Algebraic and Giroux torsion in higher-dimensional contact manifolds
AAlgebraic and Giroux torsion in higher-dimensionalcontact manifolds
Agustin Moreno
Abstract
We construct examples in any odd dimension of contact manifolds withfinite and non-zero algebraic torsion (in the sense of [LW11]), which are there-fore tight and do not admit strong symplectic fillings. We prove that Girouxtorsion implies algebraic 1-torsion in any odd dimension, which proves a con-jecture in [MNW13]. These results are part of the author’s PhD thesis [Mo2].
Contents a r X i v : . [ m a t h . S G ] M a r Introduction
In this paper, and its followup [Mo], we address the general problem of construct-ing “interesting” examples of higher-dimensional contact manifolds, and developingtechniques in order to compute SFT-type holomorphic curve invariants.We will construct examples of contact manifolds in every odd dimension, present-ing a geometric structure which is a higher-dimensional version of that of a spinalopen book decomposition or SOBD, as defined in [L-VHM-W] in dimension 3. Thetype of SOBD present in our examples, which one could call partially planar , mimicsthe notion of planar m -torsion domains as defined in [Wen2]. Indeed, it consists oftwo surface fibrations over a higher-dimensional contact base, one of them havinggenus zero fibers, glued together along a contact fibration over a Liouville domain.This geometric structure can be “detected” algebraically by algebraic torsion, aholomorphic-curve contact invariant. For suitable data, the surface fibers becomeholomorphic, and are leaves of a finite energy foliation of the symplectization R × M .The isolated ones may be counted in a suitable way, and the result is an invariantwhich “recovers” the number m . This is the idea inspiring algebraic m -torsion.We exhibit a detailed construction of an isotopy class of contact forms, which is“supported” by the SOBD, so that one may view these contact forms as “Giroux”forms. We will estimate the algebraic torsion of these examples, which we show isfinite, and, in certain cases, non-zero. In those cases, the contact manifolds are tightand admit no strong symplectic fillings.We will also relate algebraic torsion with a geometric condition, Giroux torsion .While this is a classical notion in dimension 3, the higher-dimensional version wasintroduced in [MNW13]. We will show that the geometric presence of certain tor-sion domains inside a contact manifold can be detected algebraically by SFT. Moreconcretely, Giroux torsion implies algebraic 1-torsion, in any odd dimension. Thisproves a conjecture in [MNW13].The proof of this result is carried out by interpreting the Giroux torsion domainsas being supported by a suitable SOBD, which we call a
Giroux SOBD , for which wegive a notion of a “Giroux form”. The result follows by adapting our computationsfor the above partially planar model contact manifolds.In order to carry out our computations, we need a very detailed understanding ofholomorphic curves in the symplectization of our model contact manifolds, and theSOBD structures, together with the associated finite energy foliations, are crucial to-wards this end. The key technical inputs are: transversality of the genus zero curvesin the foliation, and a uniqueness result for holomorphic curves (Theorem 2.10).Proving transversality is needed so that indeed one has a space of isolated curvesto count, whereas uniqueness is necessary to know precisely what to count. Fortransversality, a standard technique in dimension three is the automatic transversal-ity criterion of [Wen1], which consists in checking a fairly straightforward numerical2nequality involving topological data associated to a given curve. For uniqueness,one can sometimes resort to Siefring’s intersection theory for punctured holomorphiccurves in dimension four [Sie11]. In higher dimensions, things become cumbersome.To prove transversality, we resorted to a “hands-on” analytical approach of comput-ing precisely the kernel of the linearization of the Cauchy-Riemann operator, andcheck that its dimension coincides with its Fredholm index, from which transver-sality follows. For uniqueness, we resorted to a combination of energy estimates,holomorphic cascades and geometric arguments.
On the invariant.
The invariant we will use, algebraic torsion, was definedin [LW11], and is a contact invariant taking values in Z ≥ ∪ {∞} . It was intro-duced, using the machinery of Symplectic Field Theory , as a quantitative way ofmeasuring non-fillability, giving rise to a “hierarchy of fillability obstructions”, cf.[Wen2]. At least morally, 0-torsion should correspond to overtwistedness, whereas1-torsion is implied by Giroux torsion (the converse is not true). Having 0-torsion isactually equivalent to being algebraically overtwisted , which means that the contacthomology, or equivalently its SFT, vanishes (Proposition 2.9 in [LW11]). This iswell-known to be implied by overtwistedness, but the converse is still wide open.The key fact about this invariant is that it behaves well under exact symplec-tic cobordisms, which implies that the concave end inherits any order of algebraictorsion that the convex end has. Thus, algebraic torsion may be also thought ofas an obstruction to the existence of exact symplectic cobordisms. In particular, itserves as an obstruction to symplectic fillability. Moreover, there are connections todynamics: any contact manifold with finite torsion satisfies the Weinstein conjecture(i.e. there exist closed Reeb orbits for every contact form).One should mention that there are other notions of algebraic torsion in theliterature which do not use SFT, but which are only 3-dimensional (see [KM-VHM-W] for the version using Heegard Floer homology, or the appendix in [LW11] byHutchings, using ECH).
Statement of results.
For the SFT setup, we follow [LW11], where we referthe reader for more details. We will take the SFT of a contact manifold (
M, ξ ) (withcoefficients) to be the homology H SF T ∗ ( M, ξ ; R ) of a Z -graded unital BV ∞ -algebra( A [[ (cid:126) ]] , D SF T ) over the group ring R R := R [ H ( M ; R ) / R ], for some linear subspace R ⊆ H ( M ; R ). Here, A = A ( λ ) has generators q γ for each good closed Reeb orbit γ with respect to some nondegenerate contact form λ for ξ , (cid:126) is an even variable,and the operator D SF T : A [[ (cid:126) ]] → A [[ (cid:126) ]]is defined by counting rigid solutions to a suitable abstract perturbation of a J -holomorphic curve equation in the symplectization of ( M, ξ ). It satisfies3 D SF T is odd and squares to zero, • D SF T (1) = 0, and • D SF T = (cid:80) k ≥ D k (cid:126) k − , where D k : A → A is a differential operator of order ≤ k , given by D k = (cid:88) Γ + , Γ − ,g,d | Γ + | + g = k n g (Γ + , Γ − , d ) C (Γ − , Γ + ) q γ − . . . q γ − s − z d ∂∂q γ +1 . . . ∂∂q γ + s + The sum ranges over all non-negative integers g ≥
0, homology classes d ∈ H ( M ; R ) / R and ordered (possibly empty) collections of good closed Reeb orbits Γ ± = ( γ ± , . . . , γ ± s ± )such that s + + g = k . After a choice of spanning surfaces as in [EGH00] (p. 566, seealso p. 651), the projection to M of each finite energy holomorphic curve u can becapped off to a 2-cycle in M , and so it gives rise to a homology class [ u ] ∈ H ( M ),which we project to define [ u ] ∈ H ( M ; R ) / R . The number n g (Γ + , Γ − , d ) ∈ Q de-notes the count of (suitably perturbed) holomorphic curves of genus g with positiveasymptotics Γ + and negative asymptotics Γ − in the homology class d , includingasymptotic markers as explained in [EGH00], or [Wen3], and including rationalweights arising from automorphisms. C (Γ − , Γ + ) ∈ N is a combinatorial factor de-fined as C (Γ − , Γ + ) = s − ! s + ! κ γ − . . . κ γ − s − , where κ γ denotes the covering multiplicityof the Reeb orbit γ .The most important special cases for our choice of linear subspace R are R = H ( M ; R ) and R = { } , called the untwisted and fully twisted cases respectively,and R = ker Ω with Ω a closed 2-form on M . We shall abbreviate the lattercase as H SF T ∗ ( M, ξ ; Ω) := H SF T ∗ ( M, ξ ; ker Ω), and the untwisted case simply by H SF T ∗ ( M, ξ ) := H SF T ∗ ( M, ξ ; H ( M ; R )). Definition 1.1.
Let (
M, ξ ) be a closed manifold of dimension 2 n + 1 with a pos-itive, co-oriented contact structure. For any integer k ≥
0, we say that (
M, ξ )has Ω-twisted algebraic torsion of order k (or Ω-twisted k -torsion) if [ (cid:126) k ] = 0 in H SF T ∗ ( M, ξ ; Ω). If this is true for all Ω, or equivalently, if [ (cid:126) k ] = 0 in H SF T ∗ ( M, ξ ; { } ),then we say that ( M, ξ ) has fully twisted algebraic k -torsion.We will refer to untwisted k -torsion to the case Ω = 0, in which case R R = R and we do not keep track of homology classes. Whenever we refer to torsion withoutmention to coefficients we will mean the untwisted version. We will say that, if acontact manifold has algebraic 0-torsion for every choice of coefficient ring, then itis algebraically overtwisted , which is equivalent to the vanishing of the SFT, or its4ontact homology. By definition, k -torsion implies ( k + 1)-torsion, so we may defineits algebraic torsion to be AT ( M, ξ ; R ) := min { k ≥ (cid:126) k ] = 0 } ∈ Z ≥ ∪ {∞} , where we set min ∅ = ∞ . We denote it by AT ( M, ξ ), in the untwisted case.This construction is well-behaved under symplectic cobordisms: Any exact sym-plectic cobordism (
X, ω = dα ) with positive end ( M + , ξ + ) and negative end ( M − , ξ − )gives rise to a natural R [[ (cid:126) ]]-module morphism on the untwisted SFT,Φ X : H SF T ∗ ( M + , ξ + ) → H SF T ∗ ( M − , ξ − ) , a cobordism map . This implies that if ( M + , ξ + ) has k -torsion, then so does ( M − , ξ − ).There is also a version with coefficients for the case of non-exact cobordisms andfillings [LW11, Prop. 2.4].Examples of 3-dimensional contact manifolds with any given order of torsion k −
1, but not k −
2, were constructed in [LW11]. The underlying manifold is theproduct manifold M g := S × Σ, for Σ a surface of genus g which is divided into twopieces Σ + and Σ − along some dividing set of simple closed curves Γ of cardinality k , where the latter has genus 0, and the former has genus g − k + 1. The contactstructure ξ k is S -invariant and may be obtained, for instance, by a constructionoriginally due to Lutz (see [Lutz77]). Its isotopy class is characterized by the factthat every section { pt } × Σ is a convex surface with dividing set Γ. The behaviourof algebraic torsion under cobordisms then implies that there is no exact symplecticcobordisms having ( M g , ξ k ) and ( M g (cid:48) , ξ k (cid:48) ) as convex and concave ends, respectively,if k < k (cid:48) .The existence of the analogue higher dimensional contact manifolds was con-jectured in [LW11]. We will consider a modified version of their examples. Themodification we do here consists in taking the S -factor and replacing it by a closed (2 n − Y , having the special property that Y × I admits the structure ofa Liouville domain (here, I denotes the interval [ − , dα , and has disconnected contact-type boundary ∂ ( Y × I, dα ) = ( Y − , ξ − = ker α − ) (cid:70) ( Y + , ξ + = ker α + ), where Y ± coincide with Y as manifolds, but ( Y ± , ξ ± ) are not contactomorphic to each other. In fact, Y ± havedifferent orientations, and so they might not even be homeomorphic to each other(not every manifold admits an orientation-reversing homeomorphism). A Liouvilledomain of the form ( Y × I, dα ) is what we will call a cylindrical Liouville semi-filling (or simply a cylindrical semi-filling). Their existence in every odd dimensionwas established in [MNW13]. We immediately see that this generalizes the previous3-dimensional example, since S admits the Liouville pair α ± = ± dθ , which meansthat the 1-form e − s α − + e s α + is Liouville in S × R . We prove that the manifold5 P + M P - M Y M Y M Y � Y X Figure 1: The SOBD structure in M . Y × Σ indeed achieves ( k − Notation.
