Geometric criteria for overtwistedness
GGEOMETRIC CRITERIA FOR OVERTWISTEDNESS
ROGER CASALS, EMMY MURPHY, AND FRANCISCO PRESAS
Abstract.
In this article we establish efficient geometric criteria to decide whether a con-tact manifold is overtwisted. Starting with the original definition, we first relate overtwisteddisks in different dimensions and show that a manifold is overtwisted if and only if the Leg-endrian unknot admits a loose chart. Then we characterize overtwistedness in terms of themonodromy of open book decompositions and contact surgeries. Finally, we provide severalapplications of these geometric criteria. Introduction
Symplectic and contact topology intertwine the global behaviour from differential topologywith subtle rigid geometric structures [1, 21, 25, 34]. Both sides of the flexible–rigid di-chotomy [25] feature prominently in the field; the present work belongs to the flexible sideof contact topology. The main result in this article is Theorem 1.1, which characterizes thecontact structures satisfying the parametric h –principle [4, 35] in terms of the different geo-metric notions existing in the literature in contact topology, including adapted open bookdecompositions [17, 31], contact surgeries [23, 55], loose Legendrian submanifolds [44] andobstructions to fillability [47]. This work proves that the existence of these objects withadditional properties, which conjecturally led to an h –principle, does indeed imply that the h –principle is satisfied.The geometric criteria stated in Theorem 1.1 can be verified in several interesting cases, andin particular we provide the first explicit examples of overtwisted contact manifolds in higherdimensions. Theorem 1.1 has been used in a variety of contexts, such as [10, 27, 39], and westrongly believe it captures the most efficient ways to detect overtwistedness. The centralgoal of this article is to prove Theorem 1.1.1.1. The main theorem.
A contact structure on a (2 n − Y is a maximally non–integrable hyperplane distribution ξ ⊆ T Y , a succinct introductionto the properties of such hyperplane distributions can be found in [1, Chapter 4]. Thereexists a remarkable class of contact structures, which has been introduced in [4, Definition3.6] in any dimension, called the overtwisted contact structures. Generalizing the originaldefinition and results in the 3-dimensional case [22], it is shown in [4, Theorem 1.2] thatovertwisted contact structures satisfy a parametric h -principle [26], i.e. their classificationup to contact isotopy coincides with the classification of homotopy classes of almost contactstructures. This classification then becomes a strictly algebraic topological problem whichcan be solved via obstruction theory [36]. The definition of the class of overtwisted contactstructures provided in [4, Definition 3.6] will be reviewed in Section 3, but for the time beingthe reader can think of them as contact structures containing a certain contact hypersurfacegerm in the same vein that the 3–dimensional case [22].The result in the article [4, Theorem 1.1] does demonstrate the existence of overtwistedcontact structures homotopic to any almost contact structure, but a crucial drawback to theexistence proof is that the construction is not explicit. In consequence, there were no explicitexamples of closed overtwisted contact manifolds of dimension 2 n − ≥
5, and the techniquesused in [4] give no criterion in order to show that a given manifold is overtwisted, other thana direct application of the definition, which to the knowledge of the authors has never been
Mathematics Subject Classification.
Primary: 57R17. Secondary: 53D10, 53D15. a r X i v : . [ m a t h . S G ] N ov ROGER CASALS, EMMY MURPHY, AND FRANCISCO PRESAS done. The geometric criteria Theorem 1.1 entirely solves these problems by providing anumber of equivalent conditions which characterize overtwistedness. In addition, it bringstogether different geometric objects used to study contact structures, thus establishing aunifying ground for flexibility across the different facets of contact topology.Let us now state our main result. The notions and notations used in the following statementwill be explained in the rest of this Subsection 1.1.
Theorem 1.1.
Let ( Y, ξ ) be a contact manifold of dimension n − ≥ and α ot ∈ Ω ( R ) a –form such that ( R , ker α ot ) is an overtwisted contact structure.Then the following conditions are equivalent:1. The contact structure ( Y, ξ ) is overtwisted.2. There is a contact embedding of ( R × C n − , ker { α ot + λ st } ) into ( Y, ξ ) .3. The standard Legendrian unknot Λ ⊆ ( Y, ξ ) is a loose Legendrian submanifold.4. ( Y, ξ ) contains a small plastikstufe with spherical core and trivial rotation.5. There exists a contact manifold ( Y (cid:48) , ξ (cid:48) ) and a loose Legendrian submanifold Λ ⊆ ( Y (cid:48) , ξ (cid:48) ) such that ( Y, ξ ) is contactomorphic to the contact (+1) –surgery of ( Y (cid:48) , ξ (cid:48) ) along Λ .6. There exists a negatively stabilized contact open book compatible with ( Y, ξ ) . (cid:3) We will momentarily discuss the different items in the statement of Theorem 1.1. However,we first note that the second condition in the geometric criteria is actually verifiable.
Remark 1.2.
There exists a positive constant R = R ( n, α ot ) ∈ R + , which only depends onthe choice of overtwisted contact form α ot and the dimension of the contact manifold ( Y, ξ )such that the second condition in Theorem 1.1 on the existence of the contact embedding of( N ∞ , ξ ∞ ) = ( R × C n − , ker { α ot + λ st } )into the contact manifold ( Y, ξ ) is equivalent to the existence of a contact embedding of( N R , ξ R ) = ( R × D n − ( R ) , ker { α ot + λ st } )into the contact manifold ( Y, ξ ), as it follows from the h–principle [4, Corollary 1.4]. Thiscritical radius R ( n, α ot ) is a finite number, and thus in order to verify overtwistedness ofthe contact manifold ( Y, ξ ) using the second characterization in Theorem 1.1 it suffices tofind contact embeddings of ( N ρ , ξ ρ ) into ( Y, ξ ) for any finite radius ρ ∈ R + . Then choosing ρ ∈ R + such that R ( n, α ot ) ≤ ρ yields the existence of a contact embedding of the infiniteradius contact domain ( N ∞ , ξ ∞ ) into ( Y, ξ ). (cid:3) Let us now describe the elements entering in the statement of Theorem 1.1 above in moredetail. The 1–form λ st ∈ Ω ( C n − ) denotes the standard Liouville form on both the opendisk D n − ( R ) and the complex Euclidean space C n − , which in the standard coordinates( x , y , . . . , x n − , y n − ) = ( r , θ , . . . , r n − , θ n − ) is expressed as λ st = 12 n − (cid:88) i =1 ( x i dy i − y i dx i ) = 12 n − (cid:88) i =1 r i dθ i . There are infinitely many choices for an overtwisted contact form α ot for the real Euclideanspace R ( z, r, θ ), the most used in the literature is the rotationally symmetric z –invariantform α ot = cos( r ) dz + r sin( r ) dθ, but note that the statement of Theorem 1.1 only requires a contact form α ot defining an over-twisted contact structure on R , not necessarily contactomorphic to ker(cos( r ) dz + r sin( r ) dθ ).It is interesting to observe that the classification of overtwisted contact structures in R isunderstood [24], and thus such choices can be readily classified as well. EOMETRIC CRITERIA FOR OVERTWISTEDNESS 3
The Statement of Theorem 1.1.
Theorem 1.1 proves the equivalence between differ-ent notions of flexibility in contact topology, and the several geometric objects that appearin the statement of Theorem 1.1 have been introduced by many authors in different works,who we now credit.The definition of a higher–dimensional overtwisted disk featuring in the first item (1) initiallyappears in the article [4, Definition 3.6]. The 3-dimensional overtwisted disk was initiallyintroduced by Y. Eliashberg in [22, Section 1.4].The second item (2) in Theorem 1.1 features the 3-dimensional contact structure ( R , ker α ot ),as discussed above, and the contact product ( R × C n − , ker { α ot + λ st } ). This latter higher-dimensional contact manifold should be seen as an infinitely large neighborhood of the contactsubmanifold R × { } . The first breakthrough in the study of contact neighborhoods iscontained in the article [48], particularly [48, Corollary 13], where it is proven that a largeenough neighborhood of a 3-dimensional overtwisted contact manifold contains a generalizedplastikstufe.The article [48] has been of central importance for higher-dimensional contact geometry, andthe statement of the equivalence (1) ⇐⇒ (2) has its roots in [48]. In particular, the first-named author is grateful to K. Niederkr¨uger for several discussions on [48] and related topics.The third-named author is also thankful to him for many useful conversations. Remark 1.3.
The equivalence (1) ⇐⇒ (2) in Theorem 1.1 is crucial in the present proofof Theorem 1.1. Indeed, all other equivalences with the first item (1) use the equivalence(1) ⇐⇒ (2), which we shall prove first in Section 3. In this sense, the equivalence (1) ⇐⇒ (2)is the core of Theorem 1.1. (cid:3) Regarding the third item (3), the notion of a loose Legendrian submanifold was introduced inthe article [44, Definition 4.3], and the definition of a plastikstufe, appearing in the fourth item(4), was given by K. Niederkr¨uger in the article [47, Section 1]. The concept of (+1)–surgeryin the fifth item (5) is first detailed in the articles [23, 55] and the notions of a compatibleopen book and a negative stabilization appearing in the sixth item (6) were introduced byE. Giroux [17, 31].In short, we now describe and contextualize these geometric concepts before delving intotheir more technical nature in Section 2.The standard Legendrian unknot Λ ⊆ ( R n − , ξ st ) is defined to be the Legendrian sphereΛ = { y i = 0 : i = 1 , . . . , n } ∩ S n − ⊆ ( S n − , ξ st ) \ { point } ⊆ C n [ x , y , . . . , x n , y n ] , where the standard contact structure ( S n − , ξ st ) is defined by restriction of the Liouvilleform λ st to the unit sphere S n − = { ( x , y , . . . , x n , y n ) ∈ C n : (cid:107) x (cid:107) + (cid:107) y (cid:107) = 1 } ⊆ C n andwe have used the natural contactomorphism ( R n − , ξ st ) ∼ = ( S n − , ξ st ) \ { point } , detailed forinstance in [29, Prop. 2.1.8]. Then the standard Legendrian unknot Λ ⊆ ( Y, ξ ) is defined bythe inclusion of a Darboux chart Λ ⊆ ( R n − , ξ st ) ⊆ ( Y, ξ ), all of which are isotopic.The concept of a loose
Legendrian submanifold, which appears in the third item (3) is firststudied in the article [44], and the reader might also be interested in [10, 11, 16, 45]. Thefact that a Legendrian submanifold is loose is characterized by the existence of a certainpiece that the Legendrian might or might not have: Theorem 1.1 states that if the mostbasic Legendrian, the Legendrian unknot, already contains such piece, then the ambientmanifold (
Y, ξ ) is overtwisted. Thus, we are relating the flexibility h –principle exhibitedby loose Legendrian submanifolds [44] with the h –principle satisfied by overtwisted contactstructures [4, Theorem 1.2]. Note that with the techniques developed in [10, 11] it is muchsimpler to verify that the Legendrian unknot is loose than proving overtwistedness by usingthe definition. ROGER CASALS, EMMY MURPHY, AND FRANCISCO PRESAS
The plastikstufe , appearing in the fourth item (4), is an n -dimensional smooth submanifold P ⊆ ( Y, ξ ) such that the germ of the ambient contact structure ξ in an open neighborhood of P is given by an explicit local model, inspired by the definition of the overtwisted disk [22, 34] inthe three–dimensional case. The plastikstufe was first defined in the article [47] and shown tobe an obstruction to symplectic fillability in higher–dimensions, in the same manner that theovertwisted 2–disk in a contact 3–fold obstructs the existence of a 4–dimensional symplecticfilling. The technical definitions of small and trivial rotation were first introduced in [45],where the existence of a plastikstufe is studied in relation to loose charts for Legendrianssubmanifolds. Both hypotheses are technical, and according to the recent work [39] they canactually be removed. The statement of Theorem 1.1 concerning the plastikstufe thus relatesthe existence of an explicit n -dimensional contact germ P , built as a family of overtwisted2–disks, with higher–dimensional overtwistedness [4]. This has meaningful advantages, suchas the fact that there are simple geometric constructions of plastikstufes [8, 15, 48] and it isoften simpler to find the contact germ P than the higher–dimensional overtwisted contactgerm [4, Definition 3.6].In the fifth item of Theorem 1.1, overtwisted contact manifolds are characterized as thecontact manifolds which admit a contact surgery presentation in which one of the Legendrian(+1)–components of the surgery link admits a loose chart in the complement of the othercomponents. The definition of a contact (+1) –surgery along a Legendrian sphere is implicit inthe theory of Weinstein handle attachments [16, 23, 55]; it is the surgery induced by attachinga concave handle to a compact piece of the symplectization. In the higher dimensional caseit was studied in more depth in [2], where the implication (5) ⇒ (1) proven in Theorem 1.1is stated as Conjecture 9.16. The essential fact that the reader should remember regardingthis fifth item is that we prove that contact (+1)–surgery along a loose Legendrian spherealways yields an overtwisted contact manifold. The study of contact structures from thesurgery viewpoint is well–understood in the three–dimensional case [50] and it is currently adeveloping field of interest in higher–dimensions [10, 11].Finally, compatible open books appear as the sixth geometric criteria to detect overtwistedness.In order for this characterization in Theorem 1.1 to apply we also suppose that ( Y, ξ ) is aclosed contact manifold. The notion of an open book compatible with a contact structure(
Y, ξ ) was first introduced by E. Giroux in his study of the correspondance between openbooks and contact structures [17, 31]. In brief, it states that an appropriate open bookdecomposition of the smooth manifold Y determines a contact structure ξ , and converselyevery contact manifold ( Y, ξ ) admits such an adapted open book decomposition.The open books compatible with a contact structure (
Y, ξ ) admit a contact operation: theycan be positively stabilized or negatively stabilized. The resulting open books induce twocontact structures ξ + and ξ − on the smooth manifold Y . The positive stabilization ( Y, ξ + )is an operation which yields a contact structure contactomorphic to ( Y, ξ ), but the contactstructure (
Y, ξ − ) resulting from a negative stabilization is oftentimes not contactomorphicto ( Y, ξ ). In particular, the negative stabilization of a contact structure is known to havevanishing symplectic field theory [6, 7]. In particular, using [7, Theorem 1.3], Theorem 1.1implies that the contact homology of an overtwisted contact manifold vanishes.These geometric objects will be discussed in more technical detail in the subsequent Section2, but we hope that the above description provides some context for Theorem 1.1 and it helpsthe reader to navigate between the diverse range of objects in its statement.1.3.
The argument for Theorem 1.1.
First, there are six equivalences stated in Theorem1.1 and there is by no means a canonical approach nor a natural order to prove them. How-ever, we have chosen a route that in our perspective most enlightens the connection betweenthe different geometric objects and also minimizes the need for a thorough understanding ofthe article [4].
