Let a b $ (r 0 ;:::;r k1 ;r k +1). We will show a b = p 0 q 0 . We proceed by induction onk. Sayk = 0. Thendet 2 6 6 4 r 0 + 1 r 0 1 1 3 7 7 5 =r 0 + 1r 0 = 1. For the inductive step, notice thatr 1 cd = rcd c andr 1 c 0 nd 0 = rc 0 d 0 c 0 , which means that if c d and c 0 d 0 form an integral basis forr, so do the above values. Thus, sincer k andr k + 1 form an integral basis, so dor k1 1 r k andr k1 1 r k +1 . Continuing inductively we get that (r 0 ;:::;r k ) and (r 0 ;:::;r k + 1) form an integral basis, giving us thatqapb = 1. 23 Next we want to showp > a > 0;q b > 0. We proceed again by induction. Sayk = 0. Then p =r 0 >r 1 1 =a andq = 1b. Now assume c d , c 0 d 0 are both less than1,c>c 0 ,dd 0 , andr<1 is an integer. Thenr 1 c d = rcd c andr 1 c 0 d 0 = rc 0 d 0 c 0 are both less than1. Thenc>c 0 andjrc +dj>jrc 0 +d 0 j. Finally, anya,b with such properties must be unique by an application of the Farey Tesselation and properties on its connected edges. Thusa =p 0 ;b =q 0 . Now continuing the above construction, we can breakT 2 [0; 1] inton =jr k + 1j +jr k1 + 2j + +jr 0 + 2j basic slices, which we will denote byB i =T 2 [ i1 n ; i n ]. Now ifB i has slope corresponding to continued fraction expansion (t 0 ;:::;t l ) along its front face, then along the back face it has expansion (t 0 ;:::;t l + 1). By Theorem 3.2, each fraction block has exactly two tight contact structures, giving us an upper bound of 2 n (for now). In order to rene this upper bound, we look at groupings ofB i called continued fraction blocks, which we will denote byN i . N will be decomposed intok + 1 fraction blocks. Essentially, the grouping groups together rationals with continued fraction expansions of same length. More specically, if the front face of anN i has slope corresponding to (r 0 ;:::;r i + 1), its back face has slope corresponding to (r 0 ;:::;r i+1 + 1) for i = 1;:::;k 1, has front sloper 0 + 1 and back slope1 fori = 0, and front slope (r 0 ;:::;r k ) and back slope (r 0 ;:::;r k1 + 1) fori =k. Claim 3.4.2: Each fraction blockN i has at mostr i + 1 tight minimally twisting contact structures on it ifi<k and at mostr k ifi =k. Note that once we have proven this statement, our proof on the upper bound on the number of tight minimally twisting contact structures onN follows immediately. Proof. First note that we can make use of linear transformations to normalize the slopes of the front and back faces of each N i to be r i + 1 and1 respectively. For i = 0, this is trivially true. For i k, 24 we show this by the transformation A = 2 6 6 4 r 0 1 1 0 3 7 7 5 . This matrix sends (1;1) 7! (r 0 1; 1) and (p;q)7! (r 0 qp;q). This gives us that the back face ofN has dividing slope 1 r 0 +1 and front face has dividing slope q r 0 q+p . ButN 0 has dividing sloper 0 + 1 so after applyingA we have a slope of1. Then after removingN 0 from N we haveT 2 I with a back face of slope1 and a front face of slope q r 0 q+p . Note that: p q $ (r 0 ;:::;r k )) q r 0 q+p $ (r 1 ;:::;r k ) (This is easy to see once fractions are expanded and multiplied out). Thus continuing in this manner shows thatN has the normalized dividing slopes as claimed above. Now this proof follows from the following lemma. Lemma 3.4.3: Given T 2 I and multicurves i on T 2 i (i = 0; 1) with two components such that s 0 =1 ands 1 =m,m> 0 an integer, then #Tight(N; 0 [ 1 )m. Proof Sketch. As in the previous case, we break upT 2 I intom 1 basic slices to retrieve the same broad upper bound 2 m1 . We then notice that dierent regions of ourB i can be divided into regions depending on their orientations (remember how breaks up regions + and of a convex surface ). If the order of the positive/negative regions agree, we note that we can shue these regions and simply counting the number of positive/negative regions determines the number of contact structures onB i . Now that if Bypass attachments are not nested (one is not attached in the region of another), then we can add them in whichever order we want to change the orientations on a region. Thus the diculty lies in nested bypasses. These can be dealt with by adding copies of of regions containing the bypass and isotoping curves to essentially split the bypasses. This will prevent them from being nested and reduce to the previous un-nested case. 25 Combining this lemma with our normalized slopes onN i completes the proof on the number of contact structures on eachN i . This proves our proposition. Now