Throughout this paper, the symbol I will be reserved for the interval[ − , M = M g = Y × Σ into three pieces M g = M Y (cid:91) M ± P , where M Y = (cid:70) k Y × I × S , and M ± P = Y × Σ ± (see Figure 1). We have naturalfibrations π Y : M Y → Y × Iπ ± P : M ± P → Y ± , with fibers S and Σ ± , respectively, and they are compatible in the sense that ∂ (( π ± P ) − ( pt )) = k (cid:71) π − Y ( pt )While π Y has a Liouville domain as base, and a contact manifold as fiber, thesituation is reversed for π ± P , which has contact base, and Liouville fibers. This is aprototypical example of a spinal open book decomposition , or SOBD. While we willnot give a general definition of such a notion, we refer the reader to [Mo2] for atentative one.Using this decomposition, we can construct a contact structure ξ k which is asmall perturbation of the stable Hamiltonian structure ξ ± ⊕ T Σ ± along M ± P , andis a contactization for the Liouville domain ( Y × I, (cid:15)dα ) along M Y , for some small (cid:15) >
0. This means that it coincides with ker( (cid:15)α + dθ ), where θ is the S -coordinate.We will do this in detail in Section 2. 6 emark 1.2. Let us remark that, since the fibrations above are trivial, one canalways reverse their roles. More precisely, we could consider instead the “dual”SOBD: π ∗ Y : M Y → S ( π ± P ) ∗ : M ± P → Σ ± For these fibrations, we may also construct a contact form which is “supported” bythe SOBD. The resulting contact structure is isotopic to ξ k , which is what we expectfrom the point of view of a “Giroux correspondence” (in this more general setting).This is actually used for the results in [Mo].For the contact manifolds ( M g , ξ k ), we can estimate their algebraic torsion. First,recall that a contact structure is hypertight if it admits a contact form without con-tractible Reeb orbits (which we call a hypertight contact form). In particular, thereare no holomorphic disks in their symplectization, which implies that there is no0-torsion. By a well-known theorem by Hofer and its generalization to higher dimen-sions by Albers–Hofer (in combination with [BEM]), hypertight contact manifoldsare tight. Theorem 1.3.
For any k ≥ , and g ≥ k , the (2 n + 1) -dimensional contact mani-folds ( M g = Y × Σ , ξ k ) satisfy AT ( M g , ξ k ) ≤ k − . Moreover, if ( Y, α ± ) are hyper-tight, and k ≥ , the corresponding contact manifold ( M g , ξ k ) is also hypertight. Inparticular, AT ( M g , ξ k ) > , and it is tight. In fact, the examples of Theorem 1.3 admit Ω-twisted k − O := Ann( (cid:76) k H ( Y ; R ) ⊗ H ( S ; R )), the annihilator of (cid:76) k H ( Y ; R ) ⊗ H ( S ; R ) ⊆ H ( M g ; R ). Here, we take the homology of the subregion (cid:70) k Y × { } × S , lying along the region M Y where Σ ± glue together. Using [LW11,Prop. 2.4], we obtain: Corollary 1.4.
The examples of Theorem 1.3 do not admit weak fillings ( W, ω ) forwhich [ ω | M g ] is rational and lies in O . In particular, they are not strongly fillable. Remark 1.5. • By a result of Mitsumatsu in [Mit95], any 3-manifold Y which admits a smoothAnosov flow preserving a smooth volume form satisfies that Y × I can be enrichedwith a cylindrical Liouville semi-filling structure. Therefore any of these 3-manifoldscan be used in the construction of 5-dimensional contact models with AT ≤ k − k ≥ • The examples of Liouville cylindrical semi-fillings of [MNW13] satisfy the hyper-tightness condition. Then we have a doubly-infinite family of contact manifolds7ith 0 < AT ( M g , ξ k ) ≤ k −
1, in any dimension. These are then an instance ofhigher-dimensional tight but not strongly fillable contact manifolds, since they havenon-zero and finite algebraic torsion. For k = 2, this precisely computes the alge-braic torsion.The authors of [MNW13] define a generalized higher-dimensional version of thenotion of Giroux torsion. This notion is defined as follows: consider ( Y, α + , α − ) a Liouville pair on a closed manifold Y n − , which means that the 1-form β = ( e s α + + e − s α − ) is Liouville in R × Y . Consider also the Giroux π -torsion domain modeledon ( Y, α + , α − ) given by the contact manifold ( GT, ξ GT ) := ( Y × [0 , π ] × S , ker λ GT ),where λ GT = 1 + cos( r )2 α + + 1 − cos( r )2 α − + sin( r ) dθ (1)and the coordinates are ( r, θ ) ∈ [0 , π ] × S . Say that a contact manifold ( M n +1 , ξ )has Giroux torsion whenever it admits a contact embedding of (
GT, ξ GT ). In thissituation, denote by O ( GT ) ⊆ H ( M ; R ) the annihilator of R GT := H ( Y ; R ) ⊗ H ( S ; R ), viewed as a subspace of H ( M ; R ). The following was conjectured in[MNW13]: Theorem 1.6.
If a contact manifold ( M n +1 , ξ ) has Giroux torsion, then it has Ω -twisted algebraic 1-torsion, for every [Ω] ∈ O ( GT ) , where GT is a Giroux π -torsiondomain embedded in M . The proof uses the same techniques as Theorem 1.3, and the main idea is tointerpret Giroux torsion domains in terms of a specially simple kind of SOBD,which we call
Giroux SOBD .A natural corollary is the following:
Corollary 1.7.
If a contact manifold ( M n +1 , ξ ) has Giroux torsion, then it doesnot admit weak fillings ( W, ω ) with [ ω | M ] ∈ O ( GT ) and rational, where GT is aGiroux π -torsion domain embedded in M . In particular, it is not strongly fillable. This is essentially corollary 8.2 in [MNW13], which was obtained with differentmethods. Observe that if R GT = 0 then ( M, ξ ) does not admit weak fillings at all.This is in fact the condition used in [MNW13] to obstruct weak fillability.
Further work: a synopsis.
We now state a series of results, to be provenin the followup paper [Mo] (see also [Mo2]). In the following, we use the fact thatthe unit cotangent bundle of a hyperbolic surface fits into a cylindrical semi-filling[McD91]. 8 heorem 1.8.
Let ( M , ξ ) be a -dimensional contact manifold with Giroux tor-sion, and let Y be the unit cotangent bundle of a hyperbolic surface. If ( M =Σ × Y, ξ k ) is the corresponding -dimensional contact manifold of Theorem 1.3 with k ≥ , then there is no exact symplectic cobordism having ( M , ξ ) as the convexend, and ( M, ξ k ) as the concave end. In particular, we obtain
Corollary 1.9. If Y is the unit cotangent bundle of a hyperbolic surface, and ( M =Σ × Y, ξ k ) is the corresponding -dimensional contact manifold of Theorem 1.3 with k ≥ , then ( M, ξ k ) does not have Giroux torsion. Moreover, we have reasons, coming from string topology [CL09], to believe thatthe examples of Corollary 1.9 have untwisted algebraic 1-torsion (for any k ≥ Corollary 1.10.
There exist infinitely many non-diffeomorphic -dimensional con-tact manifolds ( M, ξ ) which are tight, not strongly fillable, and which do not haveGiroux torsion. To our knowledge, there are no other known examples of higher-dimensionalcontact manifolds as in Corollary 1.10. Also, we expect the above examples to havealgebraic 1-torsion.One can twist the contact structure of Theorem 1.3 close to the dividing set, byperforming the l -fold Lutz–Mori twist along a hypersurface H lying in ∂ ( (cid:70) k Y × I × S ). This notion was defined in [MNW13], and builds on ideas by Mori in dimension5 [Mori09]. The resulting contact structures are, in general, all homotopic as almostcontact structures, but in our case they are distinguishable by a suitable version ofcylindrical contact homology. By construction, all of these have Giroux torsion, soby Theorem 1.6 they have Ω-twisted 1-torsion, for [Ω] ∈ O = O ( GT ).As a corollary of Theorem 1.8, we get: Corollary 1.11.
Let Y be the unit cotangent bundle of a hyperbolic surface, andlet ( M = Σ × Y, ξ ) be the corresponding -dimensional contact manifold of Theorem1.3, with k ≥ . If ( M, ξ l ) denotes the contact manifold obtained by an l -fold Lutz–Mori twist of ( M, ξ ) , then there is no exact symplectic cobordism having ( M, ξ l ) asthe convex end, and ( M, ξ ) as the concave end (even though the underlying mani-folds are diffeomorphic, and the contact structures are homotopic as almost contactstructures). The results from [Mo] stated above make use of Richard Siefring’s intersectiontheory for holomorphic curves and hypersurfaces, as outlined in an appendix in9Mo2] written in coauthorship with Siefring, as a prequel of his upcoming work[Sie], and to appear as an independent article [MS19]. Another technical input isthe obstruction bundle technique as in Hutchings-Taubes [HT1, HT2]. The SOBD is“dualized” in the sense of Remark 1.2, and the finite energy foliation is replaced bya foliation by holomorphic hypersurfaces. Siefring’s intersection theory then impliesthat holomorphic curves with suitable asymptotic behaviour lie in the leaves of thefoliation. This, combined with symmetries in the setup and the obstruction bundletechnique, allows us to obtain our results, as well as information on the SFT of ourcontact manifolds.
Disclaimer 1.12.
Since the statements of our results make use of machinery fromSymplectic Field Theory, they come with the standard disclaimer that they assumethat its analytic foundations are in place. They depend on the abstract perturbationscheme promised by the polyfold theory of Hofer–Wysocki–Zehnder. We shall as-sume that it is possible to achieve transversality by introducing an arbitrarily smallabstract perturbation to the Cauchy-Riemann equation, and that the analogue ofthe SFT compactness theorem still holds as the perturbation is turned off. In prac-tice, this means that, in order to study curves for the perturbed data, we need toalso study holomorphic building configurations for the unperturbed one. However,we have taken special care in that the approach taken not only provides results thatwill be fully rigorous after the polyfold machinery is complete, but also gives severaldirect results that are already rigorous.
Acknowledgements.
First of all, my thanks go to my PhD supervisor, ChrisWendl, for introducing me to this project and for his support and patience through-out its duration. To Richard Siefring, for very helpful conversations and for co-authoring an appendix in [Mo2]. To Janko Latschev and Kai Cieliebak, for goingthrough the long process of reading [Mo2]. To Patrick Massot, Sam Lisi, and Mom-chil Konstantinov, for helpful conversations/correspondence on different topics.This research, forming part of the author’s PhD thesis, has been partly carriedout in (and funded by) University College London (UCL) in the UK, and by theBerlin Mathematical School (BMS) in Germany.