EOMETRIC CRITERIA FOR OVERTWISTEDNESS 5
To begin with, the argument we use to prove Theorem 1.1 is in its entirety an induction in thedimension (2 n − Y, ξ ), whichconstitutes the base case, and we will then show that if the statement is true for any smoothmanifold Y with dim( Y ) = 2 n − n − ⇐⇒ (2) is the content of Theorem 3.2, proved in Section 3.- The equivalences (1) ⇐⇒ (3) ⇐⇒ (4): the implication (3) ⇒ (1) is proven in Section 4as a consequence of Theorem 4.5. The main ingredient is Lemma 4.2, which is wherethe inductive hypothesis is used. Note that (4) ⇒ (3) follows from [45, Theorem 1.1].- The equivalence (1) ⇐⇒ (5) is proven in Section 5 where we show the implication(5) ⇒ (4).- The equivalence (1) ⇐⇒ (6) is shown in Section 6, with an argument proving (6) ⇒ (3).The implications we have emphasized above are the ones that require new ideas and tech-niques. The remaining implications needed in order to obtain the equivalences follow fromthe h –principle: the relative parametric h –principle, [4, Theorem 1.1] and [4, Theorem 1.2],does imply (1) ⇒ (2), (1) ⇒ (3) and (1) ⇒ (4), and the implications (1) ⇒ (5) and (1) ⇒ (6) arenot hard. The real effort, as in any result characterizing an h –principle, is to prove the con-verse implications by constructing an overtwisted disk from a priori weaker geometric object.Section 7 contains the proof of Theorem 1.1 gathering the equivalences above. Remark 1.4.
Here is an alternative route that two of the authors have also used in talks sinceit minimizes the use of the h –principle [4, Theorem 1.1]. First, one proves the equivalence(1) ⇐⇒ (2) with the argument in this article, and then proceeds with the following sequence:- The equivalence (3) ⇐⇒ (6) can be proven directly with the techniques we develop inSection 4. This is a self–contained relation.- The equivalence (2) ⇐⇒ (3) then can be established by proving (3) ⇒ (2) from ourcobordism argument in Section 4, and deducing the implication (2) ⇒ (3) by adaptingthe classical 3–dimensional destabilizing argument in the presence of an overtwisteddisk [45].- The implications (5) ⇒ (2) and (5) ⇒ (3) can be proven directly by studying Weinsteinhandle attachment in detail [10, 11, 27], and finally the implication (6) ⇒ (5) followsfrom the fact that the zero section in OB( T ∗ S n , id) is a loose Legendrian submanifold[11].Hence the equivalences (2) ⇐⇒ (3) ⇐⇒ (4) ⇐⇒ (5) ⇐⇒ (6) do not require the overtwisted h –principle [4, Theorem 1.2]. Nevertheless, they require the loose Legendrian h –principle [44]and the main arguments in this article. Thus, from a flexible perspective it is neater todirectly use the h –principle [4, Theorem 1.2] to immediately conclude the converses, whichalso explains our choice of strategy. (cid:3) Organization.
The article contains eight sections, which we have distributed as follows.First, Section 2 provides the required background in contact topology in order to follow thearticle. Then, Sections 3, 4, 5 and 6 contain the main results for the proof of Theorem 1.1,these results are divided in terms of the equivalences they are used to prove in Theorem 1.1.Section 7 contains the proof of Theorem 1.1. Section 8 details two applications of Theorem1.1.Section 3 proves the first equivalence (1) ⇐⇒ (2), Section 4 establishes the two equivalences(1) ⇐⇒ (3) ⇐⇒ (4), Section 5 then proves the equivalence (1) ⇐⇒ (5) and Section 6 concludeswith the proof of the equivalence (1) ⇐⇒ (6). Each of these sections also contains results ROGER CASALS, EMMY MURPHY, AND FRANCISCO PRESAS that can be of interest on their own. In particular, we believe that the connection developedin Section 6 is relevant for high–dimensional contact topology, as the subsequent work [11]hopefully illustrates. Finally, Section 8 gives some applications of Theorem 1.1 to contactsqueezing and constructions of Weinstein cobordisms. (cid:3)
Acknowledgements.
We are grateful to M.S. Borman and Y. Eliashberg for manyuseful discussions. We thank O. van Koert, O. Plamenevskaya and K. Siegel for valuableconversations and U. Varolgunes and C. Wendl for comments on the article.F. Presas is indebted to R. Casals for suggesting to study the first equivalence in Theorem1.1. He would also like to acknowledge him for pushing this project with so much insight anddetermination.R. Casals is supported by the NSF grant DMS-1841913 and a BBVA Research Fellowship andE. Murphy is supported by the NSF grant DMS-1510305 and a Sloan Research Fellowship.F. Presas is supported by the Spanish Research Projects SEV–2015–0554, MTM2016–79400–P and MTM2015–72876–EXP. (cid:3) Preliminaries
In this section we detail a number of relevant definitions and results in high–dimensionalcontact topology which are used along the article. In particular, we have included the def-initions of the geometric objects in the statement of Theorem 1.1. These preliminaries arenecessary both for the understanding of its statement as well as its proof.2.1.
Loose Legendrians.
The notion of a loose Legendrian submanifold appears in theequivalence (1) ⇐⇒ (3), where overtwistedness is characterized in terms of the Legendrianunknot being a loose Legendrian; let us now define this class of Legendrian submanifolds.First, let B ⊆ ( R , ξ st ) be the round 3–dimensional ball in a contact Darboux chart and letΛ S ⊆ ( R , ξ st ) be the 1–dimensional stabilized Legendrian arc depicted in Figure 1. Figure 1.
The front projection of a stabilized Legendrian arc.Then consider a closed manifold Q and an open neighborhood O p ( Z ) ⊆ T ∗ Q of the zerosection Z ⊆ T ∗ Q , and note that the product smooth submanifoldΛ S × Z ⊆ ( B × O p ( Z ) , ker( α st + λ st ))is a Legendrian submanifold of the contact structure ker( α st + λ st ). This is the crucial localmodel that defines looseness, as precised in the following definition. Definition 2.1.
The contact pair ( B × O p ( Z ) , Λ S × Z ) endowed with the contact structureker( α st + λ st ) is said to be a loose chart , where Z ⊆ T ∗ Q is the zero section of the cotangentbundle ( T ∗ Q, λ st ) and Q an arbitrary closed manifold. EOMETRIC CRITERIA FOR OVERTWISTEDNESS 7
Let Λ ⊆ ( Y, ξ ) be a Legendrian submanifold in a contact manifold with dim( Y ) ≥
5. TheLegendrian Λ is loose in the contact manifold (
Y, ξ ) if there is an open set V ⊆ Y such thatthe contact pair ( V, V ∩ Λ) is contactomorphic to a loose chart. (cid:3)
Loose Legendrians were classified up to Legendrian isotopy in the article [44], and althoughthe definition presented above differs slightly from the one presented in [44], both definitionsare equivalent [45, Section 4.2]. The following property, which is satisfied by loose Legendri-ans, but not by all Legendrians, will be most useful for us. It constitutes the characterizingproperty of loose Legendrians, and as a basic form of an h –principle [26] it indicates thatthis class of Legendrians is related to the flexible side of contact topology. Theorem 2.2 ([44]) . Let Λ −→ ( Y, ξ ) be a loose Legendrian submanifold with a loose chart U ⊆ ( Y, ξ ) . Let f t : Y −→ Y be a smooth isotopy such that f = id is the identity map, andthe restriction f t | Λ ∩ U = id | Λ ∩ U is the identity map on Λ ∩ U for all t ∈ [0 , .Then there exists a contact isotopy g t : Y −→ Y such that g t | Y \ U is C –close to f t | Y \ U . Theorem 2.2 is used in Proposition 2.12 below, which in turn is needed in Theorem 4.4, prov-ing the implication (3) ⇒ (1), and in Theorem 8.6, one of the main applications of Theorem1.1. Remark 2.3.
The h –principle for loose Legendrian embeddings, Theorem 2.2, does notprovide an isotopy which is C -close near the loose chart U ⊆ ( Y, ξ ). However, our argumentswill not rely on that, and Theorem 2.2 will suffice for our purposes. (cid:3)
Theorem 2.2 is the result the reader should have in mind whenever a loose Legendriansubmanifold appears along the article, and it should be read as the fact that loose Legendriansbehave according to their smooth topology [11, 44].2.2.
The plastikstufe.
The plastikstufe is a particular germ of a contact submanifold ina contact manifold, which coincides with the overtwisted 2–disk in the 3–dimensional case[22, 34, 47] . It appears in Theorem 1.1 as one of the characterizations of higher–dimensionalovertwistedness, and we now provide the details on its definition, first introduced in thearticle [47].
Remark 2.4.
The initial purpose of the plastikstufe was to provide a higher–dimensionalobject which obstructs symplectic fillings, in the same manner that the existence of an over-twisted 2–disk in a contact 3–fold prevents the existence of a 4–dimensional symplectic filling[34]. This leads to the geometric idea of considering a parametric family of overtwisted 2–disks and, with the appropriate count of dimensions of moduli spaces, the definition of theplastikstufe [47]. (cid:3)
Let O p ( D ) ⊆ ( R , ξ ot ) be a contact neighborhood of an overtwisted disk for any overtwistedcontact structure ξ ot = ker α ot . Definition 2.5.
Let Z be a closed manifold and O p ( Z ) ⊆ T ∗ Z a neighborhood of the zerosection. The contact manifold ( O p ( D ) × O p ( Z ) , ker( α ot + λ st )) is said to be a plastikstufe .The submanifold Z ⊆ O p ( D ) × O p ( Z ) is the core of the plastikstufe. (cid:3) The authors have provided constructions of the plastikstufe [10, 15], and they arise naturallyin the contact divisor sum of two contact manifolds along overtwisted contact divisors.The remarkable fact about plastikstufes is that, thanks to Theorem 1.1, not only they serveas obstructions to symplectic fillability but actually can be used to detect overtwistedness inany dimension.The proof that we provide holds for a large class of plastikstufes, but there are two technicalhypothesis that are needed in order for the argument to work. In order to state one of thesetwo hypotheses, we introduce the following notion.
ROGER CASALS, EMMY MURPHY, AND FRANCISCO PRESAS
Given a contact manifold (
Y, ξ ) and a smooth Legendrian embedding f : Λ −→ Y , the rotation class of the Legendrian embedding f , also called the rotation class of Λ, is thehomotopy class of the induced injective bundle map T f : T Λ −→ f ∗ ξ , considered as a mapin the space of Lagrangian bundle monomorphisms. See [44, Definition A.1] and [45, Section4] for a more detailed discussion on the rotation class. Equipped with this notion, the twotechnical hypothesis are given in the following definition. Definition 2.6.
Let Λ l ⊆ O p ( D ) be an open leaf of the characteristic foliation of theovertwisted disk. The plastikstufe ( O p ( D ) × O p ( Z ) , ker( α ot + λ st )) has trivial rotation ifthe open Legendrian submanifold Λ l × Z has trivial rotation class.Also, a plastikstufe O p ( D ) × O p ( Z ) ⊆ ( Y, ξ ) is said to be small if it is contained in asmooth ball in ambient manifold Y . (cid:3) Note that the rotation class of the Legendrian Λ l × Z is well defined since the hyperplane field ξ has a unique framing on the smooth ball up to homotopy. Also, observe that in the case Q = S n − , a plastikstufe O p ( D ) × O p ( Z ) ⊆ ( Y, ξ ) is both small and has trivial rotationif and only if Λ l × Z , which is a Legendrian annulus [0 , × S n − , can be included intoa Legendrian disk. Then an open neighborhood of the union of the Legendrian disk andthe plastikstufe is diffeomorphic to a smooth ball, and since a Legendrian disk has a uniqueframing it induces a trivial framing on its boundary collar.The following theorem from [45] gives in particular the implication (4) ⇒ (3). Theorem 2.7 ([45]) . Let
P ⊆ ( Y, ξ ) be a small plastikstufe with spherical core and trivialrotation, and Λ ⊆ ( Y, ξ ) a Legendrian submanifold disjoint from P .Then the Legendrian Λ ⊆ ( Y, ξ ) is a loose Legendrian submanifold. Remark 2.8.
The sphericity hypothesis of the core of the plastikstufe in Theorem 1.1 can bereadily generalized, but being able to remove the hypothesis on its smallness requires moreeffort. This has been recently achieved by Y. Huang [39] using Theorem 1.1. (cid:3)
Now that we have defined the contact geometric objects appearing in characterizations 3 and4, we must address Weinstein structures since they have a fundamental role in the proof ofthe equivalence (1) ⇐⇒ (3) in Theorem 1.1.2.3. Weinstein manifolds.
This subsection contains a succinct treatment on Weinsteincobordisms, where we state the results that will be used in the proof of Theorem 1.1 relatedto Weinstein structures. The reader is invited to study the thorough account [16] for furtherresults on these structures.First, the study of Weinstein structures aims at the understanding of contact and symplecticstructures from the Morse theoretical viewpoint. The theory of Morse functions in smoothtopology intertwines with contact and symplectic topology by requiring a compatibility con-dition between the Morse functions and the symplectic structure [16]. The objects of interestare the content of the following definition.
Definition 2.9. A Weinstein cobordism is a triple (
W, λ, f ), where the pair (
W, dλ ) is acompact symplectic manifold with boundary, f : W −→ [0 ,
1] is a Morse function such that ∂W = ∂ − W ∪ ∂ + W = f − (0) ∪ f − (1), and the vector field V λ symplectic dual to the Liouvilleform λ is a gradient–like vector field for the Morse function f . (cid:3) From the definition it follows that the 1–form λ | f − ( c ) is a contact form on the submanifold f − ( c ) for any regular value c ∈ [0 , D kp associatedto any critical point p of f satisfies λ | D kp = 0. In particular the submanifold D kp is isotropicand thus ind( p ) = k (cid:54) n = 12 dim W. EOMETRIC CRITERIA FOR OVERTWISTEDNESS 9
Critical points with index strictly less than n are called subcritical , and a subcritical Weinsteincobordism (
W, λ, f ) is one where all critical points of f are subcritical.In the case that c ∈ [0 ,
1] is a regular value, the intersection Λ cp = D kp ∩ f − ( c ) is an isotropicsubmanifold of the contact manifold ( f − ( c ) , ker λ ). In case c ∈ [0 ,
1] is a critical value with aunique critical point p ∈ W , the Weinstein cobordism ( f − ([ c − ε, c + ε ]) , λ, f ) is determined,up to homotopy through Weinstein structures, by the contact manifold ( f − ( c − ε ) , ker λ )and the (parametrized) isotropic submanifold Λ c − εp , together with a framing of the symplecticnormal bundle, which is necessarily trivial. Hence the contact manifold ( f − ( c + ε ) , ker λ ) isdetermined up to contactomorphism, and it is said to be obtained from ( f − ( c − ε ) , ker λ )by contact surgery along the isotropic sphere Λ c − εp . Notice that ( f − ( c − ε ) \ Λ c − εp , ker λ ) hasa natural contact inclusion into ( f − ( c + ε ) , ker λ ), defined by the flow of the gradient-likevector field V λ . We refer to the monograph [16] for proofs of these statements and a morecomplete discussion of Weinstein handle attachments.In the particular case in which c ∈ [0 ,
1] is a critical value of f with a unique critical point p of index n then Λ c − εp ⊆ ( f − ( c − ε ) , ker λ ) is a Legendrian submanifold. If this Legendrian isloose, we say that the critical point p is a flexible critical point. Definition 2.10.