before we move on to the next proposition, we present another theorem. Theorem3.5: Let (T 2 I;) be tight with convex boundary, # T 2 I = 2, and our slopes bes 0 and s 1 . Without loss of generality, assumings 0 s 1 , given anys2 [s 0 ;s 1 ], there exists a convex torus parallel toT 2 fptg with slopes. The proof of this theorem is omitted, but can be done by constructing an explicit model of the tight contact structures on a basic slice and perturbing ourT 2 I to make it a basic slice. Proposition 3.6: #Tight(S 1 D 2 1 ) #Tight min (N; 0 [ 1 ). Here 0 and 1 come from the denition given in basic slices. Proof. LetM = S 1 D 2 such that@M has a multicurve with slope p q withp > q > 0. LetL be a Legendrian curve isotopic toS 1 fptg with twisting number equal tom. We may assumem is largely negative. LetN 0 be a standard neighborhood ofL. LetN =MnN 0 =T 2 I. Clearly@N is convex with dividing slopes 0 = 1 m ands 1 = p q . Now note that any minimally twisting tight contact structure onM is minimally twisting when re- stricted toN. If not, there must be a convex torusT inN that is boundary parallel with dividing slope not in [ p q ; 1 m ]. But then by Theorem 3.5, we can nd a torus with any slope we want. In particular we nd a torusT inN with slope 0. The Legendrian divides inT must then bound meridianal disks in our original manifoldM. These disks must be overtwisted, contradicting our tightness assumption, so our structure must be minimally twisting onN. Now1 lies in [ p q ; 1 m ], so by Theorem 3.5, we must be able to nd a torusT inN parallel to to boundary with slope1. Then we splitM alongT in two pieces,T 0 the solid torus with slope1 and a 26 torical annulus with slopes1 and p q . By the p q =1 case dealt with earlier, we knowT 0 has a unique contact structure. Now before our nal proposition, we present a corollary and a denition of a Lens Space. Corollary3.7: Let be a tight contact structure onT =S 1 D 2 with dividing slopes one boundary denoteds. Then, for anys 0 2 (0;s], there is a convex torus parallel to the boundary with slopes 0 . Proof. Given anys 0 2 (0;s], there is somem such that 1 m > s 0 > s. Then we can nd a Legendrian curveL isotopic to the core curve ofT with twisting number equal tom by stabilizing. LetN 0 be a standard neighborhood ofL. ThenTnN 0 = T 2 I with dividing slopes 1 m ands. Then this theorem follows from Theorem 3.5. Now our nal proposition (and ultimately our lower bound) comes from the embedding ofT into a Lens Space. Denition: A Lens Space denotedL(p;q) is dened to be the 3-manifold obtained by gluing together two solid toriV 0 andV 1 by the mapA = 2 6 6 4 q q 0 p p 0 3 7 7 5 whereq 0 andp 0 are such thatqp 0 qp 0 =1. LettingC 0 be a Legendrian curve inV 0 topologically isotopic to the core curve ofV 0 , we can assume it has twisting numbern < 0 (by stabilizing if necessary). ThenV 0 can be thought of to be a standard neighborhood ofC 0 and so@V 0 is convex with two dividing curves of slope 1 n . To see@V 1 , we map@V 0 to@V 1 viaA. Thus@V 1 becomes convex with dividing slopes pnp 0 qn+q 0 . Proposition3.8: #Tight(L(p;q)) #Tight(S 1 D 2 1 ). Proof. Now by Corollary 3.7, there is a torusT 0 inV 1 isotopic to@V 1 that is convex with dividing slope 1. SplitV 0 intoT =S 1 D 2 andN =T 2 I usingT 0 . The boundary ofN must be convex with front dividing slope pnp 0 qn+q 0 and back dividing slope1. Since p 0 q 0 lies in [1; pnp 0 qn+q 0 ], we can nd another torusT 00 inN isotopic to@V 1 that is convex with dividing slope p 0 q 0 . SplittingL(p;q) alongT 00 gives us 27 V 0 andV 1 separately. Thus@V 1 must be convex with dividing slope p 0 q 0 while@V 0 must be convex with dividing slope1 (the slopes can be veried by invertingA). Then by our p q =1 case,V 0 has a unique tight contact structure with our given boundary condi- tions. Thus all other contact structures must be given byV 1 , butV 1 is simplyS 1 D 2 with boundary data given by as stated in the beginning of this section. Thus our three corollaries tell us that the number of tight contact structures on the solid torus with our given boundary conditions is bounded above byj(r 0 +1)(r 1 +1)::: (r k1 +1)r k j and bounded below by the number of tight contact structures on the Lens SpaceL(p;q). The nal step is to prove that the lower bound on the number of tight contact structures onL(p;q) is equal to our desired upper bound. This requires a deeper dive into the Lens Space which we will not do here, but the result at least will be listed below. Proposition3.9:j(r 0 + 1)(r 1 + 1)::: (r k1 + 1)r k j #Tight(L(p;q)). Given this theorem, our proposition follows, i.e. given our desired boundary conditions. #Tight(T =S 1 D 2 ; ) =j(r 0 + 1)(r 1 + 1)::: (r k1 + 1)r k j. The case for # T > 2 is still an open question. 