Guide to the document
The main construction is dealt with in Section 2. We show Fredholm regularity inSection 2.5, and uniqueness (Theorem 2.10) in Section 2.7. Theorem 1.3 is provedin Section 2.9.The proof of Theorem 1.6 is dealt with in Section 3, which is basically a refor-mulation of the previous sections, with the key input being an adaptation of theuniqueness Theorem 2.10. 10 asic notions
A contact form in a (2 n + 1)-dimensional manifold M is a 1-form α such that α ∧ dα n is a volume form, and the associated contact structure is ξ = ker α (we will assume all our contact structures are co-oriented). The Reeb vector fieldassociated to α is the unique vector field R α on M satisfying α ( R α ) = 1 , i R α dα = 0A T -periodic Reeb orbit is ( γ, T ) where γ : R → M is such that ˙ γ ( t ) = T R α ( γ ( t )), γ (1) = γ (0). We will often just talk about a Reeb orbit γ withoutmention to T , called its period, or action. If τ > γ ( τ ) = γ (0), and k ∈ Z + is such that T = kτ , we say that the covering multiplicityof ( γ, T ) is k . If k = 1, then γ is said to be simply covered (otherwise it is multiplycovered). A periodic orbit γ is said to be non-degenerate if the restriction of the time T linearised Reeb flow dϕ T to ξ γ (0) does not have 1 as an eigenvalue. More generally,a Morse–Bott submanifold of T -periodic Reeb orbits is a closed submanifold N ⊆ M invariant under ϕ T such that ker( dϕ T − ) = T N , and γ is Morse–Bott wheneverit lies in a Morse–Bott submanifold, and its minimal period agrees with the nearbyorbits in the submanifold. The vector field R α is non-degenerate/Morse–Bott if allof its closed orbits are non-degenerate/Morse–Bott.A stable Hamiltonian structure (SHS) on M is a pair H = (Λ , Ω) consisting of aclosed 2-form Ω and a 1-form Λ such thatker Ω ⊆ ker d Λ , and Ω | ξ is non-degenerate, where ξ = ker ΛIn particular, ( α, dα ) is a SHS whenever α is a contact form. The Reeb vectorfield associated to H is the unique vector field on M defined byΛ( R ) = 1 , i R Ω = 0There are analogous notions of non-degeneracy/Morse–Bottness for SHS.A symplectic form in a 2 n -dimensional manifold W is a 2-form ω which is closedand non-degenerate. A Liouville manifold (or an exact symplectic manifold) is asymplectic manifold with an exact symplectic form ω = dλ , and the associatedLiouville vector field V is defined by the equation i V dλ = λ . Any Liouville manifoldis necessarily open. A boundary component M of a Liouville manifold (endowedwith the boundary orientation) is convex if the Liouville vector field is positivelytransverse to M , and is concave, if it is so negatively. An exact cobordism from a(co-oriented) contact manifold ( M + , ξ + ) to ( M − , ξ − ) is a compact Liouville manifold( W, ω = dα ) with boundary ∂W = M + (cid:70) M − , where M + is convex, M − is concave,and ker α | M ± = ξ ± . Therefore, the boundary orientation induced by ω agrees withthe contact orientation on M + , and differs on M − . A Liouville filling (or a Liouvilledomain) of a –possibly disconnected– contact manifold ( M, ξ ) is a compact Liouville11obordism from (
M, ξ ) to the empty set. A strong symplectic cobordism and astrong filling are defined in the same way, with the difference that ω is exact only ina neighbourhood of the boundary of W (so that the Liouville vector field is definedin this neighbourhood, but not necessarily in its complement).The symplectization of a contact manifold ( M, ξ = ker α ) is the symplectic man-ifold ( R × M, ω = d ( e a α )), where a is the R -coordinate. In particular, it is a non-compact Liouville manifold. Similarly, the symplectization of a stable Hamiltonianmanifold ( M, Λ , Ω) is the symplectic manifold ( R × M, ω ϕ ), where ω ϕ = d ( ϕ ( a )Λ)+Ω,and ϕ is an element of the set P = { ϕ ∈ C ∞ ( R , ( − (cid:15), (cid:15) )) : ϕ (cid:48) > } Here, (cid:15) > ω ϕ is indeed symplectic. An H -compatible(or simply cylindrical) almost complex structure on a symplectization ( W = R × M, ω ϕ ) is J ∈ End(
T W ) such that J is R -invariant , J = − , J ( ∂ a ) = R, J ( ξ ) = ξ, J | ξ is Ω-compatibleThe last condition means that Ω( · , J · ) defines a J -invariant Riemannian metric on ξ . If J is H -compatible, then it is easy to check that it is ω ϕ -compatible, whichmeans that ω ϕ ( · , J · ) is a J -invariant Riemannian metric on R × M .To any closed T -periodic Reeb orbit ( γ, T ) one can associate an asymptoticoperator A γ . To write it down, choose a symmetric connection ∇ on M , and a H -compatible almost complex structure J , and define A γ = A γ,J : W , ( γ ∗ ξ ) → L ( γ ∗ ξ ) A γ η = − J ( ∇ t η − T ∇ η R )Alternatively, one has the expression A γ η ( t ) = − J dds (cid:12)(cid:12)(cid:12)(cid:12) s =0 dϕ − T s η ( t + s ) , for η ∈ W , ( γ ∗ ξ ), where ϕ s again is the time- s Reeb flow.Morally, this is the Hessian of a certain action functional on the loop space of M whose critical points correspond to closed Reeb orbits. It is symmetric with respectto a suitable L -product. A periodic orbit γ is non-degenerate if and only if 0 doesnot lie in the spectrum of A γ , and more generally, if γ is Morse–Bott and lies ina Morse–Bott submanifold N , then dim ker A γ = dim N −
1. Under a choice ofunitary trivialization τ of γ ∗ ξ , this operator looks like A γ : W , ( S , R n ) → L ( S , R n )12 γ = − i∂ t − S ( t ) , where S is a smooth loop of symmetric matrices (the coordinate representationof − T J ∇ R ), which comes associated to a trivialization of A γ . When γ is non-degenerate, its Conley–Zehnder index with respect to τ is defined to be the Conley–Zehnder index of the path of symplectic matrices Ψ( t ) satisfying ˙Ψ( t ) = iS Ψ( t ),Ψ(0) = . We denote this by µ τCZ ( γ ) = µ CZ (Ψ).We will consider, for cylindrical J , punctured J -holomorphic curves u : ( ˙Σ , j ) → ( R × M, J ) in the symplectization of a stable Hamiltonian manifold M , where˙Σ = Σ \ Γ, (Σ , j ) is a compact connected Riemann surface, and u satisfies the non-linear Cauchy–Riemann equation du ◦ j = J ◦ u . We will also assume that u isasymptotically cylindrical, which means the following. Partition the punctures into positive and negative subsets Γ = Γ + ∪ Γ − , and at each z ∈ Γ ± , choose a biholo-morphic identification of a punctured neighborhood of z with the half-cylinder Z ± ,where Z + = [0 , ∞ ) × S and Z − = ( −∞ , × S . Then writing u near the puncturein cylindrical coordinates ( s, t ), for | s | sufficiently large, it satisfies an asymptoticformula of the form u ◦ φ ( s, t ) = exp ( T s,γ ( T t )) h ( s, t )Here T > γ : R → M is a T -periodic Reeb orbit, the exponentialmap is defined with respect to any R -invariant metric on R × M , h ( s, t ) ∈ ξ γ ( T t ) goes to 0 uniformly in t as s → ±∞ and φ : Z ± → Z ± is a smooth embedding suchthat φ ( s, t ) − ( s + s , t + t ) → s → ±∞ for some constants s ∈ R , t ∈ S .We will refer to punctured asymptotically cylindrical J -holomorphic curves simplyas J -holomorphic curves.Observe that, for any closed Reeb orbit γ and cylindrical J , the trivial cylinderover γ , defined as R × γ , is J -holomorphic.The Fredholm index of a punctured holomorphic curve u which is asymptoticto non-degenerate Reeb orbits in a (2 n + 2)-dimensional symplectization W n +2 = R × M is given by the formula ind ( u ) = ( n − χ ( ˙Σ) + 2 c τ ( u ∗ T W ) + µ τCZ ( u ) (2)Here, ˙Σ is the domain of u , τ denotes a choice of trivializations for each of thebundles γ ∗ z ξ , where z ∈ Γ, at which u approximates the Reeb orbit γ z . The term c τ ( u ∗ T W ) is the relative first Chern number of the bundle u ∗ T W . In the case W is2-dimensional, this is defined as the algebraic count of zeroes of a generic section of u ∗ T W which is asymptotically constant with respect to τ . For higher-rank bundles,one determines c τ by imposing that c τ is invariant under bundle isomorphisms, andsatisfies the Whitney sum formula (see e.g. [Wen5]). The term µ τCZ ( u ) is the totalConley–Zehnder index of u , given by 13 τCZ ( u ) = (cid:88) z ∈ Γ + µ τCZ ( γ z ) − (cid:88) z ∈ Γ − µ τCZ ( γ z )Given a H -compatible J , and a J -holomorphic curve u in R × M , the expression u ∗ Ω is a non-negative integrand, and one can define its Ω-energy E ( u ) = (cid:90) u ∗ ΩIt is non-negative, and vanishes if and only if u is a (multiple cover of) a trivialcylinder. In this section, we construct the contact manifolds M of Theorem 1.3, making useof a cylindrical semi-filling ( Y × I, dα ). We will use the “double completion” con-struction, originally appearing in [L-VHM-W]. While very geometrically flavoured,this construction has the effect of endowing M with a contact structure and anexplicit deformation to a SHS, by viewing it as a contact-type hypersurface in anon-compact Liouville manifold. The contact form thus obtained will be degener-ate, and a standard Morse function technique as in [Bo02] will be necessary.Let Y be a closed (2 n − Y × I, dα ) is a Liouville domain,for some exact symplectic form dα ∈ Ω ( Y × I ) (recall that throughout this paper, I will denote the interval [ − , α = { α r } r ∈ I is given by a 1-parameter family of1-forms in Y , which is the case for all known examples of cylindrical semi-fillings.In particular, we get that α ( ∂ r ) = 0. We can write the symplectic form as dα = dα r + dr ∧ ∂α r ∂r The Liouville vector field V , defined to be dα -dual to α , points outwards at eachboundary component, and hence, using its flow, we can choose our coordinate r ∈ I so that V agrees with ± ∂ r near the boundary ∂ ( Y × I ) = Y ×{± } =: Y ± . Therefore,we can assume that α = e ± r − α ± on Y × [ − , − δ ) and Y × (1 − δ, δ >