A Weinstein cobordism (
W, λ, f ) is said to be flexible if every critical pointof f is either subcritical or flexible. Remark 2.11.
In dim( W ) = 4, a critical point p is called flexible if the Legendrian Λ c − εp has overtwisted complement. This dimension is however not discussed in this paper. (cid:3) By Definition 2.10, every subcritical Weinstein cobordism is flexible. The importance ofDefinition 2.10 is that flexible Weinstein manifolds are completely classified [16, Chapter 14].In our case, we use flexible Weinstein cobordisms in relation to overtwisted contact manifolds.The first result we need to prove in this direction, which will be used in Section 4 for part ofthe proof of Theorem 1.1, is the following proposition.
Proposition 2.12.
Let ( W, λ, f ) be a flexible Weinstein cobordism such that ( ∂ − W, ker λ ) isan overtwisted contact manifold. Then the contact manifold ( ∂ + W, ker λ ) is overtwisted.Proof. First, split the cobordism (
W, λ, f ) into cobordisms with a single critical point W = f − ([0 , c ]) ∪ . . . ∪ f − ([ c s , , for 0 < c < . . . < c s < . The resulting attaching spheres Λ j ⊆ ( f − ( c j ) , ker λ ), for 1 ≤ j ≤ s , are either subcriti-cal or loose Legendrians submanifolds, and we will now show by induction that each con-tact manifold ( f − ( c j ) , ker λ ) is overtwisted. The j = 1 case follows from the fact that( ∂ − W, ker λ ) is overtwisted, and the case j = s case implies the result. The contact manifold( f − ( c j +1 ) , ker λ ) is obtained from ( f − ( c j ) , ker λ ) by a single Weinstein surgery along theisotropic sphere Λ j , and any smooth isotopy of Λ j can be C –approximated by a contactisotopy. Indeed, if Λ j is subcritical this follows from the h –principle for subcritical isotropicsubmanifolds [35], and if Λ j is a loose Legendrian this is Theorem 2.2. In particular, wecan find a contact isotopy which makes the attaching isotropic sphere Λ j disjoint from anyovertwisted disk in ( f − ( c j ) , ker λ ). (cid:3) Finally, we define a vertical connected sum operation of Weinstein cobordisms. For that, let( W , λ , f ) and ( W , λ , f ) be two Weinstein cobordisms with non-empty negative boundary,and choose two points p ∈ ∂ − W and p ∈ ∂ − W which are not in the descending manifoldof any critical point. Let γ and γ be the image curves of the points p and p by theflow of the gradient-like vector fields V λ and V λ , and thus γ ⊆ W and γ ⊆ W are twocurves which intersect transversely every level set of their corresponding ambient cobordismsexactly once. We define the connected sum cobordism as the smooth cobordism W W = ( W \ O p ( γ )) ∪ ( W \ O p ( γ )) , where the union glues a collar neighborhood of ∂ O p ( γ ) to a collar neighborhood of ∂ O p ( γ )with a map that pulls back the Liouville form λ to the Liouville form λ and the Morsefunction f to the Morse function f . The smooth manifold W W then inherits a Weinsteinstructure ( W W , λ, f ), the critical set of f being the union of the critical sets of f and f ,and every regular level set ( f − ( c ) , ker λ ) being contactomorphic to the contact connectedsum ( f − ( c ) , ker λ ) f − ( c ) , ker λ ). The Weinstein manifold ( W W , λ, f ) is the verticalconnected sum of ( W , λ , f ) and ( W , λ , f ). Remark 2.13.
This operation is used in [45, Section 5] to construct contactomorphismsusing the flexible Weinstein h -cobordism theorem [16], we use the vertical connected sum inSections 4 and 8. (cid:3) The connected sum cobordism and flexible Weinstein structures have a crucial role in thecobordism arguments proving Theorem 4.4 and Theorem 8.6. Note also the Weinstein cobor-disms are the natural context in which contact surgeries, either positive or negative, arise.Further discussion on contact (+1)–surgeries appears in Section 5, but for now we move for-ward and complete the preliminaries concerning the objects appearing in Theorem 1.1, thatis, we discuss the statement of the sixth equivalence (1) ⇐⇒ (6) in Theorem 1.1, concerningopen book decompositions.2.4. Open book decompositions.
Open books compatible with a contact structure have acentral role in contact topology [31, 54]. Theorem 1.1 states that it is possible to characterizehigher–dimensional overtwistedness in terms of compatible open book decompositions. Inthis subsection we review the basic facts about open book decompositions relevant for thestatement of Theorem 1.1 and its proof.Let (
W, λ ) be a Liouville domain, i.e. an exact symplectic manifold with the Liouville vectorfield V λ outwardly transverse to the smooth boundary ∂W , and ϕ : W −→ W a compactlysupported exact symplectomorphism, such that ϕ ∗ λ = λ + dh for some compactly supportedfunction h ∈ C ∞ c ( W ). The triple ( W, λ, ϕ ) is an open book decomposition [17, 31], and theLiouville domain (
W, λ ) is referred to as its page .Every open book decomposition (
W, λ, ϕ ) canonically defines a contact manifold (
Y, ξ ), whichis constructed as the mapping torus Y = W × [0 , / ( x, ∼ ( ϕ ( x ) , ∪ ∂W × S ∂W × D ξ = ker (cid:0) ( λ + Kdθ + θdh ) ∪ ( λ | ∂W + Kr dθ ) (cid:1) . for a sufficiently large K ∈ R + . We write ( Y, ξ ) = OB(
W, λ, ϕ ) to denote this relationship,and say that (
Y, ξ ) is compatible with or supported by the open book ( W, λ, ϕ ). Notice thatthe construction readily implies the contactomorphism OB(
W, λ, ϕ ) ∼ = OB( W, λ, ψ ◦ ϕ ◦ ψ − )for any symplectomorphism ψ .The remarkable feature of open book decomposition in relation to contact structures is thatthe converse also holds. This is E. Giroux’s existence theorem [31]: Theorem 2.14 ([31]) . Every contact manifold ( Y, ξ ) can be presented as ( Y, ξ ) = OB(
W, λ, ϕ ) ,and there exists a Morse function f : W −→ [0 , such that ( W, λ, f ) is a Weinstein manifold. Hence the study of contact manifolds can be approached as the study of Weinstein structuresand their compactly supported symplectomorphisms. Let (
W, λ ) be a Liouville manifold,and suppose it contains a parametrized Lagrangian sphere L ⊆ ( W, λ ). We denote the Dehntwist [17, 51] around the Lagrangian sphere L by τ L ∈ Symp c ( W ), where we have extended EOMETRIC CRITERIA FOR OVERTWISTEDNESS 11 the Dehn twist τ L ∈ Symp c ( O p ( L )) to a compactly supported symplectomorphism of theambient Weinstein manifold ( W, λ ) by using the identity on the complement W \ O p ( L ).Note that a Lagrangian sphere L is an exact Lagrangian the moment dim( L ) ≥
2, and thus L ⊆ ( W, λ ) defines a Legendrian sphere Λ in the contact manifold OB(
W, λ, ϕ ) obtained byintegrating the exact form λ | L . We denote the relation between the exact Lagrangian L andits Legendrian lift Λ by the equality( Y, ξ,
Λ) = OB(
W, λ, ϕ, L ) . This equality is defined to contain two statements. First, the contact manifold (
Y, ξ ) isadapted to the open book decomposition OB(
W, λ, ϕ ), where (
W, λ ) is the Liouville pageand ϕ ∈ Symp c ( W, λ ) is the symplectic monodromy. Second, the Legendrian Λ ⊆ ( Y, ξ ) isLegendrian isotopic to the Legendrian lift of an exact Lagrangian L ⊆ ( W, λ ) embedded inthe Liouville page.
Remark 2.15.
The equality (
Y, ξ,
Λ) = OB(
W, λ, ϕ, L ) is not an existence theorem, i.e. it isnot meant to state that for any
Legendrian Λ ⊆ ( Y, ξ ), there exists an open book OB(
W, λ, ϕ )supporting (
Y, ξ ) and an exact Lagrangian L ⊆ ( W, λ ) whose Legendrian lift is isotopic to Λ.The equality is only used when the existence of such L ⊆ ( W, λ ) is known and the equalityis the notation we use to specify that data. (cid:3)
The conjugation invariance stated above Theorem 2.14 now readsOB(
W, λ, ϕ, L ) = OB(
W, λ, ψ ◦ ϕ ◦ ψ − , ψ ( L ))as it can be readily verified by considering Λ as being near the page θ = 0.The following proposition relates Dehn twists of exact Lagrangian on the page of an openbook ( W, λ, ϕ ) with contact surgeries on the the associated contact manifold:
Proposition 2.16 ([40]) . Suppose that ( Y, ξ,
Λ) = OB(
W, λ, ϕ, L ) , then the contact manifold OB(
W, λ, ϕ ◦ τ L ) is obtained from ( Y, ξ ) by contact surgery along Λ . Note that both the mapping class [ τ L ] ∈ π Symp c ( W ) and the contact surgery along Λdepend on a parametrizations S n − ∼ = L and S n − ∼ = Λ, which is often non-canonical. Thediffeomorphism Λ ∼ = L is however canonically given by projection to the page ( W, λ ).The remaining ingredient to be discussed in relation to compatible open books is the sta-bilization procedure. Consider a Lagrangian disk D ⊆ ( W, λ ) with Legendrian boundary ∂D ⊆ ( ∂W, ker λ ) and attach a Weinstein handle to ( W, λ ) along the Legendrian sphere ∂D ,obtaining a new Weinstein manifold ( W ∪ H, λ (cid:48) ). Let us assume that the smooth parametriza-tion of the Legendrian boundary ∂D ⊆ ( ∂W, ker λ ) is such that the Lagrangian sphere S ,whose lower hemisphere is the Lagrangian disk D and whose upper hemisphere is the coreof the handle H , is a smoothly standard sphere. See [33, Section 6.3] and [55] for furtherdetails on Weinstein handle attachments.With this assumption, the new Weinstein manifold ( W ∪ H, λ (cid:48) ) contains a Lagrangian sphere S , smoothly standard, whose lower hemisphere is the Lagrangian disk D and whose upperhemisphere is the core of the handle H . Then, the new open book decomposition ( W ∪ H, λ (cid:48) , ϕ ◦ τ L ) is said to be the positive stabilization of ( W, λ, ϕ ) along D , and ( W ∪ H, λ (cid:48) , ϕ ◦ τ − L )is referred to as the negative stabilization of ( W, λ, ϕ ) along D [17, 40].Both the positive and the negative stabilization of an open book decomposition can be de-scribed as a contact connected sum. This description is the content of the following theorem. Theorem 2.17 (E. Giroux) . Let ( Y, ξ ) = OB(
W, λ, ϕ ) be a contact manifold, D ⊆ ( W, λ ) any Lagrangian disk with Legendrian boundary ∂D ⊆ ( ∂W, ker λ ) , and consider the contactstructure ( S n − , ξ − ) = OB( T ∗ S n − , τ − ) . Then the positive and negative stabilizations of ( W, λ, ϕ ) along D are diffeomorphic to Y .The positive stabilization is contactomorphic to ( Y, ξ ) , and the negative stabilization is con-tactomorphic to the contact connected sum ( Y S n − , ξ ξ − ) . (cid:3) This result is due to E. Giroux, but there is no detailed account on it available on theliterature, it is however well–known to experts, and an outline of the proof can be foundin the article [10, Proposition 2.6]. To the knowledge of the authors, E. Giroux and hiscollaborators are currently writing a more detailed source.3.
Thick neighborhoods of overtwisted submanifolds
In this section we begin the proof of Theorem 1.1 with the equivalence (1) ⇐⇒ (2). In trans-parent terms, the equivalence states that a contact manifold ( Y, ξ ) is overtwisted if and only ifit contains an overtwisted contact submanifold (
N, ξ ot ) with an infinite contact neighborhood.In fact, as noted in Remark 1.2 above, this is equivalent to the existence of an overtwistedcontact submanifold with an arbitrarily large, but finite, contact neighborhood. This lattercharacterization is the result we prove in the main theorem of this section, Theorem 3.2.In here, we are measuring the size of a contact neighborhood in terms of the maximal radiusthat can be achieved in the normal form for the contact structure in a neighborhood of acontact submanifold [29, Section 2.5.3]. Remark 3.1.
Technically, the radius exists as a global coordinate only if the conformalsymplectic normal bundle is trivial, however in order to detect overtwistedness it suffices torestrict the symplectic normal bundle to a neighborhood of an overtwisted disk in the contactsubmanifold, in which case the normal bundle becomes trivial and the distance to the zerosection provides a well–defined radius coordinate. (cid:3)
The equivalence (1) ⇐⇒ (2) thus becomes a statement about the behaviour of overtwistedcontact manifolds after a large enough thickening. This first equivalence in Theorem 1.1 isthe content of the following theorem. Theorem 3.2.
Let ( N n − , ker α ot ) be an overtwisted contact structure. Then for a suffi-ciently large radius R ∈ R + , the contact manifold ( N × D ( R ) , ker( α ot + λ st )) is overtwisted. Theorem 3.2 and its proof require some preliminaries, including the definition of the higher–dimensional overtwisted disk [4, Definition 3.6]. This definition is reviewed in Subsection 3.1,and we provide the necessary details in this article such that the reader does not need toread [4].Theorem 3.2 is proven in Subsection 3.2 for the case n = 2, where it is proven that asufficiently large neighborhood of an overtwisted contact 3–fold is an overtwisted contact 5–fold. Then we proceed with the general case of Theorem 3.2 in Subsection 3.3; this distinctionbetween the 5–dimensional case and higher–dimensions is not essential and we could havewritten a unified proof for any n ≥
2. However, encouraged by the suggestions of readers andreferees it seems that this distinction contributes to a better understanding of the result.
Remark 3.3.
The radius R ∈ R + that appears in the statement depends on the choice ofcontact form α ot for the contact manifold ( N, ker α ot ). This dependence is to be expectedsince there is no natural distance measurement associated to a hyperplane distribution andthe usual normalization is to fix a contact form. (cid:3) Let us now start by describing the contact germ that defines an overtwisted disk in higher–dimensions, which lies at the core of Theorem 3.2.
EOMETRIC CRITERIA FOR OVERTWISTEDNESS 13
Overtwisted Disks.