28 FurtherDirections: Though not explored in this paper, each contact structure has an associated vector eld given certain conditions called the Reeb Vector Field. The dynamics of the Reeb eld can be studied to analyze contact structures in both low and high dimensions using symplectic eld theory and its many tools such as relative contact homology. Additionally, Ko Honda’s ideas on Bypasses have begun to become generalized to higher dimensions. Contact geometry’s many applications in classical mechanics, thermodynamics, integrable systems, control theory, makes it useful for study though by nature classication results may not be so easy to prove. As previously mentioned, the complete classication for # T > 2 is still an open question. Addition- ally, it has been proven that there is only a nite number of tight contact structures on any Lens Space. Though uniqueness results have been proven for Lens Spaces wherep = 0;p = 1;p = 2; as well as for whenp = 3;q = 2, the general result for anyp andq is still an open question. 29 Bibliography [1] Y. Eliashberg. “Classication of overtwisted contact structures on 3-manifolds”. In: Invent. Math. 98.3 (Oct. 1989), pp. 623–637.issn: 1432-1297.doi: 10.1007/BF01393840. [2] Y. Eliashberg. “Contact 3-manifolds in the years since J. Martinet’s work”. In: Ann. Inst. Fourier 42.1-2 (Mar. 1992), pp. 165–192.issn: 1777-5310.doi: 10.5802/AIF.1288. [3] Y. Eliashberg. “Filling by holomorphic discs and its applications”. In: Geometry of Low-Dimensional Manifolds (Donaldson and Thomas eds.) II (Jan. 1991), pp. 45–68.issn: 9780521400015.doi: 10.1017/CBO9780511629341.006. [4] J. Etnyre. Convex surfaces in contact geometry: Class notes. [5] J. Etnyre. Introductory lectures on contact geometry. Nov. 2002.doi: math/0111118v2. [6] J. Etnyre and K. Honda. “On the nonexistence of tight contact structures”. In: Annals of Math 153.3 (May 2001), pp. 749–766.issn: 0003486X.doi: 10.2307/2661367. [7] G. Frobenius. “Über das Pfasche Problem”. In: J. für Reine und Agnew. Math. 82 (Jan. 1877), pp. 230–315.issn: 1435-5345. [8] E. Giroux. “Convexité en topologie de contact”. In: Comment. Math. Helv. 66.4 (Oct. 1991), pp. 637–677.issn: 1420-8946. [9] M. Gromov. “Psuedoholomorphic curves in symplectic manifolds”. In: Invent. Math. 82.2 (June 1985), pp. 307–347.issn: 1432-1297.doi: 10.5802/AIF.1288. [10] K. Honda. “Contact 3-manifolds in the years since J. Martinet’s work”. In: Geom. Topol. 4.1 (Jan. 2000), pp. 309–368.issn: 1364-0380.doi: 10.2140/gt.2000.4.309. [11] K. Honda. Notes for Math 599: Contact Geometry. [12] K. Honda. “On the Classication of Tight Contact Structures”. In: Geom. Topol. 5.1 (Oct. 2000), pp. 309–368.issn: 1364-0380. 30

Abstract (if available)

Abstract In this thesis, we focus on establishing well-known results of tight contact structures on some manifolds, focusing in particular on low dimensional (3−dimensional) manifolds. We introduce the basic dichotomy between tight and overtwisted structures, noting the differences in classifying both structures. In particular, we focus on the classification of tight contact structures. In order to do so, we introduce and make use of some knot invariants so that we may develop the notion of the bypass, a tool developed by Ko Honda. Finally, making use of bypasses, we state many of the basic classification results of the field and end with a classification of the tight contact structures on the Solid Torus, T = S1 × D2.

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Creator Kim, David Jinkyum (author)

Core Title Tight contact structures on the solid torus

School College of Letters, Arts and Sciences

Degree Master of Arts

Degree Program Mathematics

Degree Conferral Date 2022-05

Publication Date 04/16/2022

Defense Date 04/15/2022

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag bypasses,contact geometry,differential geometry,knot theory,OAI-PMH Harvest,tight contact structures

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Advisor Crombecque, David (committee chair), Cremaschi, Tommaso (committee member), Ganatra, Sheel (committee member)

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