0. Then Y ± carries a contact structure ξ ± = ker α ± , where α ± = i V dα | T ( Y ± ) = α | T ( Y ± ) . The behaviour of V near the ends necessarily impliesthat there are values r ∈ I such that V | Y ×{ r } lies in T Y , and hence Y r := Y × { r } is not a contact type hypersurface. The slices Y r which are of contact type inherit14 =-1 r=0 rr=1 Figure 2: The qualitative behaviour of the flow of the Liouville vector field V onany cylindrical Liouville semi-filling, for which the central slice ( r = 0) is invariant.One may informally think of such a Liouville domain as being obtained by gluingtwo negative symplectizations along a “non-contact hypersurface”.a contact structure ξ r = ker α r and the resulting Reeb vector field R r satisfies R r = e ∓ r R ± in the respective components of {| r | > − δ } , where R ± is the Reebvector field of α ± = α ± . We shall assume throughout that the only non-contacttype slice is Y , so that α r is a contact form for every r (cid:54) = 0. Also, we shall makethe convention that whenever we deal with equations involving ± ’s and ∓ ’s, one hasto interpret them as to having a different sign according to the region (the “upper”sign denotes the “plus” region, and the “lower”, the “minus” region).Let now M = Y × Σ be a product (2 n + 1)-manifold, where Σ is the orientablegenus g surface obtained by gluing a connected genus 0 surface with k boundarycomponents Σ − , to a connected genus g − k + 1 > k boundarycomponents Σ + along the boundary, by an orientation preserving map. The surfaceΣ then inherits the orientation of Σ − , which is opposite to the one in Σ + . On eachboundary component of ∂ Σ ± , choose collar neighbourhoods N ( ∂ Σ ± ) = ( − δ, × S (for the same δ as before), and coordinates ( t ± , θ ± ) ∈ N ( ∂ Σ ± ), so that ∂ Σ ± = { t ± =0 } . We will consider Σ − and Σ + to be attached at each of the k boundary componentsby a cylinder I × S , so that M at this region is the disjoint union of k copies of Y × I × S , with the Y × I identified with the Liouville domain above. We write thepoints of M here as ( y, r, θ ), where the θ ∈ S coordinate can be chosen to coincidewith θ ± where the gluing takes place. We shall therefore drop the subscript ± when15alking about the θ coordinate. Denote also N ( − Y ) := Y × [ − , − δ ) × S , N ( Y ) := Y × (1 − δ, × S , in the above identification.We have M = M Y ∪ M ± P , where M Y = (cid:70) k Y × I × S is a region gluing M ± P = Y ± × Σ ± together (recall Figure1). We shall refer to them as the spine or cylindrical region , and the positive/negativepaper , respectively. We have fibrations π Y : M Y → Y × Iπ ± P : M ± P → Y ± , with fibers S and Σ ± , respectively, and hence can be given the structure of a SOBD(see [Mo2] for a definition).We now construct an open manifold containing M as a contact-type hypersurface.Denote by Σ ∞± the open manifolds obtained from Σ ± by attaching cylindrical ends ofthe form ( − δ, + ∞ ) × S at each boundary component, where the subset ( − δ, × S coincides with the collar neighbourhoods chosen above. The coordinates t ± and θ extend to these ends in the obvious way, and we shall refer to the cylindrical ends as N ( ∂ Σ ∞± ). We also consider the cylinder R × S obtained by enlarging the cylindricalregion I × S we had above. Denote then M ± , ∞ P = Y × Σ ∞± M ∞ Y = Y × R × S N ∞ ( − Y ) = Y × ( −∞ , − δ ) × S , N ∞ ( Y ) = Y × (1 − δ, + ∞ ) × S and define the double completion of E to be E ∞ , ∞ = ( −∞ , − δ ) × M − , ∞ P (cid:71) (1 − δ, + ∞ ) × M + , ∞ P (cid:71) ( − δ, + ∞ ) × M ∞ Y / ∼ , where we identify ( r, y, t − , θ ) ∈ ( −∞ , − δ ) × Y × N ( ∂ Σ ∞− ) with ( t, y, r, θ ) ∈ ( − δ, + ∞ ) × N ∞ ( − Y ) if and only if t = t − , and ( r, y, t + , θ ) ∈ (1 − δ, + ∞ ) × Y ×N ( ∂ Σ ∞ + ) with ( t, y, r, θ ) ∈ ( − δ, + ∞ ) × N ∞ ( Y ) if and only if t = t + (see Figure 4).By definition, the t coordinate coincides with the t ± coordinates, where these aredefined, so we shall again drop the ± subscripts from the variables t ± . Note alsothat the r coordinate is globally defined, whereas t is not. Denote then by E ∞ , ∞ ( t )the region of E ∞ , ∞ where the coordinate t is defined.16hoose now λ ± to be Liouville forms on the Liouville domains Σ ∞± , such that λ ± = e t dθ on N ( ∂ Σ ∞± ). This last expression makes sense in the region of E ∞ , ∞ where both θ and t are defined, and where they are not, the form λ ± makes sense.So this yields a globally defined 1-form λ ∈ Ω ( E ∞ , ∞ ) which coincides with λ ± where these are defined. Also, the same argument works for α , so that we get aglobal α ∈ Ω ( E ∞ , ∞ ).For K (cid:29) (cid:15) > L ≥
1, choose a smoothfunction σ = σ L(cid:15),K : R → R + satisfying • σ ≡ K on R \ [ − L, L ]. • σ ≡ (cid:15) on [ − L + δ, L − δ ]. • σ (cid:48) ( r ) <
0, for r ∈ ( − L, − L + δ ), σ (cid:48) ( r ) > r ∈ ( L − δ, L ).We have that the 1-form σα is Liouville on Y × R . Indeed, if dvol is a positivevolume form in Y with respect to the α − -orientation, we may write dα n = dvol ∧ dr, α r ∧ dα n − r = link ( α r ) dvol, where the last equation defines a self-linking function r (cid:55)→ link ( α r ), whose sign isopposite to that of r ∈ R . Then d ( σα ) n = σ n − ( σ − nσ (cid:48) link ( α r )) dvol ∧ dr Tracking the signs, one checks that the above expression is positive.The associated Liouville vector field is V σ := σσ + σ (cid:48) dr ( V ) V = V, on ( R \ ( − L, L )) ∪ [ − L + δ, L − δ ] σσ + σ (cid:48) ∂ r , on ( L − δ, L ) − σσ − σ (cid:48) ∂ r , on ( − L, − L + δ ) (3)Observe that V σ is everywhere positively colinear with V .After extending the form σα to E ∞ , ∞ in the natural way, one checks that λ σ := λ Lσ := σα + λ is a Liouville form on E ∞ , ∞ . Denote ω σ := ω Lσ := dλ σ = σ (cid:48) dr ∧ α + σdα + dλ, which is symplectic. Denote by X σ the associated Liouville vector field.17f X ± denotes the Liouville vector field on Σ ∞± which is dλ ± -dual to λ ± , coincidingwith ∂ t in N ( ∂ Σ ∞± ), we can define a smooth vector field on E ∞ , ∞ by X = X + , on { r > − δ } X − , on { r < − δ } ∂ t , on {| r | < } (4)Then X σ = X + V σ Denote E L,Q = E ∞ , ∞ / ( {| r | > L } ∪ { t > Q } ) , for Q ≥
0, and L ≥
1. We have its “horizontal” and “vertical” boundaries (cid:102) M L,QY := ∂ h E L,Q := { t = Q } ∩ { r ∈ [ − L, L ] } (cid:102) M L,QP := ∂ v E L,Q := (cid:102) M − ,L,QP (cid:71) (cid:102) M + ,L,QP , where (cid:102) M − ,L,QP := { r = − L } ∩ { t ≤ Q } (cid:102) M + ,L,QP := { r = L } ∩ { t ≤ Q } The manifold (cid:102) M L,Q := ∂E L,Q := ∂ h E L,Q ∪ ∂ v E L,Q , is then a manifold with corners ∂ h E L,Q ∩ ∂ v E L,Q = {| r | = L } ∩ { t = Q } One has X σ = ± σσ ± σ (cid:48) ∂ r + ∂ t in the corresponding components of the region {| r | > L − δ } ∩ { t ≥ − δ } . Thismeans that X σ will be transverse to the smoothening of ∂E L,Q that we shall nowconstruct.Choose smooth functions F ± , G ± : ( − δ, δ ) → ( − δ,
0] such that ( F + ( ρ ) , G + ( ρ )) = ( ρ, , ( F − ( ρ ) , G − ( ρ )) = (0 , ρ ) , for ρ ≤ − δ/ G (cid:48) + ( ρ ) < , F (cid:48)− ( ρ ) > , for ρ > − δ/ G (cid:48)− ( ρ ) > , F (cid:48) + ( ρ ) > , for ρ < δ/ F + ( ρ ) , G + ( ρ )) = (0 , − ρ ) , ( F − ( ρ ) , G − ( ρ )) = ( ρ, , for ρ ≥ δ/ F , G ) - - - (F , G ) - - - δδ - - - δ -- δ - - ρ r t r t ρ Figure 3: The paths ρ (cid:55)→ ( F ± ( ρ ) , G ± ( ρ )).We now smoothen the corner ∂ h E L,Q ∩ ∂ v E L,Q by substituting the region ∂E L,Q ∩ ( { t ∈ ( Q − δ, Q ] } ∪ {| r | ∈ ( L − δ, L ] } ) , which contains the corners, with the smooth manifold M ± ,L,QC := { ( r = ± L + F ± ( ρ ) , t = Q + G ± ( ρ ) , y, θ ) : ( ρ, y, θ ) ∈ ( − δ, δ ) × Y × S } The smoothened boundary can then be written as M L,Q := M ± ,L,QP (cid:91) M ± ,L,QC (cid:91) M L,QY , where M ± ,L,QP = (cid:102) M ± ,L,QP ∩ { t < Q − δ/ } M L,QY = (cid:102) M L,QY ∩ {| r | < L − δ/ } The Liouville vector field X σ is transverse to this manifold, so that we get a contactstructure on M L,Q given by ξ L,Q = ker( λ Lσ | T M
L,Q )Observe that M L,Q is canonically diffeomorphic to M . So, this actually yields acontact structure on M . By construction, we have non-empty intersections M ± ,L,QP ∩ M ± ,L,QC = { r = ± L } ∩ { t ∈ ( Q − δ, Q − δ/ } M L,QY ∩ M ± ,L,QC = { t = Q } ∩ {| r | ∈ ( L − δ, L − δ/ } We shall construct a stable Hamiltonian structure on M L,Q which arises as adeformation of the above contact structure, such that both coincide on M L,QY , asfollows. 19hoose a smooth function β : R → [0 ,
1] such that β ( t ) = 0 for t ≤ − δ + δ/ β ( t ) = 1 for t ≥ − δ/ − δ/
9, and β (cid:48) ≥
0. Set Z = V σ + β ( t ) X, in the region E ∞ , ∞ ( t ) where t is defined σσ + σ (cid:48) ∂ r , in E ∞ , ∞ ( t ) c ∩ { r > − δ }− σσ − σ (cid:48) ∂ r , in E ∞ , ∞ ( t ) c ∩ { r < − δ } , (5)which yields a smooth vector field on E ∞ , ∞ , a deformation of X σ . Then Z is stilltransverse to M L,Q and is stabilizing, so that the pair H := H L,Q := (Λ
L,Q = i Z ω σ | T M
L,Q , Ω L,Q = ω σ | T M
L,Q )yields a stable Hamiltonian structure on M L,Q . For Q = 0, ( M L, Y , Λ L, ) can be seenas the contactization of the Liouville domain ( Y × I, (cid:15)dα ).Along M L,QY the Reeb vector field is given by R L,Q = ∂ θ e Q , which is degenerate, andthe space of Reeb orbits is identified with Y × [ − L, L ]. We consider two perturbationapproaches: Morse, and Morse-Bott. In the first approach we choose H L : Y × [ − L, L ] → R ≥ to be Morse, depending only in r near r = ± L , satisfying ∂ r H L ≤ r = L , ∂ r H L ≥ r = − L , and vanishing as one approaches r = ± L . Inthe second approach, we choose H L to depend only on r globally , with respect towhich it is a Morse function.If t (cid:55)→ Φ tZ denotes the flow of Z , choose (cid:15) > M (cid:15) ; L,Q := { Φ (cid:15)H L ( x ) Z ( x ) ∈ E ∞ , ∞ : x ∈ M L,Q } is still transverse to Z . We have a stable Hamiltonian structure H (cid:15) := H L,Q(cid:15) := (Λ
L,Q(cid:15) , Ω L,Q(cid:15) ) := ( i Z ω σ | T M (cid:15) ; L,Q , ω σ | T M (cid:15) ; L,Q ) , and a decomposition M (cid:15) ; L,Q = M (cid:15) ; L,QY ∪ M (cid:15) ; ± ,L,QC ∪ M (cid:15) ; ± ,L,QP , where each component is the perturbation of the corresponding component of M L,Q .Along the region M (cid:15) ; ± ,L,QC the new coordinates are r = F (cid:15) ; L ± ( ρ ) = Φ V σ ( (cid:15)H L ( · , ± L + F ± ( ρ )) , ± L + F ± ( ρ )) t = G (cid:15) ; L,Q ± ( ρ ) = Q + G ± ( ρ ) + (cid:15)H L ( · , ± L + F ± ( ρ )) , where Φ V σ ( s, · ) is the time s flow of V σ . 20 t rH M P - M P + M Y M C + M C - Figure 4: The double completion, and a Morse function H along the spine.We then haveΛ L,Q(cid:15) = Λ L,Q = Kα ± , in M (cid:15) ; ± ,L,QP = M ± ,L,QP e (cid:15)H L ( (cid:15)α + e Q dθ ) , in M (cid:15) ; ,L,QY σ ( F (cid:15) ; L ± ( ρ )) e ± F (cid:15) ; L ± − L α ± + β ( G (cid:15) ; L,Q ± ( ρ )) e G (cid:15) ; L,Q ± ( ρ ) dθ, in M (cid:15) ; ± ,L,QC One can similarly write down Ω
L,Q(cid:15) explicitly.The Reeb vector field R L,Q(cid:15) associated to this stable Hamiltonian structure is R L,Q(cid:15) = R L,Q = R ± K , in M (cid:15) ; ± ,L,QP e − (cid:15)H L − Q (cid:0) (1 + (cid:15)α ( X H L )) ∂ θ − e Q X H L (cid:1) , in M (cid:15) ; L,QY (cid:15) ; L,Q ± (cid:16) e ∓ F (cid:15) ; L ± + L ( G (cid:15) ; L,Q ± ) (cid:48) R ± − e − G (cid:15) ; L,Q ± ( σ (cid:48) ± σ )( F (cid:15) ; L ± )( F (cid:15) ; L ± ) (cid:48) ∂ θ (cid:17) , in M (cid:15) ; ± ,L,QC where X H L is the Hamiltonian vector field on Y × I associated to H L , defined by i X HL dα = − dH L , andΦ (cid:15) ; L,Q ± ( ρ ) = σ ( F (cid:15) ; L ± ( ρ ))( G (cid:15) ; L,Q ± ) (cid:48) ( ρ ) − β ( G (cid:15) ; L,Q ± ( ρ ))( σ (cid:48) ± σ )( F (cid:15) ; L ± ( ρ ))( F (cid:15) ; L ± ) (cid:48) ( ρ ) (6)One can check that Φ (cid:15) ; L,Q ± has sign which is opposite to its subscript. Observethat critical points ( y, r ) of H give rise to closed Reeb orbits of the form γ p := { p } × S ⊆ crit( H L ) × S . If we are taking the Morse approach, we have onlya finite number of such orbits, and they are non-degenerate. Choosing H L to be C -small has the effect of making the vector field X H L also small, so that the closedorbits which do not arise from critical points of H L have large period, including the21nes not contained in M L,QY . So, taking any large (but fixed) T (cid:29)
0, we can choose H L small enough so that all the periodic Reeb orbits up to period T are of the form γ lp , for p ∈ crit( H L ), and l ≤ N , for some covering threshold N depending on T .For the Morse–Bott case, we obtain Y -families of Morse-Bott orbits for each criticalpoint of H L . Remark 2.1.