In order to define an overtwisted disk in an arbitrary dimension[4, Section 3] we first consider cylindrical coordinates( z, u , . . . , u n − , ϕ , . . . , ϕ n − ) ∈ R n − = R × ( R ) n − with each pair ( √ u i , ϕ i ) ∈ R being polar coordinates, and note that the standard contactstructure ( R n − , ξ st ) is given by the kernel of the 1–form α st = dz + n − (cid:88) i =1 u i dϕ i = dz + udϕ, where u := n − (cid:88) i =1 u i , and udϕ := n − (cid:88) i =1 u i dϕ i . The aim is to define a germ of a contact structure along a (2 n − n − ε ∈ R + begiven, consider the contact subdomains of ( R n − , ξ st ) given by∆ cyl = { z ∈ [ − , − ε ] , u ∈ [0 , } , ∆ ε = { z ∈ [ − ε, − ε ] , u ∈ [0 , − ε ] } ⊆ ∆ cyl , and define the subset B = { z = − , u ∈ [0 , } ∪ { z ∈ [ − , − ε ] , u = 1 } ⊆ ∂ ∆ cyl of theboundary of ∆ cyl . These three contact domains ∆ cyl , ∆ ε and B are shown in Figure 2. Figure 2.
The domains ∆ cyl in yellow, ∆ ε in blue and B in red. The domains arerotationally symmetric along the z –axis and we implicitly consider the coordinatesof the angle ϕ as included in the graphic representations of these domains. In a nutshell, the contact germ will be defined as the restriction of an ambient contactstructure in a neighborhood of a hypersurface describe as the graph of a particular functionin the domain ∆ cyl . Let k ε : R −→ R be the piecewise linear function defined by k ε ( x ) := (cid:40) x (cid:54) − εx − (1 − ε ) x (cid:62) − ε. and fix a piecewise smooth function K ε : ∆ cyl −→ R of the form K ε ( z, u , ϕ , . . . , u n − , ϕ n − ) := (cid:40) k ε ( | z | ) + k ε ( u ) ( z, u , ϕ , . . . , u n − , ϕ n − ) ∈ ∆ cyl \ Int(∆ ε ) < z, u , ϕ , . . . , u n − , ϕ n − ) ∈ Int(∆ ε ) . Let us denote q = ( z, u , ϕ , . . . , u n − , ϕ n − ), then the function K ε defines the following twoembeddings of two (2 n − = { ( q, v, t ) ∈ ∆ cyl × T ∗ S : t ∈ S , v = K ε ( q ) } ⊆ (∆ cyl × T ∗ S , ker( α st + vdt ))Σ = { ( q, v, t ) ∈ ∆ cyl × C : q ∈ B, t ∈ S , v ∈ [0 , K ε ( q )] } ⊆ (∆ cyl × C , ker( α st + vdt )) . In the description of Σ the pair of coordinates ( v, t ) represents linear coordinates in T ∗ S whereas in the definition of Σ the coordinates ( √ v, t ) represent polar coordinates on thecomplex plane C . Remark 3.4.
The homonymous notation [4, Section 2] for these two distinct pairs of coor-dinates is genuinely useful once interiorized, and as the notation suggests we then implicitlyidentify the open subset { v > } ⊆ T ∗ S with the open subset C ∗ = {√ v > } ⊆ C . (cid:3) Notice that the function K ε : ∆ cyl −→ R satisfies K ε | B > B ⊆ ∂ ∆ cyl and thusthe hypersurface Σ is well–defined as a subset of ∆ cyl × C . Each of the two hypersurfaces Σ and Σ defines a germ of a contact structure ( O p Σ , η ) and ( O p Σ , η ) inherited from itsrespective ambient contact domains (∆ cyl × T ∗ S , ker( α st + vdt )) and (∆ cyl × C , ker( α st + vdt )).By using the contact identification of the two respective subsets { ( v, t ) ∈ T ∗ S : v > } ⊆ ( T ∗ S , λ st ) , { ( √ v, t ) ∈ C : v > } ⊆ ( C , λ st )in the two ambient contact domains (∆ cyl × T ∗ S , ker( α st + λ st )) and (∆ cyl × C , ker( α st + λ st )),the union Σ ∪ Σ of the two hypersurfaces is a piecewise smooth disk in a contact domain andthus, by restriction of the contact structure, we obtain a contact germ in (a neighborhood of)this disk. Let us denote the disk endowed with this germ of a contact structure by ( D K ε , η K ε ). Remark 3.5.
Note that the dependence of the contact germ on the constant ε ∈ R + isgeometrically meaningful. Intuitively it describes the amount of rotation that the contactstructure is allowed to have in the boundary ∂ ∆ cyl , and this quantity features crucially inthe argument for the existence h –principle [4, 22]. (cid:3) Let us move to the definition of the overtwisted disk. In the article [4, Definition 3.6],an overtwisted disk ( D K univ , η K univ ) is defined to be a certain contact germ η K univ along apiecewise smooth (2 n − D K univ , where the definition of the function K univ is neitherconstructive nor canonical. However in this article we can take the function to be K univ = K ε : ∆ cyl −→ R for any sufficiently small ε < ε univ , where ε univ is a fixed constant depending only on dimen-sion. We then have the following definition: Definition 3.6.
An overtwisted disk ( D ot ε , η ot ε ) is any contact germ along a disk of the form( D K ε , η K ε ) where the constant ε ∈ R + satisfies ε < ε univ . (cid:3) In practice, this implies that finding an overtwisted disk is tantamount to finding a neigh-borhood of a disk with the contact germ ( D K ε , η K ε ) for an arbitrarily small ε ∈ R + . Notealso that the contact structure is defined as a contact germ on the disk, and thus the smoothregularity of the disk, as a hypersurface, is not a concern from the smooth topology perspec-tive: the contact structure is defined in a smooth open neighborhood of a disk, which is stilla smooth neighborhood even if the disk we consider is piecewise smooth [4]. Definition 3.7.
A contact manifold (
Y, ξ ) is overtwisted if there exists a piecewise smoothembedding D n − ⊆ Y such that the contact germ ( D n − , ξ | n − D ) is an overtwisted disk. (cid:3) The reader should now be equipped to understand the statement of Theorem 3.2 and thusthe statement of the equivalence (1) ⇐⇒ (2) in Theorem 1.1.Let us proceed with the proof of Theorem 3.2 in the case that ( N, ξ ot ) is a 3–dimensionalovertwisted contact manifold, which corresponds to the characterization (1) ⇐⇒ (2) in thecase where ( Y, ξ ) is a 5–dimensional contact manifold in the statement of Theorem 1.1.3.2.
The 5-dimensional case.
The initial step in order to prove Theorem 3.2 for the n = 2case is to substitute the general overtwisted contact 3–fold ( N, ξ ot ) by an explicit overtwistedlocal model ( M, ker α M ) and prove Theorem 3.2 for this particular 3–dimensional overtwistedcontact domain ( M, ker α M ). EOMETRIC CRITERIA FOR OVERTWISTEDNESS 15
The description and motivation of this contact domain ( M, ker α M ) strongly use the biva-lent coordinates ( v, t ) that appear in Subsection 3.1, which allows us to neatly describe thetransition from ( T ∗ S , λ st ) to the complex plane ( C , λ st ).In this 5-dimensional case, the overtwisted disk ( D , ξ ot ) is equivalent to the 2-dimensionalovertwisted disk introduced by Y. Eliashberg [22, Section 1.4] with the singular characteristicfoliation as depicted in [29, Section 4.5]. The model for ( D , ξ ot ) that we have in mind inthe present article is [29, Figure 4.9], i.e. an embedded 2-disk ( D , ξ ot ) whose characteristicfoliation contains a unique singular point in the interior and the characteristic foliation of( D , ξ ot ) is singular along ∂D . In particular, ∂D is a Legendrian curve with vanishingThurston-Bennequin invariant. Remark 3.8.
In Definition 3.7 for the overtwisted disks we expressed the contact germ in aparticular disk ( D ot ε , η ot ε ) = Σ ∪ Σ , which is described as the union of two pieces Σ and Σ .These two pieces are both defined in terms of a function K ε : ∆ cyl −→ R as explained above:the first piece Σ is precisely the graph of the function { v = K ε } ⊆ ∆ cyl × T ∗ S , whereas thesecond piece Σ is instead the sublevel set { v (cid:54) K ε | B } ⊆ ∆ cyl × C .For the first piece Σ , it is essential that the coordinates ( v, t ) belong to the cotangent space( v, t ) ∈ T ∗ S , and not the complex plane, since the function K ε : ∆ cyl −→ R attains bothpositive and negative values. In contrast, for the second piece Σ it is essential that thecoordinates ( √ v, t ) actually define polar coordinates on the complex plane ( C , λ st ), sincewe can then define the sublevel set { v (cid:54) K ε } correctly. This hopefully emphasizes theimportance of the varying domains that the coordinates ( v, t ) are defining. (cid:3) Let us now describe a local model ( M, ker α M ) which is contained in any overtwisted contact3-fold ( N, ξ ot ) and has a crucial role in the proof of Theorem 3.2. The domain M is diffeo-morphic to a compact 3-ball with a piecewise smooth boundary and it admits coordinates( z, v, t ), where ( v, t ) are coordinates in the sense of Remarks 3.4 and 3.8 above. In thesecoordinates ( z, v, t ) the contact form α M reads α M = dz + vdt. It is our duty to be precise with the meaning of the pair ( v, t ): in this case the coordinate z ∈ ( − − ε,
1] dictates the domain of definition of the pair of coordinates ( v, t ). This goesas follows, the symplectic submanifolds { z = constant } belong to one of these three types: Figure 3.
The overtwisted contact ball ( M, ker α M ) .a. For z ∈ [ − ε , v, t ) ∈ [ − , × S . Thus in this range the submanifolds { z = constant } are exact symplectomorphic to the unit disk bundle ( D ∗ S , λ st ) inside( T ∗ S , λ st ) since the restriction of α equals the canonical Liouville form. b. For z ∈ ( − ε , − ε ), we let t ∈ S and v ∈ [ ε, { ε ≤ v ≤ } ⊆ ( T ∗ S , λ st ). Notice that these fibers are alsoequal to the standard Liouville structure on D (1) \ D ( ε ) ⊆ C , where we equip( C , λ st ) with polar coordinates ( √ v, t ).c. For z ∈ [ − − ε, − ε ), we define the fibers { z = constant } to be equal to the unitdisk ( D (1) , λ st ) ⊆ ( C , λ st ), with ( √ v, t ) continuing to represent polar coordinates.The choice of the dependence of the domain of ( v, t ) on the z -coordinate allows for a moreflexible notation, which hopefully helps the reader. It is possible to alternatively define ( v, t )to be global coordinates independent of z and work with a contact form α M = dz + ρ ( v ) dt, and domain M depending on the choice of a Hamiltonian ρ : R −→ R . This is the notationthat is followed in [4, Section 2]. In our notation, the dependence of ( v, t ) absorbs keepingtrack of such Hamiltonian ρ , which we prefer. Remark 3.9.
It might help the reader to understand the contact domain (
M, α M ) as asymplectic foliation where the leaves are parametrized by the interval in the z –coordinate.This fibration viewpoint has been fruitful in contact topology [9, 14] and it provided us withthe right insight to prove Theorem 3.2. (cid:3) The contact domain ( M, ker α M ) is depicted in Figure 3, where the reader can see howthe dependence of the domain of the coordinates ( v, t ) varies according to the value of the z –coordinate. Let us now analyze the two fundamental contact properties of ( M, ker α M ):1. First, the 3–dimensional contact domain ( M, ker α M ) is overtwisted.This can be proven by direct inspection and finding an overtwisted 2–disk. Instead, wecan note that the Legendrian circle { ( z, v, t ) ∈ M : 6 z = − ε, v = 0 } ⊆ ( M, α M ) isan unknotted Legendrian with zero Thurston-Bennequin number, which proves thatthe contact model ( M, α M ) is overtwisted [22].2. Second, the contact domain ( M, ker α M ) serves as a local model in any overtwisted3-manifold. Indeed, if ( Y, ker α ) is any overtwisted contact 3-manifold, possibly open,then ( M, ker α M ) admits a contact embedding f : ( M, ker α M ) −→ ( Y, ker α )due to Eliashberg’s classification theorem [22]. Even better, defining the positivesmooth function c f : M −→ R given by the conformal factor f ∗ α = c f α M , weconclude by compactness that there exists a constant R ∈ R + such that c f < R . Inconsequence, the contact product ( M × D (1) , ker( α M + λ st )) embeds into the contactproduct ( Y × D ( R ) , ker( α + λ st )).It follows from the second property and Definition 3.7 that in order to prove Theorem 3.2 inthis 5–dimensional case it suffices to find an overtwisted disk in the contact product manifold( M × D (1) , ker( α M + λ st )) . This is our goal now, which we achieve by first proving the technical Lemma 3.11.Let us define the 3–dimensional contact domain (cid:101) ∆ = ( − − ε, × D (1) endowed withcoordinates ( z, u , ϕ ) and the standard contact form dz + u dϕ , where z ∈ ( − − ε,
1) and( √ u , ϕ ) are polar coordinates on D (1). Consider the map π : ( M × D (1) , α M + u dϕ ) −→ ( (cid:101) ∆ , dz + u dϕ ) , EOMETRIC CRITERIA FOR OVERTWISTEDNESS 17 ( z, v, t, u , ϕ ) (cid:55)−→ π ( z, v, t, u , ϕ ) = ( z, u , ϕ ) , whose fibers are Liouville surfaces symplectomorphic to subdomains of ( T ∗ S , λ st ) or ( C , λ st ).The contact domain (cid:101) ∆ contains two different subdomains in terms of the fibers of the smoothmap π , which we can define as (cid:101) ∆ = { ( z, u , ϕ ) ∈ (cid:101) ∆ : z ∈ ( − ε/ , − ε ] } ⊆ (cid:101) ∆ (cid:101) ∆ = { ( z, u , ϕ ) ∈ (cid:101) ∆ : z ∈ [ − , − ε/ } ⊆ (cid:101) ∆ . Therefore the first subdomain (cid:101) ∆ ⊆ (cid:101) ∆ corresponds to those values of ( z, u , ϕ ) such thatthe fiber is given by ( v, t ) ∈ ( T ∗ S , λ st ) for ( z, v, t, u , ϕ ) ∈ M × D (1). Similarly, the secondsubdomain (cid:101) ∆ ⊆ (cid:101) ∆ corresponds to those values in (cid:101) ∆ where the fiber of the projection π isequivalent to ( C , λ st ). Remark 3.10.
Following Definition 3.7 and the discussion above, the proof of Theorem 3.2in this 5–dimensional case consists in finding a 4–dimensional overtwisted disk in the contactmodel ( M × D (1) , ker( α M + λ st )). The contact germ of an overtwisted disk is given in termsof a domain of definition ∆ cyl , and observe that there is a natural embedding ∆ cyl −→ (cid:101) ∆.However, and that is the difficulty that needs to be solved at this point, it is not true thatwe have the inclusions ∆ ε ⊆ (cid:101) ∆ and B ⊆ (cid:101) ∆ . Note that if this were the case the contactmodel above would readily be overtwisted. (cid:3) The exact relation between the contact domain (cid:101) ∆ and ∆ cyl needed in order to prove Theorem3.2 in this n = 2 case is established in the following lemma. Lemma 3.11.