1. One can check that λ σ ( R (cid:15) ) = 1 (recall that λ σ | M (cid:15) ; L,Q is the primitive of Ω (cid:15) = ω σ | M (cid:15) ; L,Q ). Therefore, for compatible almost complex structure J and an asymptot-ically cylindrical J -holomorphic curve u with positive/negative punctures Γ ± , theΩ (cid:15) -energy of u is (cid:90) u ∗ Ω (cid:15) = (cid:90) u ∗ dλ σ = (cid:88) z ∈ Γ + T z − (cid:88) z ∈ Γ − T z , (7)where T z is the action of the Reeb orbit corresponding to the puncture z . In partic-ular, if the positive punctures correspond to critical points of H L , then so will thenegative ones.2. By inspecting the expressions of the Reeb vector field we see that there are nocontractible closed Reeb orbits for the SHS, if we assume this same condition for R ± .Moreover, the direction of the Reeb vector field does not change after perturbingback to sufficiently close contact data (cf. Section 2.8 below), and so this also holdsfor the latter data. It follows that the isotopy class defined by the resulting contactstructure is hypertight, and this shows the hypertightness condition of Theorem 1.3. Construction
We set L = 1, and Q = 0, and drop the superscripts L and Q fromall of the notation. We now define a suitable, though non-generic, almost complexstructure J = J (cid:15) on the symplectization W (cid:15) = R × M (cid:15) , where M (cid:15) = M (cid:15) ;1 , = M (cid:15)Y (cid:91) M (cid:15) ; ± P (cid:91) M (cid:15) ; ± C It will be compatible with the stable Hamiltonian structure H (cid:15) , and the fibers Σ ± of our fibration π ± P , the “pages”, will lift as holomorphic curves. We will blur thedistinction between M = M , and its diffeomorphic perturbed copy M (cid:15) (as wellas for W and W (cid:15) ), so that we are actually working on a fixed M with a SHS whichdepends on (cid:15) .Denote by ξ (cid:15) := ker Λ (cid:15) . We will define J on ξ (cid:15) , in an R -invariant way, and thensimply set J ( ∂ a ) := R (cid:15) . 22hoose a dα -compatible almost complex structure J on Y × I , which is cylin-drical in the cylindrical ends of Y × I , so that, along these, it coincides with a dα ± -compatible almost complex structure J ± on ξ ± , and maps the Liouville vec-tor field V to R ± . Observe that the vector field ∂ θ is transverse to ξ (cid:15) along M Y .Therefore, we may then define J Y = π ∗ Y J on ξ (cid:15) | M Y .Along the regions Σ ± / N ( ∂ Σ ± ) × Y ⊆ M ± P , and { t ∈ ( − δ, − δ/ } × Y ⊆ M ± P ,the restriction of the projection π ± P : M ± P → Σ ± induces an isomorphism dπ ± P : ξ (cid:15) /ξ ± (cid:39) −→ T Σ ± . Choose j ± to be a dλ -compatible almost complex structures onΣ ± , so that j ± ( ∂ t ) = K∂ θ in N ( ∂ Σ ± ). Define J = J ± ⊕ ( π ± P ) ∗ j ± on ξ (cid:15) | M ± P .In M ± C , we have ξ (cid:15) = ξ ± ⊕ (cid:104) v , v ± (cid:105) , where v = ∂ ρ , v ± = a ± ( ρ ) R ± + b ± ( ρ ) ∂ θ , (8)Here, a ± = − β ( G (cid:15) ± ) e ± F (cid:15) ± − Φ (cid:15) ± b ± = σ ( F (cid:15) ± ) e G (cid:15) ± Φ (cid:15) ± where Φ (cid:15) ± is defined in (6). In the overlaps M ± C ∩ M Y , one computes that J ( v ) = g Y ± ( ρ ) v ± where g Y ± := ± e ∓ F(cid:15) ± ( ρ )+1 a ± ( ρ ) = ∓ Φ (cid:15) ± β ( G (cid:15) ± ) , which is always positive. Similarly, in M ± P ∩ M ± C , we have J ( v ) = e ∓ ρ v ± . Since g Y ± and e ∓ ρ are both positive, we can nowtake any smooth positive functions h ± : ( − δ, δ ) → R + which coincide with e ∓ ρ near ρ = ± δ and with g Y ± near ρ = ∓ δ . We glue the two definitions by setting J ( v ) = h ± ( ρ ) v ± := w ± , (9)and we make J agree with J ± on ξ ± .This gives a well-defined cylindrical J in R × M . Compatibility.
One can check that J is H (cid:15) -compatible by straightforward com-putations [Mo2]. Remark 2.2.
We observe that over R × M ± P , where Λ (cid:15) = Kα ± , we have d Λ (cid:15) ( v, J v ) ≥ v ξ ± = 0, so that the projection to T Y of v lies in thespan of R ± . 23 .3 Finite energy foliation We will now consider the symplectization of our stable Hamiltonian manifold ( M, H (cid:15) ),given by ( W = R × M, ω ϕ(cid:15) = d ( ϕ ( a )Λ (cid:15) ) + Ω (cid:15) ), where ϕ ∈ P . We will construct afinite-energy foliation of W by J -holomorphic curves, consisting of three distincttypes, which we describe in the next theorem. This is an adaptation of the con-struction in [Wen4]. Theorem 2.3.
There exists a finite energy foliation of the symplectization ( W, ω ϕ(cid:15) ) by simple J -holomorphic curves of the following types: • trivial cylinders C p , corresponding to Reeb orbits of the form γ p = { p } × S for p ∈ crit ( H ) , and which may be parametrized by C p : R × S → R × M Y = R × ( Y × I ) × S C p ( s, t ) = ( s, p, e − (cid:15)H ( p ) t ) • flow-line cylinders u aγ , parametrized by u aγ : R × S → R × M Y = R × ( Y × I ) × S u aγ ( s, t ) = ( a ( s ) , γ ( s ) , θ ( s ) + t ) (10) for a proper function a : R → R , a function θ : R → S , and a map γ : R → Y × I satisfying ˙ γ = ∇ H ( γ ( s ))˙ a ( s ) = e (cid:15)H ( γ ( s )) , a (0) = a ˙ θ ( s ) = − (cid:15)α ( ∇ H ( γ ( s ))) , θ (0) = 0 (11) Here, the gradient is computed with respect to the metric g dα,J := dα ( · , J · ) .They have for positive/negative asymptotics the Reeb orbits corresponding to p ± := lim s →±∞ γ ( s ) ∈ crit ( H ) • positive/negative page-like holomorphic curves u ± y,a , which consist of atrivial lift at symplectization level a of a page P ± y := (Σ ± \N ( ∂ Σ ± )) × { y } , for y ∈ Y , glued to k cylindrical ends which lift the smoothened corners and thenenter the symplectization of M Y , asymptotically becoming a flow-line cylinder.They have k positive asymptotics at Reeb orbits of the form γ p , exactly one foreach component of M Y . The positive curves have genus g − k + 1 > and k punctures, whereas the negative curves have genus , and also k punctures. t rspine neg.paper pos.paper Figure 5: The double completion E ∞ , ∞ , containing M and its perturbed version M (cid:15) as contact type hypersurfaces. The foliation by holomorphic curves is shown ingreen (for the non-trivial curves) and blue (for the trivial cylinders). Remark 2.4.
In the Morse-Bott case, one can show that α ( ∇ H ) = 0 for a suitablemetric on Y × I ([Mo2], Remark 2.8), so that the function θ vanishes identically.Figure 5 summarizes the situation. We shall not distinguish the curves u ± y,a and u aγ from the simple holomorphic curves that they parametrize, and we will drop the a for the notation whenever we wish to refer to the equivalence class of the curvesunder R -translation. A short computation shows that the flow-line cylinders areindeed holomorphic (see e.g. [Mo2, Sie16]). We now construct the page-like curves.The pages P ± y := (Σ ± \N ( ∂ Σ ± )) × { y } , y ∈ Y , clearly lift to a holomorphicfoliation of the region R × M ± P ⊆ W , which takes the form {{ a }× P ± y : a ∈ R , y ∈ Y } .We now glue cylindrical ends to these lifts.We have that J v = h ± v ± and R (cid:15) are both linear combinations of the vectorfields R ± and ∂ θ along M ± C , with the coefficients only depending on ρ . Since theseare not colinear, we have smooth functions B, C : ( − δ, δ ) → R such that ∂ θ = BR (cid:15) + CJ v One can in fact compute that the following expressions hold: B = e G (cid:15) ± β ( G (cid:15) ± ) C = e G(cid:15) ± ( G (cid:15) ± ) (cid:48) h ± (12)25e have that J ∂ θ = − B∂ a − Cv = − B∂ a − C∂ ρ (13)We conclude that (cid:104) ∂ θ , B∂ a + C∂ ρ (cid:105) = (cid:104) ∂ θ , J ∂ θ (cid:105) It follows that the distribution above has integral submanifolds which are un-parametrized holomorphic curves. We can actually find holomorphic parametriza-tions given by w ± y,a : ( − δ, δ ) × S → R × M ± C = R × Y × ( − δ, δ ) × S ( s, t ) (cid:55)→ ( b ( s ) , y, ρ ( s ) , t ) , for some fixed y ∈ Y , and functions b, ρ : ( − δ, δ ) → R satisfying˙ ρ ( s ) = C ( ρ ( s )) , ρ ( − δ ) = − δ ˙ b ( s ) = B ( ρ ( s )) , b ( − δ ) = a The curve w ± y,a is indeed holomorphic.The curves w ± y,a glue with curves u aγ , which look like u aγ ( s, t ) = ( a ( s ) , y ( s ) , r ( s ) , t )for some y ( s ) ∈ Y such that y ( s ) ≡ y near r = ± s → + ∞ y ( s ) = y + , andsome r : R → I with lim s → + ∞ r ( s ) = r + , so that ( y + , r + ) ∈ crit ( H ). We may thendefine a J -holomorphic curve u ± y,a := P ± y,a (cid:91) w ± y,a (cid:91) u aγ which asympote k Reeb orbits γ ± y,a ; i = { p ± y,a ; i } × S , where p ± y,a ; i ∈ crit ( H ), for i = 1 , . . . , k , and which have genus g ( u − y,a ) = 0 and g ( u + y,a ) = g − k + 1. In this section, we compute the Fredholm index of the curves in the foliation.
Theorem 2.5.
1. After a sufficiently small Morse perturbation making Reeb orbits along M Y non-degenerate, we can find a natural trivialization τ of the contact structure along γ p (inducing a trivialization τ l along all of its covers γ lp ), and N ∈ N , which dependson H and grows as H gets smaller, such that the Conley–Zehnder index of γ lp isgiven by µ τ l CZ ( γ lp ) = ind p ( H ) − n, for l ≤ N . . In the Morse approach, the Fredholm indexes of the curves in our finite energyfoliation are given byind ( u − y,a ) = 2 n (1 − k ) + k (cid:88) i =1 ind p − y,a ; i ( H ) ind ( u + y,a ) = 2 n (1 − g − k ) + k (cid:88) i =1 ind p + y,a ; i ( H ) ind ( u aγ ) = ind p + ( H ) − ind p − ( H ) (14) Proof.
See [Mo2].
Remark 2.6.