There exists a strict contact embedding f : (∆ cyl , ξ st ) −→ ( (cid:101) ∆ , ξ st ) , i.e. suchthat f ∗ α st = α st , with the property that f (∆ ε ) ⊆ (cid:101) ∆ and f ( B ) ⊆ (cid:101) ∆ . Lemma 3.11 will be proven momentarily, but let us first conclude Theorem 3.2 for an over-twisted contact 3–fold ( N, ker α ot ) assuming such contact embedding exists. Proof of Theorem 3.2 for dim( N ) = 3 . It suffices to show that the 5–dimensional domain( M × D (1) , ker( α M + u dϕ ))is overtwisted, as we have discussed above. In order to do that, let ε ∈ R + be a fixed butsmall enough constant such that ε < ε univ and consider the contact embedding f : ∆ cyl −→ (cid:101) ∆provided in Lemma 3.11. The claim is now that the preimage π − ( O p ( f (∆ cyl )) ⊆ ( M × D (1) , ker( α M + u dϕ ))contains an overtwisted disk. Indeed, define the function (cid:101) K ε : f (∆ cyl ) −→ R as the pull–back (cid:101) K ε = K ε ◦ f − : f (∆ cyl ) −→ R . and consider the two hypersurfaces (cid:101) Σ = { ( z, v, t, u , ϕ ) : ( z, u , ϕ ) ∈ f (∆ cyl ) , t ∈ S , v = (cid:101) K ε ( z, u , ϕ ) } ⊆ f (∆ cyl ) × T ∗ S , (cid:101) Σ = { ( z, v, t, u , ϕ ) : ( z, u , ϕ ) ∈ f ( B ) , t ∈ S , v ∈ [0 , (cid:101) K ε ( z, u , ϕ )] } ⊆ f ( B ) × C . Notice that the first hypersurface (cid:101) Σ is a well–defined subset of M × D (1) precisely be-cause f (∆ ε ) ⊆ (cid:101) ∆ , and similarly the second hypersurface (cid:101) Σ is well–defined because theinclusion f ( B ) ⊆ (cid:101) ∆ is satisfied. This construction now exhibits an overtwisted disk inour 5–dimensional domain ( M × D (1) , ker( α M + u dϕ )): the contact germ of the 4–disk D = (cid:101) Σ ∪ (cid:101) Σ obtained as the union of the two hypersurfaces is an overtwisted disk. Indeed,since the 3–dimensional contactomorphism f : ∆ cyl −→ (cid:101) ∆ preserves the contact form, theextended contactomorphism in 5–dimensions F : ( O p ( D ot ε ) , η ot ε ) ∼ = ( O p (Σ ∪ Σ ) , η ot ε ) −→ ( M × D (1) , α M + u dϕ ) ( z, u , ϕ , v, t ) (cid:55)−→ F ( z, u , ϕ , v, t ) = ( f ( z, u , ϕ ) , v, t ) . maps the contact germ ( D ot ε , η ot ε ) to the contact germ ( D , ker( α M + λ st )), as required. (cid:3) This concludes the proof of Theorem 3.2 in the case that dim( N ) = 3 modulo the constructionof the contactomorphism in Lemma 3.11, which we now prove. Figure 4.
The contact domains appearing in Lemma 3.11 . Proof of Lemma 3.11:
Let g : [0 , −→ [0 ,
2] be a smooth and increasing function which is C –close to the piecewise linear function defined by u (cid:55)−→ (cid:40) u ∈ [0 , − ε ] ε ( u − ε ) if u ∈ [1 − ε , , and consider the diffeomorphism f : ∆ cyl −→ (cid:101) ∆ defined by f ( z, u , ϕ ) = (cid:18) z − g ( u ) , u , ϕ − (cid:90) u g (cid:48) ( u ) u du (cid:19) . The map f : ∆ cyl −→ (cid:101) ∆, which is depicted in Figure 4, has the desired properties fromthe statement. Indeed, the diffeomorphism f is a C –approximation of a piecewise smoothcontactomorphism which acts by taking the region { u ≥ − ε } and shearing its z –coordinatefar to the left, thus conforming to the required properties. (cid:3) General dimensions.
In this section we prove Theorem 3.2. The reader is stronglyencouraged to have understood the case n = 2, proven in the previous Subsection 3.2. Theargument we use in order to conclude Theorem 3.2 for an arbitrary overtwisted contactmanifold ( N, ξ ot ) contains the same steps as the 5–dimensional case above, but the generalhigher–dimensional versions of the boundary piece B ⊆ ∆ cyl and Lemma 3.11 contain moreinformation.The first difference between the general and 5–dimensional cases is that the contact embed-ding f : ∆ cyl −→ (cid:101) ∆ that we use in the general case, generalizing Lemma 3.11, is no longerstrict, and thus a conformal factor must be accounted for when constructing the domains towhich we push–forward the function K ε : ∆ cyl −→ R . This conformal factor is the reason forthe appearance of the constant ρ ∈ R + in the following definition of the local model ( M, α M ).Consider two positive reals constants ε, ρ ∈ R + , where ε is to be small and ρ quite large.Define the (2 n − I = ( − ρ, × D n − ( ρ )with coordinates z ∈ ( − ρ,
1) and ( u, ϕ ) = ( u , ϕ , . . . , u n − , ϕ n − ) are polar coordinates onthe ball D n − ( ρ ). The domain I generalizes the z –coordinate interval in the proof of the5–dimensional case. Following the first step of the proof in Subsection 3.2, we describe a EOMETRIC CRITERIA FOR OVERTWISTEDNESS 19 (2 n − M, α M ), which is endowed with coordinates( z, u, ϕ, v, t ) and the contact form given globally by α M = dz + udϕ + vdt. Hopefully, the reader noticed that the domain of the coordinates ( v, t ) must at least dependon the z –coordinate, as in Subsection 3.2. Indeed, the variables ( z, u, ϕ ) ∈ I belong to thefixed domain I but the domain of the variables ( v, t ) will either be the unit disk bundle D ∗ S , the positive part D (1) \ D ( ε ) or the unit disk D (1) depending on the coordinates( z, u, ϕ ) ∈ I . This precise dependence is given as follows:a. For ( z, | u | ) ∈ I := [ − ε , × [0 , − ε ] ⊆ I , we have ( v, t ) ∈ [ − , × S . In thisrange, the symplectic submanifolds { ( z, u, ϕ ) = constant } are exact symplectomor-phic to ( D ∗ S , λ st ).b. For ( z, | u | ) ∈ I / := ( − ε , − ε ) × [0 , − ε ) ∪ ( − ε , × (1 − ε , − ε ) ⊆ I ,we consider ( v, t ) ∈ [ δ, × S . The symplectic submanifolds { ( z, u, ϕ ) = constant } are symplectomorphic to ( D (1) \ D ( δ ) , λ st ), where δ ∈ R + is a small constant whichwill be chosen in the proof of Theorem 3.2. The constant δ does not have a crucialrole, and thus we do not include it in the notation.c. For { ( z, | u | ) ∈ I := [ − ρ, − ε ] × [0 , ρ ] ∪ [ − ρ, × [1 − ε , ρ ] } ⊆ I , we declare( v, t ) ∈ ( D (1) , λ st ) to be polar coordinates ( √ v, t ) in the unit disk.This (2 n − M, ker α M ) has the two properties of its3–dimensional analogue in Subsection 3.2. First, the contact manifold ( M, ker α M ) is over-twisted if we choose ρ ∈ R + large enough. Second, for any choice of positive constants ρ, δ and ε ∈ R + , this contact local model ( M, ker α M ) exists in every overtwisted (2 n − N, ker α ot ) by the isocontact embedding h –principle [4, Corollary 1.4], andthe fact that the scaling factor between α ot and α M is bounded because M is compact. Hencein order to conclude Theorem 3.2 it remains to prove that the (2 n + 1)–dimensional contactdomain ( M × D ( R ) , ker( α M + λ st ))contains an overtwisted 2 n –disk when R ∈ R + is sufficiently large.Let us introduce the domain (cid:101) ∆ and its relatives, following the steps in Subsection 3.2. Weconsider the contact domain (cid:101) ∆ = ( − ρ, − ε ) × D n − ( ρ ) × D ( R ) , with coordinates ( z, u, ϕ, u , ϕ ) , which contains the two (2 n − (cid:101) ∆ = I × D ( R ) ⊆ (cid:101) ∆ , (cid:101) ∆ = I × D ( R ) ⊆ (cid:101) ∆ . These two subdomains have the same role as their homonymous domains have in the 5–dimensional case discussed in Subsection 3.2. Indeed, the reason for considering the twosubdomains (cid:101) ∆ and (cid:101) ∆ is that the symplectic type of the fibers of the projection map π : M × D (1) −→ (cid:101) ∆ , ( z, u, ϕ, v, t, u , ϕ ) (cid:55)−→ π ( z, u, ϕ, v, t, u , ϕ ) = ( z, u, ϕ, u , ϕ )depends on the point of the domain (cid:101) ∆. Indeed, over the region (cid:101) ∆ the fiber of the map π isthe unit cotangent bundle ( D ∗ S , λ st ), whereas over the region (cid:101) ∆ the fiber is the unit disk( D (1) , λ st ).In the same vein than Lemma 3.11, we now need to compare the two (2 n − (cid:101) ∆ , ξ st ) and (∆ n − , ξ st ), and contact embed the domain ∆ cyl inside (cid:101) ∆ insuch a manner that the images of the subdomain ∆ ε and the boundary piece B lie in the appropriate regions of the target domain (cid:101) ∆. This is the content of the following lemma,which generalizes Lemma 3.11. Figure 5.
The contact domain ∆ cyl = ( z, u , ϕ , u, ϕ ). Lemma 3.12.
For any constant ε ∈ R + , there exist constants ρ, R ∈ R + such that there isa contact embedding f : (∆ cyl , ξ st ) −→ ( (cid:101) ∆ , ξ st ) , satisfying f (∆ ε ) ⊆ (cid:101) ∆ and f ( B ) ⊆ (cid:101) ∆ .Proof. First, the two (2 n − n − and (cid:101) ∆ are contact subdomains ofthe ambient contact space ( R n − , ξ st ) ∼ = ( R n − × D ( R ) , ξ st ) and we are using coordinates( z, u, ϕ, u , ϕ ), where ( √ u , ϕ ) are polar coordinates of the D ( R ) factor.In comparison to Lemma 3.11, additional effort must be invested when working with the set B ⊆ ∆ cyl , which can be described by the union B = { z = − } ∪ { u + | u | = 1 } . The readeris encouraged to visualize the subset B in Figure 5, where we have depicted the coordinates( z, u, u ).In order to achieve the condition f ( B ) ⊆ (cid:101) ∆ we have the choice of either decreasing the z –coordinate below the value − ε/ u –coordinate beyond the value 1 − ε/ B ⊆ ∆ cyl we find ourselves in.The decomposition of the set B = B ∪ B we consider is defined as follows B := { ( z, u, ϕ, u , ϕ ) ∈ B : | u | ≤ ε / } , B := { ( z, u, ϕ, u , ϕ ) ∈ B : | u | ≥ ε / } ⊆ B. The required contactomorphism f : (∆ n − , ξ st ) −→ ( (cid:101) ∆ , ξ st ) will be obtained as the composi-tion of two contactomorphisms g : ∆ cyl −→ (cid:101) ∆ and g : ( R n − , ξ st ) −→ ( R n − , ξ st ), both ofwhich will restrict to the identity g | ∆ ε = g | g (∆ ε ) = id | ∆ ε in the region ∆ ε = g (∆ ε ) ⊆ ∆ cyl .In geometric terms, the contactomorphism g will decrease the z –coordinate in the region B in order to contact embed it into (cid:101) ∆ , and the contactomorphism g will increase the modulus u and embed the region B into (cid:101) ∆ . Let us start with g , which already featured in the5–dimensional case.Consider the contactomorphism h : (∆ , ξ st ) −→ ( (cid:101) ∆ , ξ st ) constructed in Lemma 3.11 anddefine the contactomorphism g : (∆ n − , ξ st ) −→ ( (cid:101) ∆ n − , ξ st ) , g ( z, u , ϕ , u, ϕ ) = ( h ( z, u , ϕ ) , u, ϕ ) . This contactomorphism satisfies g ( B ) ⊆ (cid:101) ∆ since a point ( z, u , ϕ , u, ϕ ) ∈ B must have u = 1 − | u | ≥ − ε / h ( z, u , ϕ ) , u, ϕ ) has a z –coordinate belowthe value − ε/ EOMETRIC CRITERIA FOR OVERTWISTEDNESS 21
Let us now describe the contactomorphism g , where we will push the remaining piece B ⊆ B into the region (cid:101) ∆ . Consider the contact vector field X = u∂ u + u ∂ u + z∂ z on ∆ n − andcut–off its contact Hamiltonian H ∈ C ∞ (∆ n − ) to a Hamiltonian (cid:101) H ∈ C ∞ (∆ n − ) suchthat its associated contact vector field (cid:101) X satisfiesa. (cid:101) X vanishes in the region { z ≥ − ε , u + u ≤ − ε } .b. (cid:101) X coincides with the radial vector field X in the region { z ≤ − ε , − ε ≤ u + u } . Figure 6.
Cross section ( u, u ) for the expanded domain h τ (cid:101) X (∆ cyl ). Denote by h τ (cid:101) X : ( R n − , ξ st ) −→ ( R n − , ξ st ) be the τ –time contact flow of the contact vectorfield (cid:101) X : near the region B the contact flow h τ (cid:101) X acts as radial expansion, as depicted in Figure6. The contactomorphism g is defined to be h τ (cid:101) X for a large enough time τ ∈ R + , and weclaim that for such g : ( R n − , ξ st ) −→ ( R n − , ξ st ) the composition f = g ◦ g : (∆ cyl , ξ st ) −→ ( (cid:101) ∆ , ξ st )satisfies the properties in the statement of the lemma. First, we do have the inclusion f (∆ ε ) = g ( g (∆ ε )) = g (∆ ε ) = ∆ ε ⊆ (cid:101) ∆ since both g and g are the identity in ∆ ε byconstruction. Second, we need to verify that the inclusion f ( B ) = g ( g ( B )) ⊆ (cid:101) ∆ issatisfied. Indeed, since the u –coordinate on the set g ( B ) is bounded below by a positivenumber and the contact flow g expands the coordinate u exponentially by construction, weconclude that for large τ ∈ R + the inclusion g ( g ( B )) ⊆ { u > − ε/ } ⊆ (cid:101) ∆ holds. (cid:3) Remark 3.13.
The contact embedding f : (∆ cyl , ξ st ) −→ ( (cid:101) ∆ , ξ st ) in Lemma 3.12 is not astrict, in contrast to the contact embedding in Lemma 3.11. (cid:3) Lemma 3.12 is the technical ingredient in order to prove Theorem 3.2, which we now do. Thestructure of the proof is the same as for its 5–dimensional analogue proven in Subsection 3.2.Let us now provide the details.