Since ind p + y,a ; i ( H ) ≤ n for every i , then ind ( u + y,a ) ≤ n (1 − g ) ≤ g ≥
1. This means these curves cannot possibly achieve transversality, and,after a perturbation making J generic, they will disappear. In this section, we shall prove that the curves we have constructed are Fredholmregular.In the Morse case, regularity of unbranched covers of flow-line cylinders can bereduced to the Morse–Smale condition for H . This fact is known to experts, and weshall omit the details (see [Mo2]).For regularity of the other curves, we will assume the Morse–Bott situation,and prove regularity of the genus zero curves in the foliation. We will use the factfrom [Wen1] that Fredholm regularity is equivalent to the surjectivity of the normalcomponent of the linearized Cauchy-Riemann operator, which is again a Fredholmoperator. After some assumptions on our choice of coordinates and the Morse-Bottfunction H (which we can always assume hold), and a suitable choice of normalbundle, we will explicitly write down an expression for this operator. We will obtaina set of PDEs whose solutions are precisely the elements in its kernel, for whichwe can check that curves in the foliation which are nearby a fixed leaf correspondto solutions. By splitting the operator, and using automatic transversality [Wen1],we show that these are all possible solutions. This will imply that the index of thenormal operator coincides with the dimension of its kernel, from which surjectivityfollows. From the implicit function theorem, we also obtain regularity for Morsedata chosen sufficiently close to Morse-Bott data, which is enough for our purposes.In order to do computations with linearized operators, we will choose a suitableconnection on W = R × M . It will be given by the Levi–Civita connection of asuitable metric g and hence symmetric. 27 onstructing a symmetric connection Given an almost complex structure J which is compatible with a symplectic form ω , we will denote by g J,ω = ω ( · , J · ) theassociated Riemannian metric.Define, in the regions R × Y × Σ ± \N ( ∂ Σ ± ), the metric g = g J ± P ,dα ± g j ± ,dλ ± , where we are using the splitting T ( R × Y × Σ ± \N ( ∂ Σ ± )) = (cid:104) ∂ a (cid:105) ⊕ ξ ± ⊕ (cid:104) R (cid:15) (cid:105) ⊕ T Σ ± = (cid:104) ∂ a (cid:105) ⊕ T Y ⊕ T Σ ± We extend it to R × M ± C by replacing g j ± ,dλ ± by the identity in the basis { ∂ ρ , ∂ θ } in the above matrix. Along R × Y × I × S , set g = g J ,dα
00 0 1 , where we use the splitting T ( R × Y × I × S ) = R × T ( Y × I ) × (cid:104) ∂ θ (cid:105) We set ∇ = ∇ g its Levi–Civita connection, which shall be the connection we willuse to write down all our linearised Cauchy–Riemann operators. In this section, we fix a genus zero curve u := u − y,a in the foliation, and denote by u Y := u − ( R × M Y ), u C := u − ( R × M − C ), and u P := u − ( R × M − P ). We will showthat dim ker D Nu = ind D Nu , where D Nu is the normal component of the linearizedCauchy-Riemann operator.We will deal with the Morse–Bott case, where H depends only on r . In this case,the operator we need to look at is given by D u : W , ,δ ( u ∗ T W ) ⊕ V Γ ⊕ X Γ → L ,δ ( Hom C (( ˙Σ − , j − ) , ( u ∗ T W, J )) D u η = ∇ η + J ( u ) ◦ ∇ η ◦ j − + ( ∇ η J ) ◦ du ◦ j − Here, δ is a small weight making the operator Fredholm, V Γ is a 2 k -dimensionalvector space of smooth sections asymptotic to constant linear combinations of R (cid:15) ∂ a , and X Γ is a k (2 n − u C ∪ u Y (a disjoint union of its cylindrical ends, which we denote u i , i = 1 , . . . , k ), and are constant equal to a vector in T y Y along u Y . We also havethe operator D ( j − ,u ) ∂ J : T j − T ⊕ W , ,δ ( u ∗ T W ) ⊕ V Γ ⊕ X Γ → L ,δ ( Hom C (( ˙Σ − , j − ) , ( u ∗ T W, J ))(
Y, η ) (cid:55)→ J ◦ du ◦ Y + D u η, where T is a Teichm¨uller slice through j − (see e.g. [Wen1] for a definition of this).The curve u is said to be regular whenever D ( j − ,u ) ∂ J is surjective. By a result in[Wen1], this is equivalent to the surjectivity of its normal component. In this case,this operator is D Nu : W , ,δ ( N u ) ⊕ X Γ → L ,δ ( Hom C ( T ˙Σ − , N u )) η (cid:55)→ π N D u = π N ( ∂ J η + ( ∇ η J ) ◦ du ◦ j − ) , where π N is the orthogonal projection to the normal bundle N u . Recall that thelatter is any choice of J -invariant complement to the tangent space to u , whichcoincides with the contact structure at infinity. Riemann-Roch gives ind D Nu = 2 n .The operator D u and D Nu are of Cauchy-Riemann type, which means in particularthat they satisfy the Leibnitz rule D u ( f η ) = f D u η + ∂f ⊗ η (15) Splitting over the paper
We will think of the punctured surface ˙Σ − as be-ing obtained abstractly from the surface with boundary Σ − \N ( ∂ Σ − ) by attachingcylindrical ends. Over the region Y × Σ − \N ( ∂ Σ − ), we have a splitting u ∗ T ( R × M ) = T Σ − ⊕ T ( a,y ) ( R × Y )Since J preserves this splitting, this gives an identification N u = T ( a,y ) ( R × Y ) =( ξ − ) y ⊕ (cid:104) ∂ a , R (cid:15) ( y ) = R − ( y ) /K (cid:105) of the normal bundle of u along this region. Usingthat constant vectors in N u give holomorphic push-offs of u in the foliation, weobtain D u = (cid:18) D id ∂ j − D Nu (cid:19) , where the normal Cauchy–Riemann operator D Nu splits as D Nu = (cid:76) ∂ .29 ome technical assumptions In order to be able to write down a manageableexpression for D Nu over the rest of the regions, we will assume, without loss ofgenerality, that: Assumptions 2.7. A. H has a unique critical point away from a neighbourhood ( − δ, δ ) ⊂ I where J is non-cylindrical. Choose, say, H (cid:48) ( − δ ) = 0.B. Choose our coordinate r so that the Liouville vector field V coincides with ± ∂ r on the complement of ( − δ, δ ).These assumptions are only used in this section to show regularity, and do notaffect other sections. Therefore we will, for simplicity, lift them in the rest of thesections. Choosing a suitable normal bundle.
We now specify how we will extendour normal bundle N u to u along its cylindrical ends.Since J was chosen on ξ (cid:15) | M Y so that the identification ( ξ (cid:15) , J ) → ( T ( Y × I ) , J )is holomorphic, where J may be any dα -compatible almost complex structure in Y × I which is cylindrical at the ends, we may identify the bundles. Observe that J is always θ -independent. And here we use assumptions A and B: we can choose J so that it is cylindrical in the complement of ( − δ, δ ), so that J is r -independent ( r is the Liouville coordinate).We then choose N u by the global expression N u = (cid:104) v, R − ( y ) /K (cid:105) ⊕ ( ξ − ) y = (cid:104) v (cid:105) ⊕ T y Y, where v := v ρ = − J ( R − ( y ) /K ) along the corner M − C = Y × ( − δ, δ ) × S , inter-polating between ∂ a = v − δ and ∂ r = v δ . Observe that N u is a trivial J -complexbundle. Writing down the normal Cauchy-Riemann operator globally.
We nowcompute an asymptotic expression for D Nu . In the Morse-Bott case, one shows that D Nu can be written asymptotically (i.e in the cylindrical coordinates ( s, t ) along u Y )as D η ( s, t ) = ∂η ( s, t ) + S H ( s, t ) η ( s, t ) , where S H ( s, t ) = ∇ J is a symmetric matrix such that S H ( s, · ) → − π ∇ − δ H as s → + ∞ , uniformly in the second variable.By construction, ∇ and J are both independent of the coordinates a, r, and θ along R × M Y . Then S H = νe , e denotes the matrix with a 1 at the (1 , ν = − πH (cid:48)(cid:48) ( − δ ) >
0. As we traverse the smoothened corner u C , we pick up asmooth path ( − δ, δ ) (cid:51) ρ (cid:55)→ S H ( ρ ) = s H ( ρ ) e , where s H ( − δ ) = 0 and s H ( δ ) = ν , sothat D Nu = ∂ + s H e Computing the kernel.
We write any section of N u as η = ( η , η ξ , η Γ ) ∈ W , ,δ ( (cid:104) v, R − ( y ) /K (cid:105) ) ⊕ W , ,δ (( ξ − ) y ) ⊕ X Γ , with η = ( η , η ) ∈ W , ,δ ( (cid:104) v (cid:105) ) ⊕ W , ,δ ( (cid:104) R − ( y ) /K (cid:105) ). We denote X R Γ := π R X Γ , X ξ Γ := π ξ X Γ , where π R : T y Y → (cid:104) R − ( y ) /K (cid:105) π ξ : T y Y → ( ξ − ) y are the orthogonal projections with respect to the metric g . Then we write η Γ = ( η R Γ , η ξ Γ ) ∈ X R Γ ⊕ X ξ Γ , where η R Γ = π R η Γ , η ξ Γ = π ξ η Γ .Then, η ∈ ker D Nu if and only if ∂η = − s H e η , which in holomorphic coordinates( s, t ) is ∂ s η − ∂ t (cid:0) η + η R Γ (cid:1) = − s H η ∂ t η + ∂ s (cid:0) η + η R Γ (cid:1) = 0 ∂ (cid:16) η ξ + η ξ Γ (cid:17) = 0 (16)Observe that the η Γ terms all disappear away from u C . It is straightforwardto check that all the nearby holomorphic curves in the foliation satisfy the aboveequations. We will show that these are indeed the unique solutions.We have shown that the operator D Nu splits into a direct sum D Nu := D ξu ⊕ D Ru , where D ξu = ∂ : W , ,δ (( ξ − ) y ) ⊕ X ξ Γ → L ,δ ( Hom C ( ˙Σ − , ( ξ − ) y ))and D Ru : W , ,δ ( (cid:104) v, R − ( y ) /K (cid:105) ) ⊕ X R Γ → L ,δ ( Hom C ( ˙Σ − , (cid:104) v, R − ( y ) /K (cid:105) ))We will show that both operators D ξu and D Ru are surjective, and this finishesthe proof. 31he first summand has kernel the (2 n − ξ − ) y . Its index is 2 n −
2, and it follows that it is surjective. The secondsummand satisfies D Ru = ∂ + s H e In order to show that it is surjective, we use automatic transversality [Wen1]. Weneed to check that ind D Ru > c ( D Ru ), where c ( D Ru ) denotes the adjusted first Chernnumber , defined by 2 c ( D Ru ) = ind D Ru − g + even , where g is the genus, and Γ even is the set of punctures with even Conley-Zehnderindex.The Conley-Zehnder index of D Ru at each puncture is 1, and thereforeind D Ru = 2 − k + k = 2On the other hand, the adjusted first Chern number is2 c ( D Ru ) = ind D Ru − even = 0This finishes the proof of regularity. In this section, we fix a Morse perturbation scheme. First, choose H to be given by H ( y, r ) = f ( r ) where f is a sufficiently small and positive Morse function on I andhas a unique critical point at 0 with Morse index 1 (which yields the Morse–Bottsituation). And then, choose a sufficiently small positive Morse function g on Y andextend it to a neighbourhood of Y × { } to make a further perturbation (obtainingthe Morse case). Therefore, H ( y, r ) = f ( r ) + γ ( r ) g ( y ) , where γ : I → [0 ,
1] is a smooth bump function satisfying γ = 0 in the region {| r | > δ } and γ = 1 on {| r | ≤ δ } .We view the Morse case as a deformation of the Morse–Bott one, via H t ( y, r ) = f ( r ) + tγ ( r ) g ( y ) for t ∈ [0 , J t . In the case where g is chosensmall, from the implicit function theorem we obtain: Theorem 2.8 (Fredholm regularity in the nearby Morse case) . For Morse datasufficiently close to Morse-Bott data, all the genus zero curves in the finite energyfoliation are Fredholm regular.
In order to simplify the torsion computation of Section 2.9, we will choose g tohave a unique maximum max and minimum min . Both scenarios are depicted inFigure 6 in the case of Y = S . 32igure 6: The Morse–Bott and the Morse scenarios, respectively, in the case where Y = S . The “evil twins” are shown in blue (see Section 2.9). In this section, we prove that the family of curves we constructed above are theunique curves (up to reparametrization and multiple covers) that asymptote Reeborbits of the family { γ p : p ∈ crit( H ) } , and with positive asymptotics in differentcomponents of M Y . We do this in both the Morse/Morse–Bott situations. Weassume that H has a unique critical point in the interval direction, at r = 0 (andperhaps other critical points contained in Y ×{ } , in the Morse case). In the Morse–Bott case, we denote by γ y := γ ( y, = { ( y, } × S the simply covered Reeb orbitcorresponding to y ∈ Y . Lemma 2.9.
Assume either the Morse or Morse–Bott cases. Let N ∈ N > , and let u be a J -holomorphic curve with positive asympotics of the form γ p = { p } × S , forwhich the number of positive punctures is bounded above by N . Then, we can findsufficiently small (cid:15) > (depending only on N ), such that − ( u ) ≤ + ( u ) , where Γ ± ( u ) denotes the set of positive/negative punctures of u . Here, we countpunctures with the covering multiplicity of their corresponding asymptotic.Proof. This follows easily from Remark 2.1, and by computing the Ω (cid:15) -energy [Mo2].33 heorem 2.10.
Assume either the Morse or Morse–Bott scenario.Let u : ˙Σ → R × M be a (not necessarily regular) J -holomorphic curve defined onsome punctured Riemann surface ˙Σ which is asymptotically cylindrical, and asymp-totes simply covered Reeb orbits of the form γ p = { p } × S at its positive ends.Assume that any two of the positive ends of u lie in distinct components of M Y .Then, for sufficiently small and uniform (cid:15) > , we have that, if u is not a trivialcylinder over one of the γ p ’s, then u is a curve of the form u ± y,a for some y ∈ Y , ora flow-line cylinder u γ (in the Morse scenario).Proof. We will consider two cases: either u is completely contained in the region R × M Y (case A), or it is not (case B). Case A.
This case is easily dealt with in the Morse–Bott scenario. By as-sumption, we have that u has a unique positive end. Since the γ p ’s are not con-tractible/nullhomologous inside M Y , Lemma 2.9 implies that u is has one positiveand one negative end, both simply covered, corresponding to Reeb orbits γ p ± . Butthen the Ω (cid:15) -energy of u vanishes, and u is necessarily is a cover of a trivial cylinder.In the Morse case, we show that u is a flow line cylinder. Again, Lemma 2.9implies that u is has one positive and one negative end, both simply covered, cor-responding to Reeb orbits γ p ± . Observe that, a priori, u is not even necessarily acylinder, since it may have positive genus.In the degenerate case when H ≡
0, we the projection π Y : R × M Y → Y × I isholomorphic. Then, if u is a holomorphic curve for this data, then so is v := π Y ◦ u .Since the asymptotics of u are covers of the S -fibers of π Y , the map v extends to aholomorphic curve in the closed surface Σ. But Y × I is exact, so that v has to beconstant by Stokes’ theorem. This means that u is necessarily a multiple cover of atrivial cylinder.We then see that the space of stable holomorphic cascades [Bo02] in R × Y × I × S ,the objects one obtains as limits of honest curves when one turns off the function H , consists of finite collections of flow-line segments and covers of trivial cylinders.If we take H t = tH , and we assume we have a sequence { u n } of J t n -holomorphicmaps with t n → J t is the almost complex structure corresponding to H t ), with one positive and negative simply covered orbits corresponding to criticalpoints p ± , then we obtain a stable holomorphic cascade u H ∞ as a limiting object.Since the positive end of u n is simply covered, Lemma 2.9 applied to H = 0 impliesthat every Reeb orbit appearing in u H ∞ is simply covered, and therefore every ofits holomorphic map components cannot be multiply covered. These can then onlybe trivial cylinders, but stability of the cascade means that it does not have trivial34ylinder components. We conclude that the space of holomorphic cascades whichglue to curves as in our hypothesis consists solely of flow-lines, which are regular bythe Morse–Smale condition, and come in a ( ind p + ( H ) − ind p − ( H ) − d Morse families of curves for the non-degenerate perturbation and index d regularholomorphic cascades, which in our situation is exactly what proves that our curve u is a flow-line cylinder. Case B.