Proof of Theorem 3.2:
First, we choose a constant ε ∈ R + such that ε < ε univ and considerthe contact embedding f : (∆ n − , ξ st ) −→ ( (cid:101) ∆ n − , ξ st ) constructed in Lemma 3.12. De-note by ρ, R ∈ R + the homonymous constants appearing in its statement and consider theconformal factor c f ∈ C ∞ (∆ cyl ) defined by f ∗ α st = c f α st . Now we can proceed as in the5–dimensional case by defining the Hamiltonian (cid:101) K : f (∆ n − ) −→ R , (cid:101) K = ( c f · K ) ◦ f − . The statement of the theorem will be proven if we can find a 2 n –dimensional overtwisteddisk in the (2 n + 1)–dimensional contact domain (( M, α M ) × D ( R ) , ker( α M + λ st )). In orderto exhibit the disk, we consider the two domains (cid:101) Σ = { ( z, v, t, u, ϕ ) : ( z, u, ϕ ) ∈ f (∆ cyl ) , t ∈ S , v = (cid:101) K ( z, u, ϕ ) } (cid:101) Σ = { ( z, v, t, u, ϕ ) : ( z, u, ϕ ) ∈ f ( B ) , t ∈ S , v ∈ [0 , (cid:101) K ( z, u, ϕ )] } . Notice that for a sufficiently small δ ∈ R + , which appears in the definition of the contactdomain ( M, α M ), the hypersurface (cid:101) Σ is a well–defined subset of M × D (1) since f (∆ ε ) ⊆ (cid:101) ∆ .The second hypersurface (cid:101) Σ is also well–defined since we have the inclusion f ( B ) ⊆ (cid:101) ∆ . Nowthe union D n = (cid:101) Σ ∪ (cid:101) Σ of the two hypersurfaces (cid:101) Σ and (cid:101) Σ endowed with the ambientcontact structure is contactomorphic to an overtwisted disk since the map F : ( O p ( D ot ε ) , η ot ε ) ∼ = ( O p (Σ ∪ Σ ) , η ot ε ) −→ ( M × D (1) , α M + u dϕ ) , ( z, u , ϕ , u, ϕ, v, t ) (cid:55)−→ F ( z, u , ϕ , u, ϕ, v, t ) = ( f ( z, u , ϕ , u, ϕ ) , v · c f ( z, u , ϕ , u, ϕ ) , t )maps the local model ( D ot ε , η ot ε ) to the contact germ ( D n , ker( α M + λ st )). This concludesthe proof of Theorem 3.2. (cid:3) Weinstein cobordism from overtwisted to standard sphere
The main goal of this section is proving the equivalence (1) ⇐⇒ (3) ⇐⇒ (4) in Theorem 1.1,which is concluded in Theorems 4.4 and 4.5 . First, we state an application of the previoussection which will be used in their proofs. The contact branched cover [30, Theorem 7.5.4]along with Theorem 3.2 yield the following class of examples of overtwisted contact structures. Theorem 4.1.
Let ( Y, ξ ) be a contact manifold and ( D, ξ | D ) a codimension–2 overtwistedcontact submanifold. A k–fold contact branched cover of ( Y, ξ ) along ( D, ξ | D ) is overtwistedfor k large enough. Theorem 4.1 follows immediately from Theorem 3.2 since a branch cover increases the productneighborhood width of the branch locus; this latter observation has been successfully used in[48, Section 1] for producing obstructions to symplectic fillability. In a concise manner, thereason a contact branched cover increases the size of a contact neighborhood of the branchlocus is the following. Locally, the contact form near a codimension–2 submanifold D ⊆ ( Y, ξ )with trivial normal bundle can be assumed to be of the form α Y = α D + r dθ, where α D is a contact form for the contact submanifold ( D, ξ | D ), and we have smoothlyidentified O p ( D ) ∼ = D × D δ ( r, θ ) for some δ ∈ R + . In this model a k –fold branched coveralong D is given by the branched map( p, ρ, ϑ ) (cid:55)−→ ( p, r, θ ) = π ( p, ρ, ϑ ) = ( p, ρ, kϑ ) , ( p, r, θ ) ∈ D × D δ , where ( p, ρ, ϑ ) denote the upstairs coordinates. Thus the pull–back of the contact form is π ∗ α Y = α D + ( √ kr ) dθ, which is increasing the contact radius r ∈ [0 , δ ) to a radius of size ρ ∈ [0 , √ kδ ), which explainsTheorem 4.1. That being said, we now apply Theorem 4.1 to prove the following theorem. Theorem 4.2.
In every dimension, there is a Weinstein cobordism ( W, λ, ϕ ) such that theconcave end ( ∂ − W, λ ) is overtwisted and the convex end ( ∂ + W, λ ) ∼ = ( S n − , ξ st ) . Theorem 4.2 is proven assuming the equivalence (1) ⇐⇒ (2) in Theorem 1.1 which has beenproven in Section 3, and it also uses the inductive hypothesis in the dimension n . TheWeinstein cobordism ( W, λ, ϕ ) in the statement of Theorem 4.2 is smoothly non–trivial andit is constructed such that ∂ − W is a standard smooth sphere. EOMETRIC CRITERIA FOR OVERTWISTEDNESS 23
Proof of Theorem 4.2.
Let us construct a Weinstein cobordism ( W n , λ, ϕ ) of finite type froman overtwisted contact structure ( S n − , ξ ot ) to the standard contact sphere ( S n − , ξ st ). Inorder to do that, consider the A n − k –Milnor fibre obtained as an A k –plumbing of k copiesof the Weinstein manifold ( T ∗ S n − , λ st , ϕ st ), with its induced Weinstein structure. Theconstruction of the Weinstein cobordism ( W, λ, ϕ ) now has two steps.First, we prove that the contact manifold( S n − , ξ k ) = OB( A n − k − , τ − ◦ . . . ◦ τ − k − )is overtwisted for k large enough, and second, we construct the Weinstein cobordism to thestandard contact sphere ( S n − , ξ st ).Let us first prove overtwistedness of ( S n − , ξ k ) for k large enough. The right–veering cri-terion [37] shows that ( S , ξ ) is an overtwisted contact 3–fold, which can also be provenexplicitly by finding an overtwisted 2–disk, and thus ( S , ξ k ) are overtwisted for all k . Nowthe inductive hypothesis on the dimension and the equivalence (1) ⇐⇒ (5) in Theorem 1.1 for(2 n − S n − , ξ ) = OB( T ∗ S n − , τ − ) isan overtwisted contact manifold.In addition, the contact manifold ( S n − , ξ ) also admits a contact embedding into the contactmanifold ( S n − , ξ ) = OB( A n − , τ − ) compatible with the open book decomposition whichcorresponds to the cotangent bundle of an unknotted equatorial S n − ⊆ S n − . Then Theorem4.1 implies that the contact k –branched cover ( Y k , ζ k ) of the contact structure ( S n − , ξ )along the contact divisor ( S n − , ξ ) is an overtwisted contact manifold for k large enough.Note that Y k is diffeomorphic to the standard smooth sphere S n − because the smoothsubmanifold S n − is smoothly unknotted.Let us now show that the contact structure ( Y k , ζ k ) = ( S n − , ζ k ) is supported by the openbook decomposition OB( A n − k − , τ − ◦ . . . ◦ τ − k − ) and hence it is contact isotopic to ( S n − , ξ k ).First note that the projection map for the open book OB( A n − , τ − ) is given by argumentof the map f : S n − ⊂ C n −→ C , f ( z , . . . , z n ) = z + . . . + z n . Then the overtwisted submanifold ( S n − , ξ ) is cut out by the equation { z = 0 } and the k –branched cover along it can be realized by the map z (cid:55)−→ z k . Thus the contact structure( Y k , ζ k ) is supported by the open book induced by the argument of the map f : S n − ⊂ C n −→ C , f ( z , . . . , z n ) = z k + z + . . . + z n , which is OB( A n − k − , τ − ◦ . . . ◦ τ − k − ). This proves the contactomorphism( S n − , ξ k ) ∼ = ( S n − , ζ k ) , and hence the fact that ( S n − , ξ k ) is overtwisted for k large enough.The second step is to argue that ( S n − , ξ k ) is Weinstein cobordant to ( S n − , ξ st ), whichthen constructs the required cobordism in the statement of Theorem 4.2 by taking k largeenough. Notice that by Theorem 2.17 we have the contactomorphismOB( A n − k − , τ ◦ . . . ◦ τ k − ) = ( S n − , ξ st ) , since this open book is just the trivial open book ( D n − , id) positively stabilized (2 k − A n − k − , giving a total of (4 k −
2) critical handleattachments, which construct the Weinstein cobordism from ( S n − , ξ k ) to ( S n − , ξ st ). (cid:3) The following proposition is the remaining ingredient before we are able to conclude theequivalence (1) ⇐⇒ (3) from the above Theorem 4.2. Proposition 4.3.
Let ( Y, ξ ) be a contact manifold, and suppose that the standard Legendrianunknot is a loose Legendrian submanifold in ( Y, ξ ) . Let ( W, λ, ϕ ) be an arbitrary Weinsteincobordism and let SY be the symplectization of Y . Then connected sum cobordism W SY is always a flexible Weinstein cobordism.Proof. Since the symplectization SY is a Weinstein trivial product, the critical points of theWeinstein cobordism W SY are the same as the critical points of the cobordism ( W, λ, ϕ ).Let p be a critical point in W SY of index n , M = ϕ − ( ϕ ( c ) − ε ) a level set of W , andΛ ⊆ M Y the Legendrian attaching sphere of the critical point p .Let Λ ⊆ ( Y, ξ ) be the standard Legendrian unknot, and U ⊆ M Y the union of a Darbouxchart containing Λ and a loose chart for Λ . Since Λ is loose as a Legendrian in ( Y, ξ )and the Legendrian Λ is the descending sphere of a critical point of (
W, λ, ϕ ), we knowthat Λ is disjoint from U , and therefore their Legendrian connected sum Λ is a looseLegendrian, even with the same loose chart as Λ . Then, the Legendrian connected sumΛ is Legendrian isotopic to Λ since the Legendrian Λ is the standard Legendrian unknot,and thus follows that the Legendrian Λ is loose. (cid:3) This allows us to prove the equivalence (1) ⇐⇒ (3), which is the following theorem. Theorem 4.4.
Let Λ be the standard Legendrian unknot inside a contact manifold ( Y, ξ ) .If Λ is a loose Legendrian then ( Y, ξ ) is overtwisted.Proof. Let (
W, λ, ϕ ) be the cobordism constructed in Theorem 4.2, and apply Proposition 4.3to conclude that the vertical connected sum W SY is a flexible Weinstein cobordism. Theconcave end of the cobordism ∂ − ( W SY ) = ∂ − W Y is an overtwisted contact manifoldsince the contact boundary ∂ − W is overtwisted itself, and thus Proposition 2.12 implies thatthe contact convex end ∂ + ( W SY ) = ( S n − , ξ st ) Y, ξ ) ∼ = ( Y, ξ )is an overtwisted contact manifold as well. (cid:3)
Theorem 4.4 also implies the equivalence (1) ⇐⇒ (4). Indeed, the standard unknot in ( Y, ξ ) isdefined by the inclusion of a small Darboux chart in (
Y, ξ ) and thus if the contact manifoldcontains a small plastikstufe with trivial rotation, the unknot must be in the complement.Therefore, Theorems 2.7 and 4.4 imply
Theorem 4.5.
Let ( Y, ξ ) be a contact manifold containing a small plastikstufe with sphericalcore and trivial rotation. Then ( Y, ξ ) is overtwisted. Thus far in the article we have proven the equivalences (1) ⇐⇒ (2) ⇐⇒ (3) ⇐⇒ (4) in Theorem1.1. The following two sections are respectively dedicated to the proofs of the two remainingequivalences, that is, the characterization in terms of surgeries (1) ⇐⇒ (5), and the criterionin terms of open book decompositions (1) ⇐⇒ (6).5. (+1)–surgery on loose Legendrians In this section we prove the equivalence (1) ⇐⇒ (5) in Theorem 1.1 by using the charac-terization given by Theorem 4.5. For our purpose, we use the following model of contact(+1)–surgery on a Legendrian sphere, defined in the article [2, Section 9].Let Λ ⊆ ( Y, ξ ) be a Legendrian sphere in a contact manifold. A neighborhood of the Legen-drian Λ can be identified with a neighborhood of the zero section in the first–jet space( J ( S n − ) , ker α st ) = ( T ∗ S n − × R ( z ) , ker( dz − λ st )) . EOMETRIC CRITERIA FOR OVERTWISTEDNESS 25
Consider the smooth manifold Y (cid:48) obtained by removing the piece D ∗ S n − × (0 ,
1) from Y , and then gluing the boundary to itself with the identification ( x, ∼ ( τ − ( x ) ,
1) and( x, t ) ∼ ( x, t (cid:48) ) for x ∈ ∂D ∗ S n − , where τ : T ∗ S n − −→ T ∗ S n − denotes the Dehn twist alonga zero section [51]. Note that Y (cid:48) is smooth manifold since the diffeomorphism τ is compactlysupported, and it has a canonical contact structure ξ (cid:48) because the gluing diffeomorphism τ is a symplectomorphism. Definition 5.1.
The contact manifold ( Y (cid:48) , ξ (cid:48) ) obtained with the procedure above is said tobe the contact (+1) –surgery of ( Y, ξ ) along Λ. (cid:3) Remark 5.2.
Given a Legendrian sphere Λ ⊆ ( Y, ξ ), the contactomorphism type of the con-tact surgery ( Y (cid:48) , ξ (cid:48) ) depends on the chosen parametrization f : S n − −→ Λ of the Legendriansubmanifold. In fact [18, Theorem A] shows that the class [ τ ] ∈ π Symp( T ∗ S n − ) genuinelydepends on this parametrization. However, in our context we are able to dismiss this tech-nical distinction since any two parametrizations of loose Legendrian spheres are ambientlycontact isotopic [44, Theorem 1.2]. (cid:3)
Remark 5.3.
Since the symplectomorphism τ does not preserve the Liouville form thegluing above should be technically performed in the region of the contactization given by { ≤ z ≤ f } ⊆ T ∗ S n − × R , where f ∈ C ∞ ( T ∗ S n ) is a positive primitive of λ − τ ∗ λ . (cid:3) In this surgery model introduced in Definition 5.1, we can prove the equivalence (4) ⇐⇒ (5)in Theorem 1.1, which also establishes [2, Conjecture 9.16]. Theorem 5.4.
Let Λ ⊆ ( Y, ξ ) be a loose Legendrian submanifold. Then the contact (+1) –surgery of ( Y, ξ ) along Λ contains a small plastikstufe with spherical core and trivial rotation. Figure 7.
The overtwisted disk inside ( M x , ξ (cid:48) ). Here we are viewing M x as presented by surgery in the front projection of J ( S ). In particular, thetransverse curve S × R = { p = 0 , q ∈ S } . Since the boundary of D andthe surgery curve Λ both have positive slope in the front, we can choose D to lie in the region p >
0, in particular making it disjoint from S × R . Proof.
Since the Legendrian sphere Λ is loose, we can choose a Legendrian sphere (cid:101)
Λ whosespherical stabilization gives the Legendrian Λ [44]. Choose coordinates in a neighborhoodof the Legendrian (cid:101)
Λ identifying it with a neighborhood of the zero section in the jet space( T ∗ S n − × R , ker α st ), and we can then represent the original Legendrian Λ as the zerosection stabilized over the equator S n − ⊆ S n − . For a fixed point x ∈ S n − in the equator,define the circle S x ⊆ S n − to be the unique meridian passing through the point x and thenorth and south poles, and consider the submanifold J ( S x ) ⊆ T ∗ S n − × R . The jet space J ( S x ) is a 3-dimensional contact submanifold contactomorphic to T ∗ S × R , and under thiscontactomorphism the intersection Λ ∩J ( S x ) is given as the stabilization of the zero section. Note also that for x (cid:54) = y , we can identify J ( S x ) ∩ J ( S y ) ∼ = S × R where S is the unionof the north and south poles.Because the Dehn twist τ : T ∗ S n − −→ T ∗ S n − , which is used to perform the contact surgery,is a symplectomorphism defined using the geodesic flow on the sphere and the meridian S x is a geodesic submanifold, it necessarily preserves the submanifold T ∗ S x . Now, if we let q : ( Y \ O p (Λ) , ξ ) −→ ( Y (cid:48) , ξ (cid:48) )be the quotient map realizing the contact (+1)–surgery on Λ, the image q ( J ( S x )) is acontact submanifold M x which is itself contactomorphic to the contact (+1)–surgery of the1–jet space J ( S x ) along the stabilized Legendrian Λ ∩ J ( S x ). Then the contact manifold( M x , ξ (cid:48) ) is overtwisted for every x ∈ S n − , even in the complement of the submanifold S × R .See [19, Theorem 1.2] and [50, Exercise 11.2.10] for details on an overtwisted disk for ( M x , ξ (cid:48) ),and see Figure 7 for a schematic depiction. The entire picture is symmetric about x ∈ S n − ,and thus the construction defines a plastikstufe P with spherical core.It remains to show that this plastikstufe P has trivial rotation class and that it is containedin a smooth ball. We prove these claims simultaneously by showing that an open leaf of P iscontained in a Legendrian disk. Indeed, an open leaf of P is given as the union of Legendrianarcs in M x and we can consider an isotopy between this arc and a small Legendrian arc inΛ ∩J ( S x ) disjoint from the two vertical lines S × R . Then by considering this symmetricallywith respect to the point x ∈ S n − , we get an isotopy from an open leaf of P to an annulus S n − × [0 , ⊆ Λ, and since the Legendrian Λ is a sphere this annulus extends to a Legendriandisk inside the Legendrian Λ. (cid:3)
This concludes the equivalence (1) ⇐⇒ (5) in Theorem 1.1. This equivalence already sufficesto prove the two applications Proposition 8.5 and Corollary 8.6 on the existence of Weinsteincobordisms with an overtwisted concave end, which we explain in Section 8. However, wefollow the natural order and proceed with the remaining equivalence in the statement ofTheorem 1.1. 6. Stabilization of Legendrians and open books
In this section we conclude the proof of Theorem 1.1, by proving the equivalence (3) ⇐⇒ (6).To do this, in Subsection 6.3 we will relate two known procedures in contact topology: thestabilization of a Legendrian submanifold and the stabilizations of a compatible open book.The link between these two procedures can be established through Lagrangian surgery [46],also referred to as Polterovich surgery, the details of which are first explained in Subsection6.2. The results in Subsections 6.2 and 6.3 imply the following result. Theorem 6.1.
Let ( S n − , ξ − ) be the contact manifold supported by the open book whosepage is ( T ∗ S n − , λ st ) and whose monodromy is the left handed Dehn twist along the zerosection. Then the standard Legendrian unknot in ( S n − , ξ − ) is loose. In light of Theorem 2.17, Theorem 6.1 implies (3) ⇐⇒ (6) and thus Theorem 1.1. Indeed, thefact that any overtwisted contact manifold admits a negatively stabilized open book followsquickly from known results as we now explain.Let ( Y, ξ ) be an overtwisted contact structure, and note that the set of almost contact struc-tures on the sphere forms a group under connected sum [36, Chapter 4.3]. Now the existence h –principle [4, Theorem 1.2] implies that there is an overtwisted contact structure ( Y, η ) suchthat the contact connected sum ( Y S n − , η ξ − ) is in the same homotopy class of almostcontact structures as the given contact manifold ( Y, ξ ), and since the contact structures ξ and η ξ − are both overtwisted, they are necessarily isotopic. Now E. Giroux’s existence Theorem2.14 states that the contact structure ( Y, η ) is compatible with an open book (
W, λ, ϕ ) and,by using his Theorem 2.17, the negative stabilization of the open book (
W, λ, ϕ ) supports the
EOMETRIC CRITERIA FOR OVERTWISTEDNESS 27 contact structure (
Y, η ξ − ), which is isotopic to ( Y, ξ ). This shows the implication (1) ⇒ (6),and therefore Theorem 6.1 is the main remaining ingredient in order to prove the equivalence(1) ⇐⇒ (6). Let us then move towards the proof of Theorem 6.1.6.1. Legendrians in open books.
In order to prove Theorem 6.1, we develop some com-binatorics for describing Legendrian submanifolds in adapted open books decompositions.Let (
Y, ξ ) = OB(
W, λ, ϕ ), and recall that if L ⊆ ( W, λ ) is an exact Lagrangian, it determinesa Legendrian Λ ⊆ Y as noted in Subsection 2.4. The relationship was denoted by theequality ( Y, ξ,
Λ) = OB(
W, λ, ϕ, L ), and we emphasize that the Legendrian OB(
W, λ, ϕ, L )is contactomorphic to the Legendrian defined by OB(
W, λ, ψ ◦ ϕ ◦ ψ − , ψ ( L )), and typicallydistinct from the Legendrian defined by OB( W, λ, ψ ◦ ϕ ◦ ψ − , L ). In particular, the LegendrianOB( W, λ, ϕ, L ) is contactomorphic to OB(
W, λ, ϕ, ϕ ( L )).These observations are relevant to the proof and understanding of Theorem 6.1. The nextsubsection contains the results expressing Lagrangian surgery on two Lagrangians in termsof Legendrian connected sums of their Legendrian lifts.6.2. Lagrangian Surgery and Legendrian Sums.
The Dehn–Seidel twists [51][ChapterI.2] along exact Lagrangian spheres are an important class of compactly supported exactsymplectomorphisms of a Liouville domain (
W, λ ). Given a contact manifold, an adaptedopen book decomposition precisely consists of a Liouville domain, the page, and a symplecticmonodromy, which oftentimes consists of Dehn–Seidel twists. From this viewpoint, it is rele-vant for the study of contact topology to reinterpret the action of Dehn twists on Lagrangiansin terms of their Legendrian lifts. This is the aim of this subsection.We focus on the case where L ⊆ ( W, λ ) is an exact Lagrangian and S ⊆ W is a Lagrangiansphere transversely intersecting L in one point. In this case, the Dehn twist of L around S can be interpreted as the Polterovich surgery [28, 46] of L and S , denoted by L + S . Thedefinition and details of the Polterovich surgery will be given momentarily, after Remark 6.4below. For now, we state its relation to Dehn twists: Theorem 6.2 ([52]) . The Lagrangian surgery S + L is Lagrangian isotopic to τ S ( L ) .The Lagrangian surgery L + S is Lagrangian isotopic to τ − S ( L ) . (a) The Legendrian lift of τ S ( L ). (b) The Legendrian lift of τ − S ( L ). Figure 8.
The statement of Theorem 6.3.We now model this operation in terms of the fronts of Legendrian lifts Λ and Σ of the exactLagrangians L and S . The main technical result in this section is the following theorem. Theorem 6.3.
Let
L, S ⊆ ( W, λ ) be two exact Lagrangians transversely intersecting at apoint p = L ∩ S , and consider the contactization ( Y, ξ ) = ( W × R ( z ) , ker { dz − λ } ) of ( W, λ ) .There exists a Darboux chart in ( Y, ξ ) centered at p ∈ ( W, λ ) such that the front projectionof the Legendrian lift of S + L is as depicted in Figure 8.A.There exists a Darboux chart in ( Y, ξ ) centered at p ∈ ( W, λ ) such that the front projectionof the Legendrian lift of L + S is as depicted in Figure 8.B. Remark 6.4.
Figure 8 depicts the following situation. The lower horizontal sheet is the liftof a Lagrangian disk D L contained in the exact Lagrangian L centered at p , whereas theupper horizontal sheet is the lift of a Lagrangian disk D S contained is S also centered at p .Note that there exists a unique Reeb chord connecting the Legendrian lifts of the Lagrangiansdisks D L and D S , corresponding to the intersection point p = L ∩ S in the Lagrangianprojection. Then Figures 8.A and B are obtained by respectively substituting this uniquelocal Reeb chord by either a rotationally symmetric cusp or the rotationally symmetric cone.The Legendrian isotopy class of the fronts in Figure 8.A and 8.B are respectively referred toas the cusp-sum and cone-sum, or the cusp and the cone, of Λ and Σ along the Reeb chordover the intersection point p = L ∩ S . (cid:3) Let us now review L. Polterovich’s Lagrangian surgery [46] and prove Theorem 6.3.Consider local coordinates ( q , . . . , q n − , p , . . . , p n − ) ∈ R n − such that the Lagrangians L and S are locally expressed as L = { p = 0 , . . . , p n − = 0 } , S = { q = p , . . . , q n − = p n − } and the Liouville form reads λ st = n − (cid:88) i =1 p i dq i . The Lagrangian surgeries S + L and L + S are respectively described in terms of two La-grangian handles Γ ± [46]. These Lagrangian handles are depicted in Figure 9, and in orderto parametrize them we use coordinates t = ( t , . . . , t n − ) ∈ R n − . (a) The Lagrangian handle Γ + . (b) The Lagrangian handle Γ − . Figure 9.
The Lagrangian handles Γ ± ⊆ R n − ( q, p ).First, we consider the case of the positive Lagrangian handle Γ + ; it can be described via theparametrization Γ + : R n − \ { } −→ R n − defined asΓ + ( t , . . . , t n − ) = (cid:0) ( µ + µ − ) t , . . . , ( µ + µ − ) t n − , µt , . . . , µt n − (cid:1) where µ = n − (cid:88) i =1 t i . Note that we have the two asymptotics lim µ →∞ Γ + ⊆ S and lim µ → Γ + ⊆ L . By definition, thePolterovich surgery S + L is obtained by gluing the above positive Lagrangian handle Γ + tothe Lagrangian L at the limit µ = 0, and to the Lagrangian S at the limit µ = ∞ .Analogously, the Polterovich surgery L + S is obtained by using the negative Lagrangianhandle Γ − : R n − \ { } −→ R n − parametrized byΓ − ( t , . . . , t n − ) = (cid:0) ( µ − µ − ) t , . . . , ( µ − µ − ) t n − , µt , . . . , µt n − (cid:1) . This parametrization satisfies the asymptotics lim µ →∞ Γ − ⊆ S and lim µ → Γ − ⊆ L , and can beglued to L and S in the asymptotic limits, thus constructing the Lagrangian L + S . EOMETRIC CRITERIA FOR OVERTWISTEDNESS 29
Remark 6.5.
The Lagrangian handles Γ ± can be parametrized to be not only asymptoticto L and S but actually coincide with them in the local model. This is a matter of introduc-ing the appropriate cut–off functions, and the Lagrangian isotopy type of the constructionremains unchanged. (cid:3) Proof of Theorem 6.3.
In the contactization ( R n − ( q, p ; z ) , ker( dz − λ st )) of the standardexact Weinstein manifold ( R n − ( q, p ) , λ st ), the Lagrangian L described above lifts to theLegendrian Λ = { ( q , . . . , q n − , , . . . ,
0; 0) } and the Lagrangian S lifts to the LegendrianΣ = { ( q , . . . , q n − , q , . . . , q n − ; ( q + . . . + q n − ) / } . We can lift the exact Lagrangian Γ + to the contactization via z = z ( t , . . . , t n − ): dz ( t ) = n − (cid:88) i =1 µt i d (cid:0) ( µ + µ − ) t i (cid:1) = n − (cid:88) i =1 ( µ + 1) t i dt i + n − (cid:88) i =1 µt i (1 − µ − ) dµ == n − (cid:88) i =1 ( µ + 1) t i dt i + ( µ − dµ Hence the partial derivatives of z ( t ) are: ∂z ( t ) ∂t i = ( µ + 1) t i + ( µ − t i = (3 µ − t i . Thus the z –coordinate of the lift is parametrized by z ( t ) = ( µ − µ ) and in the frontprojection R n ( q , . . . , q n − , z ) we obtain a rotationally symmetric cusp. Part of the frontprojections in dimensions 3 and 5 are depicted in Figures 10 and 11. Figure 10.
Front projection to R ( q , z ) of the Legendrian lift of the positiveLagrangian handle Γ + ⊆ R ( q , p , z ) for t ∈ [ − . , − . ∪ [0 . , . S + L in terms of the cusp-sum of the two LegendriansΛ and Σ respectively lifting L and S , and concludes the first statement of Theorem 6.3.Regarding the Legendrian lift of the Polterovich surgery L + S , the z –coordinate of the liftto the contactization satisfies dz ( t ) = n − (cid:88) i =1 µt i d (cid:0) ( µ − µ − ) t i (cid:1) = n − (cid:88) i =1 ( µ − t i dt i + ( µ + 1) dµ. Thus we conclude that the partial derivatives of z ( t ) are given by ∂z ( t ) ∂t i = (3 µ + 1) t i dt i Figure 11.
Front projection to R ( q , q , z ) of the Legendrian lift of Γ + ⊆ R with ( t , t ) in the range [ − . , − . × [ − . , − . ∪ [0 . , . × [0 . , . z ( t ) = ( µ + µ ) provides a lift for Γ − . The front projection is depicted in Figures 12and 13 in the 3–dimensional and 5–dimensional cases. Figure 12.
Front projection to R ( q , z ) of the Legendrian lift of the handleΓ − ⊆ R ( q , p , z ) with t ∈ [ − . , − . ∪ [0 . , . (cid:3) Loose Legendrians in open books.
In order to show that the Legendrian unknot inthe contact manifold ( S n − , ξ − ) = OB( T ∗ S n − , τ − ) is a loose Legendrian submanifold, weneed an understanding of looseness and the standard unknot in the open book framework.This is the content of Propositions 6.6 and 6.7, which we use in order to prove Theorem 6.1. Proposition 6.6.
Let ( Y, ξ ) = OB(
W, λ, ϕ ) be a contact manifold and ( W ∪ H, λ, ϕ ◦ τ S ) apositive stabilization. The Legendrian lift of S to ( Y, ξ ) is the standard unknot. EOMETRIC CRITERIA FOR OVERTWISTEDNESS 31
Figure 13.
Front projection to R ( q , q , z ) of the Legendrian lift of Γ − ⊆ R with parameters ( t , t ) ∈ [ − . , − . × [ − . , − . ∪ [0 . , . × [0 . , . Proposition 6.7.