We first assume the Morse–Bott case, and deal with the Morse casevia the gluing results in [Bo02]. The approach in this situation is to estimate theΩ (cid:15) -energy of u , and to use a suitable branched cover argument. The details are asfollows.Assume the Morse–Bott case. Since every positive puncture of u corresponds toa critical point, Remark 2.1 implies that so does every negative one. Let us denoteby Γ ± the set of positive/negative punctures of u , and for z ∈ Γ ± , let γ κ z p z be theReeb orbit corresponding to z , where p z ∈ crit( H ), κ z ≥
1, and κ z = 1 for z ∈ Γ + .By assumption, we have that + ≤ k , the number of components of M Y .The Ω (cid:15) -energy then has the following upper bound: E ( u ) = (cid:90) ˙Σ u ∗ Ω (cid:15) =2 πe (cid:15)H (0) (cid:32) + − (cid:88) z ∈ Γ − κ z (cid:33) =2 πe (cid:15)H (0) ( + − − ) ≤ π + || e (cid:15)H || C ≤ πk || e (cid:15)H || C (17)By construction, we have that over the region Σ ± \N ( ∂ Σ ± ), the almost complexstructure J splits. This implies that the projection π ± P : R × Y × Σ ± \N ( ∂ Σ ± ) → Σ ± \N ( ∂ Σ ± )is holomorphic. Moreover, this is still true if we extend this region by adding asmall collar { t = ∓ ρ ∈ ( − δ, − δ + δ/ } , as one can check. Denote by B δ ± :=Σ ± \N ( ∂ Σ ± ) ∪ { t ∈ ( − δ, − δ + δ/ } .For each w ∈ B δ ± , the hypersurface E w := R × Y × { w } is J -holomorphic, and π ± P has E w as fiber over w . Since the asymptotics of u are away from B δ ± , theintersection of u with any of the E w is necessarily a finite set of points, since theyare restricted to lie in a compact part of the domain of u .35ssuming WLOG that u indeed has a non-empty portion lying over the “plus” re-gion Σ + \N ( ∂ Σ + ), by positivity of intersections we have that u necessarily intersectsevery E w for w ∈ B δ ± . By Sard’s theorem, we may then find a t ≥ − δ , so that π + P ( u )is transverse to the circle { t } × S ⊆ B δ + (over all k components of the collar). If wedenote B δ ; t + := B δ + \{ t ∈ ( t , − δ + δ/ } , we have that S + := ( π + P ◦ u ) − ( B δ ; t + ) ⊆ ˙Σis a surface with boundary ∂S + = ( π + P ◦ u ) − ( ∂B δ ; t + ). The upshot of the discussionabove is that the map F u := ( π + P ◦ u ) | S + : S + → B δ ; t + is a holomorphic branched cover , having as degree the (positive) algebraic intersec-tion number of u with any of the E w (which is independent of w ∈ B δ ; t + ). Call thisdegree deg + ( u ) := deg( F u ). We wish to show that deg + ( u ) = 1, and so this mapwill be actually a biholomorphism.Let us write ∂S + = (cid:83) li =1 C i , where C i is a simple closed curve whose imageunder F u wraps around one of the circles { t } × S , with winding number n i . Byholomorphicity of F u , we have n i >
0. Observe that necessarily one has that l ≥ k ,since u intersects every E w at least once, and in particular for every w on the k circles { t } × S .By counting preimages under this projection of a point in each the k circles { t } × S , we obtain that (cid:90) ∂S + u ∗ dθ = 2 π l (cid:88) i =1 n i = 2 πk deg + ( u ) (18)Using expression Ω (cid:15) = Kdα ± + dλ ± , equation (18), the fact that t ≥ − δ , u ∗ dα + ≥
0, and Stokes’ theorem, we have the following energy estimate: E ( u ) ≥ (cid:90) S + u ∗ Ω (cid:15) = (cid:90) S + u ∗ ( Kdα + + dλ + ) ≥ (cid:90) S + u ∗ dλ + = (cid:90) ∂S + u ∗ ( e t dθ )=2 πke t deg + ( u ) ≥ πke − δ deg + ( u ) (19)36f we combine this with the inequality (17), we obtain2 πk || e (cid:15)H || C ≥ πe (cid:15)H (0) (cid:0) + − − (cid:1) ≥ πke − δ deg + ( u ) ≥ πke − δ (20)Now, since we have the freedom to choose − δ and || (cid:15)H || C as close to zero as wished,one can easily see that + = k, − = 0 , deg + ( u ) = 1This proves that u has no negative ends and precisely k positive ends, and that F u gives a biholomorphism between S + and B δ ; t + . It also follows from our energyestimates that (cid:90) S + u ∗ dα + = 0Since the integrand is non-negative, we get that u ∗ dα + = 0, so that u ( S + ) liesentirely in the almost complex 4-manifold U := R × γ × Σ + ⊆ R × Y × Σ , where γ is a (not necessarily closed) R + -Reeb orbit.Choose now a point y in the projection of u ( S + ) to γ , in this region. We shallprove that the projection of u ( S + ) to γ consists of just the point y , using the Morse–Bott assumption.Using the Reeb flow along γ we have a local coordinate y = y ( s ), such that y (0) = y . Assume by contradiction that there is some s (cid:54) = 0 such that y ( s ) belongsto the projection of u ( S + ) to γ , and consider the family of curves { u + y ( s ) ,a : a ∈ R } .By choosing a suitable a , we obtain an intersection of u + y ( s ) ,a and u , and, by positivityof intersections, we may assume that y ( s ) is such that γ y ( s ) is not an asymptotic orbitof u . Then the total intersection of these curves in U is at least 1.On the other hand, the set R × ( u M ∩ ( u + y ( s ) ,a ) M ), where u M is the projection to M of u , must be bounded in the R -direction, since the corresponding Reeb orbits arebounded away from each other. This means that the standard intersection pairingis homotopy invariant (there are no contributions coming from infinity), but a priorithe intersection points might move along with a given homotopy. If we choose tohomotope by translating in the R -direction, this can only happen if the projectionof both curves to M intersect the asymptotics of each other. Since the projectionof the curve u y ( s (cid:48) ) ,a (cid:48) to M does not intersect the asymptotics of u , we can homotopethe intersections away, a contradiction. This proves the claim that u is constant in Y . We have obtained that the portion of u which lies in the over Σ + \N ( ∂ Σ + ) isactually contained in the 3-manifold M y := R × { y } × Σ + \N ( ∂ Σ + ), as is the corre-sponding portion of the curve u + y,a . But for dimensional reasons, this manifold has37 unique 2-dimensional J -invariant distribution, given by T M y ∩ J T M y . Thereforethe tangent space to u must coincide with the tangent space of some u + y,a at everypoint in this region. The unique continuation theorem finally yields u = u + y,a .This proves the theorem in the Morse–Bott situation. The proof in the Morsecase follows from uniqueness in the Morse–Bott one, and the gluing results in [Bo02]. For computations in SFT we need non-degenerate Reeb orbits and contact data.Therefore, we need to perturb the SHS H (cid:15) = (Λ (cid:15) , Ω (cid:15) ) to a nearby contact structure. Perturbation to contact data
Recall that we have defined an exact symplec-tic form ω σ on the double completion, and we denote by V σ its associated Liouvillevector field. We also have the “vertical” Liouville vector field X associated to dλ ,defined by expression (4), and the stabilizing vector field Z for the SHS H (cid:15) , definedby (5).For s ∈ [0 , Z s = V σ + ((1 − s ) β ( t ) + s ) X, in the region E ∞ , ∞ ( t ) where t is defined σσ + σ (cid:48) ∂ r + sX + , in E ∞ , ∞ ( t ) c ∩ { r > − δ }− σσ − σ (cid:48) ∂ r + sX − , in E ∞ , ∞ ( t ) c ∩ { r < − δ } We have that Z = X σ is the Liouville vector field associated to ω σ and Z = Z .This yields a family of SHS’s given by H (cid:15),s = (Λ (cid:15),s , Ω (cid:15) ) s ∈ [0 , = ( i Z s ω σ | T M (cid:15) , ω σ | T M (cid:15) ) s ∈ [0 , One can see that Λ (cid:15),s is a contact form for s >
0. By Gray’s stability, as long as (cid:15) and s are positive and sufficiently close to zero, the isotopy class of ξ (cid:15),s is independenton parameters. Holomorphic curves for the contact data
Since we have shown that thegenus zero holomorphic curves are regular for the SHS data, the implicit functiontheorem implies that they will survive a small perturbation to contact data, andwill still be regular. 38 .9 Proof of Theorem 1.3
Once all the technical tools are in place, we will prove Theorem 1.3. We fix theparameters (cid:15) and s , so that we work in the SFT algebra A (Λ (cid:15),s )[[ (cid:126) ]] whose homologyis H SF T ∗ ( M, ξ (cid:15),s ) (which is independent on parameters). We take coefficients in R Ω = R [ H ( M ; R ) / ker Ω], where Ω ∈ Ω ( M ) defines an element in the annihilator O :=Ann( (cid:76) k H ( Y ; R ) ⊗ H ( S ; R )). Here, we view (cid:70) k Y × S = (cid:70) k Y × S × { } as sitting in M Y ⊆ M . We will show that ( M, ξ (cid:15),s ) has Ω-twisted ( k − ∈ O , so that in particular it has ( k − untwisted version ofthe SFT algebra.Let us recall that for SFT to be defined, we need to introduce an abstract pertur-bation of the Cauchy–Riemann equation. We shall be doing our computation prior to introducing this perturbation, and prove by the end the section that this is areasonable thing to do. See the end of this section for more details. Computation of algebraic torsion
The index formula (2) implies that curves u − y,a which asymptote index 2 n critical points (maxima) at k − k positiveends, and one index 1 critical point at the remaining one, have index 1.Given e , . . . , e k − maxima and h an index 1 critical point, denote by γ e i and γ h the corresponding non-degenerate Reeb orbits. Consider the moduli space M = M ( W, J (cid:15),s ; ( γ e , . . . , γ e k − , γ h ) , ∅ )of R -translation classes of k -punctured genus zero J (cid:15),s -holomorphic curves in W = R × M , which have no negative asymptotics, and have γ e , . . . , γ e k − , γ h as positiveasymptotics. The uniqueness Theorem 2.10 implies that every element in M is agenus zero curve in our foliation. Since every such curve is regular, after a choiceof coherent orientations as in [BoMon04], this moduli space is an oriented zerodimensional manifold, which can therefore be counted with appropiate signs. Now,a choice of coherent orientations for the moduli space of Morse flow lines induces acoherent orientation for the moduli space containing the curves u aγ , and we fix sucha choice here onwards. We choose our function H : Y × I → R ≥ as made explicit in Section 2.6. In particular, there are no index zero critical points,and the only index 1 critical point is given by ( min, min is the uniqueminimum of g . We shall denote by M ( H ; p − , p + ) the moduli space of positive un-parametrized flow lines γ connecting p − to p + . Assuming the Morse–Smale conditionfor H , we have then that the zero dimensional moduli spaces correspond to criticalpoints satisfying ind p + ( H ) = ind p − ( H ) + 1.39e then fix e , . . . , e k − , h = ( min,