Let ( W ∪ H, λ, ϕ ◦ τ S ) be a positively stabilized open book and L ⊆ W an exact Lagrangian which transversely intersects S in one point. Then the Legendrian ( W ∪ H, λ, ϕ ◦ τ S , L ) is contactomorphic to the Legendrian ( W ∪ H, λ, ϕ ◦ τ S , τ − S ( L )) and theLegendrian ( W ∪ H, λ, ϕ ◦ τ S , τ S ( L )) is loose.Proof. Choose a Legendrian lift Λ for the Lagrangian L which has angle θ = 0 at the inter-section point L ∩ S , and a Legendrian lift for S with angle θ = ε for a small constant ε ∈ R + .Theorem 6.3 implies that the Legendrian lifts of τ S ( L ) and τ − S ( L ) are represented by thecusp and cone sums Legendrian fronts. Indeed, since they intersect in one point, we know byTheorem 6.2 that τ S ( L ) = S + L and τ − S L = L + S . Then by Theorem 6.3 the Legendrianlift of S + L corresponds to the cusp-sum, and the Legendrian lift of L + S corresponds totheir cone-sum. Note that the Legendrian lift of S is the Legendrian unknot contained in aDarboux ball which is disjoint from the Legendrian Λ, and since any two Darboux balls arecontact isotopic we have that cone or cusp summing with the unknot is a local operation onthe Legendrian Λ. Let us now discuss the two cases.For the Legendrian OB( W ∪ H, λ, ϕ ◦ τ S , τ − S ( L )), we note that cone-summing a Legendrianwith a small Legendrian unknot does not change the Legendrian isotopy type since this isjust the S n − –spinning of the first Legendrian Reidemeister move. Therefore Legendrian Λis Legendrian isotopic to the Legendrian lift of the exact Lagrangian L + S = τ − S ( L ).In contrast, the situation is different for the Legendrian OB( W ∪ H, λ, ϕ ◦ τ S , τ S ( L )). Indeed,observe that the cusp-sum of a Legendrian submanifold with a small Legendrian unknot ex-plicitly creates a loose chart [16, 44] and therefore the Legendrian lift of the exact Lagrangian τ S ( L ) = S + L is actually a loose Legendrian. (cid:3) Propositions 6.6 and 6.7 are the ingredients needed to prove Theorem 6.1.
Proof of Theorem 6.1.
Consider the contact manifold( S n − , ξ − ) = OB( T ∗ L, λ st ; τ − L )obtained by negatively stabilizing the contact open book ( S n − , ξ st ) = OB( D n − , λ st ; id),where we have denoted L ∼ = S n − for the zero section of the stabilized Weinstein page.Let us choose a cotangent fiber in the Weinstein page ( T ∗ L, λ st ) and positively stabilizethe compatible open book above along this cotangent fiber. The Weinstein page ( W, λ ) = T ∗ S n − ∪ H of the resulting open book is a plumbing of two copies of the Weinstein structure( T ∗ S n − , λ st ) whose exact Lagrangian zero sections L and S intersect in one point.First, the Legendrian Λ = OB( W, λ, τ − L ◦ τ S , S ) is the standard Legendrian unknot byProposition 6.6. And second, the Legendrian submanifold Λ (cid:96) = OB( W, λ, τ − L ◦ τ S , τ S ( L ))is a loose Legendrian by Proposition 6.7. In consequence, suffices to show that these twoLegendrians are contactomorphic, which follows from the fact that the Legendrian Λ iscontactomorphic to the LegendrianOB( W, λ, τ − L ◦ τ S , ( τ − L ◦ τ S )( S ))and the exact Lagrangian isotopy ( τ − L ◦ τ S )( S ) = τ − L ( S ) = S + L = τ S ( L ). (cid:3) Proof of Theorem 1.1
In this section we formally prove Theorem 1.1 using the results in Sections 3, 4, 5, andSection 6. First, the h -principle [4, Theorem 1.2] directly gives the implications (1) ⇒ (2)and (1) ⇒ (4). The implication (1) ⇒ (3) also follows directly from [4, Theorem 1.2],or alternatively using [45, Theorem 1.1], which states (4) ⇒ (3). The same h -principle[4, Theorem 1.2] gives the implication (1) ⇒ (6), as explained in Section 6 right after thestatement of Theorem 6.1. Finally, the implication (1) ⇒ (5) follows from the implication(6) ⇒ (5), which itself follows from the relation between Dehn twists in the symplecticmonodromy of an adapted open book and contact surgeries, see for instance [40, Theorem4.4] and [11, Section 3].By the above paragraph, the implications (1) ⇒ (2) , (3) , (4) , (5) , (6) hold. Let us now usethe results in this article to conclude the converse. Indeed, Theorem 3.2 shows (2) ⇒ (1).The implication (3) ⇒ (1) is the content of Theorem 4.5. The implication (4) ⇒ (1) followsfrom the now proven implication (3) ⇒ (1) and (4) ⇒ (3), which holds by [45, Theorem 1.1].The implication (5) ⇒ (1) follows Theorem 5.4, which proves (5) ⇒ (4) and the implication(4) ⇒ (1). Finally, (6) ⇒ (1) follows from Theorem 6.1, which proves (6) ⇒ (3), andTheorem 4.5, which shows (3) ⇒ (1). (cid:3) Let us now provide two applications of Theorem 1.1 to contact topology.8.
Applications
In this section we explore consequences of Theorem 1.1. Subsection 8.1 discusses neigh-borhoods in contact topology in relation to Theorem 1.1 and Subsection 8.2 constructs aWeinstein concordance between an overtwisted contact structure on the (2 n − S n − , ξ st ).8.1. Neighborhood size and contact squeezing.
Theorem 1.1 emphasizes in its firstequivalence 1 = 2 the importance of the size of a neighborhood of a contact submanifold.In this direction it is relevant to understand the dichotomy between tight and overtwistedcontact structures in terms of small and large neighborhoods.
Theorem 8.1.
Let ( Y, ker α ) be an overtwisted contact manifold. There exists a radius R ∈ R + such that for any R > R , there exists a compactly supported contact isotopy f t : ( Y × C , ker { α + λ st } ) −→ ( Y × C , ker { α + λ st } ) EOMETRIC CRITERIA FOR OVERTWISTEDNESS 33 such that f = id and f ( Y × D ( R )) ⊆ Y × D ( R ) . This follows immediately from the (1) ⇐⇒ (2) equivalence in Theorem 1.1 together with the h –principle for isocontact embeddings into overtwisted manifolds [4, Corollary 1.4]. Theorem8.1, being a contact squeezing result, relates to non–orderability [4, 13, 20, 32]. The radius R in the statement of Theorem 8.1 can be taken to be any radius greater than the minimalradius R c such that the contact manifold ( Y × D ( R c ) , ker { α + λ st } ) is overtwisted. Thus inTheorem 8.1 we can take R to be, for instance, twice R c .In contrast with Theorem 8.1, there are instances of contact non–squeezing: Proposition 8.2.
Let ( Y, ker α ) be a contact -manifold. Then there exists a small radius δ ∈ R + such that for any R > δ there exists no contact embedding ( Y × D ( R ) , ker { α + λ st } ) −→ ( Y × D ( δ ) , ker { α + λ st } ) . This proposition follows from [15, Proposition 11] and known obstructions to fillability [48].
Remark 8.3.
Proposition 8.2 also holds in higher–dimensions for any weakly fillable contactstructure ( Y n − , ker α ), as it follows by combining F. Bourgeois’ construction [5] of contactstructures in Y × T and the observation [42, Example 1.1] that the construction preservesweak fillability. (cid:3) In addition, we observe that the equivalence (1) ⇐⇒ (2) shows that contactomorphism typeis sensitive to dimensional stabilization. Corollary 8.4.
There exist closed smooth manifolds Y with two non–isomorphic contactstructures ker α and ker α such that ( Y × C , ker { α + λ st } ) and ( Y × C , ker { α + λ st } ) arecontactomorphic.Proof. For instance, we can consider ξ = ker α and ξ = ker α to be two different over-twisted contact structures on any integral homology 3-sphere M . Then the almost contactstructures on the smooth manifold M are classified by homotopy classes of sections of a SO (3) /U (1)–bundle over M , and the obstruction classes thus live in H ( M, Z ) [36, Chapter4.3]. The same computation shows that the set of homotopy classes of almost contact struc-ture in the 5–fold M × C is determined by the first Chern c ∈ H ( M × C , Z ) ∼ = H ( M, Z ) ∼ = 0,and thus there exists a unique class of almost contact structures on M × C . In consequencethe two hyperplane fields ker { α + λ st } and ker { α + λ st } become homotopic as almost contactstructures in Y × C , and since both of these contact structures are overtwisted at infinity,they are isotopic contact structures [4, 24]. (cid:3) Notice that the homotopy class of a compatible almost complex structure structure distin-guishes the symplectizations of two different overtwisted contact structures on S , and thusthe symplectizations are not symplectomorphic. Hence Theorem 8.4 shows that the contac-tizations of two non–isomorphic complete symplectizations can be contactomorphic.8.2. Weinstein cobordisms with overtwisted concave end.
In this subsection we con-struct a smooth concordance with a Weinstein structure between an overtwisted contactstructure on S n − and its standard contact structure ( S n − , ξ st ) for the higher dimensionsdim( S n − ) ≥
5. This contrasts with the fact that such a concordance does not exist fordim( S ) = 3 and also provides the general existence result stated in Theorem 8.6. Proposition 8.5.
Suppose that n ≥ , then there is a Weinstein structure ( M, λ, f ) onthe smoothly trivial cobordism M ∼ = [0 , × S n − such that ( ∂ + M, λ ) ∼ = ( S n − , ξ st ) and ( ∂ − M, ker( λ )) is the unique overtwisted contact sphere in the almost contact class of ξ st . Proof.
Let ( S n − , ξ ot ) be the overtwisted contact sphere in the standard almost contact classand let M be its symplectization. The standard Weinstein structure on M can be homotopedto one with a cancelling pair of critical points, one of index n − n . Considera middle contact level ( Y, ξ ) between these two critical points. Then we can view Y either asa subcritical isotropic surgery on ( S n − , ξ ot ) induced by the bottom half of the cobordism M , or as the result of a (+1)–surgery along a Legendrian sphere Λ ⊆ ( S n − , ξ ot ) inducedby the top half of the cobordism M . Note that the contact structure ( Y, ξ ) can be obtainedas a subcritical surgery on an overtwisted manifold and thus it is overtwisted. See Figure 14for a schematic picture of the forthcoming argument.Now let Λ ⊆ ( S n − , ξ st ) be the loose Legendrian sphere which is in the same formal Leg-endrian isotopy class as the Legendrian sphere Λ ⊆ ( S n − , ξ ot ). Note that ( S n − , ξ ot )and ( S n − , ξ st ) are in the same almost contact class, hence the formal Legendrian isotopyclasses are canonically identified once we fix a diffeomorphism realizing the almost contactequivalence. Then performing a (+1)–surgery along the loose Legendrian Λ gives a contactmanifold ( Y, ξ (cid:48) ) which is almost contact equivalent to (
Y, ξ ). Theorem 1.1 implies that thecontact structure (
Y, ξ (cid:48) ) is overtwisted, and therefore the contact structure ξ is isotopic to ξ (cid:48) .Let M b be the bottom half of the Weinstein cobordism M and M t the Weinstein cobordismfrom ( Y, ξ (cid:48) ) to ( S n − , ξ st ) induced by the above (+1)–surgery on Λ . Then the glued cobor-dism (cid:102) M := M b ∪ Y M t is a Weinstein cobordism from ( S n − , ξ ot ) to ( S n − , ξ st ) which isdiffeomorphic to the smooth concordance M . (cid:3) Figure 14.
On the left, the symplectization of ( S n − , ξ ot ) with a cancellingpair of critical points. On the right, Weinstein concordance from ( S n − , ξ ot )to ( S n − , ξ st ) obtained using (+1)–surgery on a loose Legendrian.Proposition 8.5 describes a strictly higher–dimensional phenomenon in contact topology.Indeed, it follows from the functoriality of Seiberg–Witten invariants that there exists nosuch Weinstein concordance in the case n = 2 [43], [38, Theorem 2.3], [41, Chapter 7].The Weinstein concordance among spheres constructed in Proposition 8.5 can now be gluedto any Weinsten cobordism, thus proving the existence of all Weinstein cobordisms withan overtwisted concave boundary and arbitrary convex end which are not prohibited bytopological restrictions: Theorem 8.6.
Let ( Y − , ξ ot ) and ( Y + , ξ ) be coorientable contact manifolds with the contactstructure ( Y − , ξ ot ) being overtwisted and dim( Y − ) = dim( Y + ) ≥ .Suppose there exists a smooth cobordism W from Y − to Y + such that EOMETRIC CRITERIA FOR OVERTWISTEDNESS 35 a. The relative homotopy type of W with respect to its boundary deformation retractsonto a half-dimensional CW complex. b. W admits an almost complex structure J such that the restriction J | Y + , resp. J | Y − ,is homotopic through almost contact structures to ξ , resp. ξ ot .Then there exists a Weinstein cobordism ( W, λ, ϕ ) with concave boundary ∂ − ( W, λ ) = ( Y − , ξ ot ) and convex boundary ∂ + ( W, λ ) = ( Y + , ξ ) .Proof. Let ( Y + , ξ +ot ) be the overtwisted contact structure which is in the same almost contacthomotopy class as ξ . The existence theorem for flexible Weinstein cobordisms [16] providesa flexible Weinstein structure ( λ f , ϕ f ) on W such that the almost complex structure J iscompatible with the symplectic 2–form dλ f after homotopy, and ∂ − ( W, λ f , ϕ f ) = ( Y − , ξ ot ).Since the Weinstein cobordism ( λ f , ϕ f ) is flexible by construction it follows that the contactstructure ∂ + ( W, λ f ) in the convex end is overtwisted by using Proposition 2.12. Note also thatthe contact structure in this convex end ∂ + ( W, λ f ) is in the same almost contact homotopyclass as the initial overtwisted contact structure ξ +ot since both are homotopic to J | Y + . Inconsequence, we obtain the contactomorphism ∂ + ( W, λ f ) ∼ = ( Y + , ξ +ot ).Let us now consider the Weinstein concordance M = ([0 , × S n − , λ, f ) constructed inProposition 8.5 and the symplectization S ( Y + , ξ ) of the contact structure ( Y + , ξ ). Then theconnected sum cobordism M S ( Y + , ξ ) is a Weinstein cobordism which is diffeomorphic tothe concordance [0 , × Y and satisfies ∂ − M S ( Y + , ξ ) ∼ = ( Y + , ξ +ot ) , ∂ + M S ( Y + , ξ ) ∼ = ( Y + , ξ ) . Thus, we can concatenate the Weinstein cobordism (
W, λ f , ϕ f ) to this Weinstein smoothconcordance M S ( Y + , ξ ) along their common contact boundary ∂ + ( W, λ f ) ∼ = ( Y + , ξ +ot ) ∼ = ∂ − M S ( Y + , ξ )and thus construct the Weinstein cobordism ( W, λ, ϕ ) with the desired properties. (cid:3)
Remark 8.7.
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