0) as above, and we let q e i and q h be thegenerators in SFT corresponding to the Reeb orbits associated to these criticalpoints. Let Q = q e . . . q e k − q h , which is an element of A (Λ (cid:15),s )[[ (cid:126) ]]. In order to compute its differential, we need tocount all of the rigid holomorphic curves which asymptote at its positive ends anyof the Reeb orbits appearing in Q . Claim . The holomorphic curves contributing to the differential of Q are of eitherthe two following types: • A holomorphic sphere u − y,a ∈ M , which is in fact unique. • A holomorphic cylinder u γ inside r = 0, connecting an index 2 n − e i .Indeed, using Theorem 2.10, it follows that only the somewhere injective curvesin our foliation are involved in the computation of this differential (see Proposition2.11 below). Using that there are no index zero critical points, we see that thereare only two possible ways to approach the critical point ( min, Y × I , and that there is a unique R -classof the form u − y which has γ h as a positive asymptotic (see Figure 6). Moreover, byobserving that the generic behaviour is hitting a maxima, by a generic and smallperturbation of the Morse function H along different components of the spine, we canarrange that this curve actually defines an element of M . The uniqueness Theorem2.10 above shows that in fact this is the only element in M . Finally, ruling out thecurves coming from the positive side (which have the wrong index and hence arenot counted by SFT), we are left only with the curves listed in our claim.In order to count the curves of type u γ , we observe that positive flow lines goingfrom an index 2 n − p to an index 2 n critical point q come in “eviltwins” pairs γ ↔ γ : by definition of the Morse index, we have only one positiveeigendirection for the Hessian of the Morse function at p , and the flow lines approachthis point on either side of this direction. Since H n ( Y × I ) = H n ( Y ) = 0 ( Y beinga (2 n − M ( H ; p, q ) = 0.Fix Ω a closed 2-form so that [Ω] ∈ O . For d ∈ H ( M ; R ), we denote by d ∈ H ( M ; R ) / ker Ω the class it induces, by u the unique R -translation class in40 , and, for a rigid holomorphic curve v , we denote by (cid:15) ( v ) the sign of v assignedby our choice of coherent orientations. In particular, we know that (cid:15) ( u γ ) = − (cid:15) ( u γ ).Observe that, for the i -th component of M Y , Reeb orbits corresponding to criticalpoints all define the same homology class [ S ] i ∈ H ( M ). We take as canonicalrepresentative of this class the 1-cycle given by the Reeb orbit γ e i over the uniquemaxima e i . For every index 2 n − p lying in the i -th component,fix γ an index 1 flow line joining p to the maxima e i . Choose the spanning surfaceof γ p to be F p := − γ × S , satisfying ∂F p = γ p − γ e i . Then, for this choice ofspanning surfaces, the homology class associated to the holomorphic cylinder u γ is[ u γ ] = [ F p ∪ u γ ]. One thinks of F p as being attached to u γ at the negative end,corresponding to γ p . Let T γ,γ denote the 2-torus γ ∪ γ × S . Observe that[ u γ ] − [ u γ ] = [ T γ,γ ] ∈ H ( Y ) ⊗ H ( S ) ⊆ ker ΩTherefore, we have [ u γ ] = [ u γ ].According to Proposition 2.11 below, the image of Q under the differential D (cid:15),s : A (Λ (cid:15),s )[[ (cid:126) ]] → A (Λ (cid:15),s )[[ (cid:126) ]] is then given by D (cid:15),s ( Q ) = (cid:15) ( u ) z [ u ] (cid:126) k − + (cid:88) i =1 ,...,k − ind p ( H )=2 n − (cid:88) γ ∈M ( H ; p,e i ) (cid:15) ( u γ ) z [ u γ ] q p ∂Q∂q e i = (cid:15) ( u ) z [ u ] (cid:126) k − + 12 (cid:88) i =1 ,...,k − ind p ( H )=2 n − (cid:88) γ ∈M ( H ; p,e i ) ( (cid:15) ( u γ ) + (cid:15) ( u γ )) z [ u γ ] q p ∂Q∂q e i = (cid:15) ( u ) z [ u ] (cid:126) k − , (21)which proves that our model has ( k − R Ω . We haveused that all orbits are simply covered, so no combinatorial factors appear, and weneed not worry about asymptotic markers. Why the computation works
We now justify the computation above. Letus recall first the fact that the abstract polyfold machinery for SFT requires theintroduction of an abstract perturbation to the Cauchy–Riemann equation makingevery holomorphic curve of positive index regular. The basic facts about this pertur-bation scheme, which comes from the polyfold theory of Hofer–Wysocki–Zehnder,are that • Every Fredholm regular index 1 holomorphic curve gives rise to a unique so-lution to the perturbed problem, if the perturbation is sufficiently small.41
If solutions to the perturbed problem with given asymptotic behaviour existfor small perturbations, then as the perturbation is switched off they give riseto a subsequence of curves which converge to a holomorphic building with thesame asymptotic behaviour.Therefore, the index 1 curves in our foliation survive and are counted, but weneed to make sure there are no extra curves which need to be taken into the count.In what follows, J will denote the original J (cid:15), we constructed in the Morse case. Proposition 2.11.
The space of connected J -holomorphic stable buildings of index 1which may become curves contributing to the differential of Q = q e . . . q e k − q h afterintroducing an abstract perturbation, or after perturbing the SHS to a sufficientlynearby contact structure, have only one level, no nodes, and are somewhere injectiveand regular: they actually consist exactly of either an index 1 curve u γ for somepositive flow line γ , or a curve u − y,a for some y ∈ Y , a ∈ R .Proof. It follows by inductively applying Theorem 2.10 [Mo2].
In this section, we address a conjecture in the paper [MNW13] (Conjecture 4.14).
Definition 3.1 (Giroux) . Let Σ be a compact 2 n -manifold with boundary, ω asymplectic form on the interior ˚Σ of Σ, and ξ a contact structure on ∂ Σ. The triple(Σ , ω, ξ ) is an ideal Liouville domain if there exists an auxiliary 1-form β on ˚Σ suchthat • dβ = ω on ˚Σ. • For any smooth function f : Σ → [0 , ∞ ) with regular level set ∂ Σ = f − (0),the 1-form β = f β extends smoothly to ∂ Σ such that its restriction to ∂ Σ isa contact form for ξ .The 1-form β is called a Liouville form for (Σ , ω, ξ ).In [MNW13]-terminology, we say that an oriented hypersurface H in a contactmanifold ( M, ξ ) is a ξ -round hypersurface modeled on some closed contact manifold( Y, ξ ) if it is transverse to ξ and admits an orientation preserving identification with S × Y such that ξ ∩ T H = T S ⊕ ξ .Given an ideal Liouville domain (Σ , ω, ξ ), the Giroux domain associated to it isthe contact manifold Σ × S endowed with the contact structure ξ = ker( f ( β + dθ )),42here f and β are as before, and θ is the S -coordinate. Away from V ( f ) = ∂ Σ,the vanishing locus of f , this contact structure coincides with the contactization ker( β + dθ ). Over V ( f ) it is just given by ξ ⊕ T S , so that V ( f ) × S is a ξ -roundhypersurface modeled on ( ∂ Σ , ξ ).One may find a collar neighbourhood of the form [0 , × H ⊆ Σ × S , on which ξ is given by the kernel of a contact form β + sdθ , where s is the coordinate onthe interval, where H ⊆ ∂M corresponds to s = 0, θ the coordinate in S , and β a contact form for ξ [MNW13, Lemma 4.1]. Using these collar neighbourhoods,one has a well-defined notion of gluing of two Giroux domains along boundarycomponents modeled on isomorphic contact manifolds (see Section 3.1 below).We also have a blow-down operation for round hypersurfaces lying in the bound-ary. If H is a ξ -round boundary component of ( M, ξ ), with orientation opposite theboundary orientation, consider the collar neighborhood [0 , × H as before. Let D be the disk of radius √ (cid:15) in R . The map Ψ : ( re iθ , y ) (cid:55)→ ( r , θ, y ) is a diffeomorphismfrom ( D \{ } ) × Y to (0 , (cid:15) ) × S × Y which pulls back β + sdt to the contact form β + r dθ . Thus we can glue D × Y to M \ H to get a new contact manifold in which H has been replaced by Y , and the S -component of H has been capped off.Given a contact embedding of the interior of a Giroux domain G Σ := Σ × S insidea contact manifold ( M, ξ ), we shall denote by O ( G Σ ) ⊆ H ( M ; R ) the annihilatorof H (Σ; R ) ⊗ H ( S ; R ), when the latter is viewed as a subspace of H ( M ; R ). If N ⊆ ( M, ξ ) is a subdomain resulting from gluing together a collection of Girouxdomains G Σ , . . . , G Σ k , we shall denote O ( N ) = O ( G Σ ) ∩ · · · ∩O ( G Σ k ) ⊆ H ( M ; R ).The following theorem implies Theorem 1.6 (see Example 3.3 below): Theorem 3.2.
If a contact manifold ( M n +1 , ξ ) admits a contact embedding of asubdomain N obtained by gluing two Giroux domains G Σ − and G Σ + , such that Σ + has a boundary component not touching Σ − , then ( M n +1 , ξ ) has Ω -twisted algebraic1-torsion, for every Ω ∈ O ( N ) . Moreover, it is also algebraically overtwisted if N contains any blown down boundary components. The motivating example of an explicit model of such subdomain is the following:
Example 3.3.
Consider (
Y, α + , α − ) a Liouville pair on a closed manifold Y n − . Asin the introduction, consider the Giroux π -torsion domain modeled on ( Y, α + , α − ),given by the contact manifold ( GT, ξ GT ) := ( Y × [0 , π ] × S , ker λ GT ), where λ GT = 1 + cos( r )2 α + + 1 − cos( r )2 α − + sin( r ) dθ (22)and the coordinates are ( r, θ ) ∈ [0 , π ] × S .We may write λ GT = f ( α + dθ ) , where f = f ( r ) = sin( r ) , = 12 ( e u ( r ) α + + e − u ( r ) α − ) ,u ( r ) = log 1 + cos( r )sin( r ) , Here, u : (0 , π ) → R is an orientation reversing diffeomorphism, whereas u :( π, π ) → R is an orientation-preserving one, which is used to pull-back the Li-ouville form ( e u α + + e − u α + ) defined on Y × R via a map of the form v = id × u .This means that we may view ( GT, ξ GT ) as being obtained by gluing two Girouxdomains of the form GT − = Y × [0 , π ] × S and GT + = Y × [ π, π ] × S , along theircommon boundary, an ξ GT -round hypersurface modeled on ( Y, α − ).Therefore, from Theorem 3.2 we obtain Theorem 1.6 as a corollary. We consider a specially simple kind of spinal open book decompositions (SOBDs),which arise on manifolds which have been obtained by gluing a family of Girouxdomains along a collection of common boundary components, each a round hyper-surface modeled on some contact manifold. Such is the case of the Giroux 2 π -torsiondomains GT . Basically, these SOBDs are obtained by declaring suitable collar neigh-bourhoods of each gluing hypersurface to be paper components, whereas the spine components are the complement of these neighbourhoods. They have the desirablefeatures that the fibrations are trivial, and that they have 2-dimensional pages (see[Mo2] for definitions). Construction of the SOBD.
Let ( G ± = Σ ± × S , ξ ± = ker( f ± ( β ± + dθ ))) be twoGiroux domains which one wishes to glue along a round-hypersurface H = Y × S modeled on some contact manifold ( Y, ξ ) and lying in their common boundaries. Fixchoices of collar neighbourhoods N ± ( H ) of H inside G ± , of the form Y × S × [0 , s ± ∈ [0 ,
1] so that H = { s ± = 0 } , and θ ∈ S , such that the contactstructures ξ ± are given by the kernel of the contact forms β + s ± dθ . Here, β isa contact form for ξ . In these coordinates, β ± = β /s ± and f = f ( s ± ) = s ± , andthe corresponding ideal Liouville vector fields are V ± = − s ± ∂ s ± . We glue N ± ( H )together in the natural way, by taking a coordinate s ∈ [ − , s = − s − and s = s + , and s = 0 corresponds to H . We thus obtain a collar neighbourhood N ( H ) = N + ( H ) (cid:83) Φ N − ( H ) (cid:39) H × [ − , M := G + (cid:91) Φ G − = (cid:32) Σ + (cid:91) Φ Σ − (cid:33) × S , M = M P ∪ M Σ , where M P (the paper ) is the disjoint union of the collar neighbourhoods of the form N ( H ) for each of the gluing round-hypersurfaces H , and M Σ (the spine ) is theclosure of its complement in M . We have a natural fibration structure π P on M P ,which is the trivial fibration over the disjoint union of the hypersurfaces H , so thatthe pages (the fibers of π P ) are identified with the annuli P := S × [ − , S -fibration π Σ on M Σ , which is also trivial. It has as base the disjointunion of Σ + and Σ − minus the collar neighbourhoods, which we denote by Σ. Letus assign a sign (cid:15) ( G ± ) := ± G ± ⊆ M .We can also blow-down boundary components in the Giroux subdomains, andin this case we may extend our SOBD by declaring the D × Y we glued in the blow-down operation to be part of the paper M P (so that its pages are disks). We declarethe blown-down components to be part of the paper.We shall refer to the SOBDs obtained by the procedure described above as Giroux
SOBDs, where are also allowing the number of
Giroux subdomains involved (whichis the same as the number of spine components) to be arbitrary.We will also fix collar neighbourhoods of the components of ∂M , which corre-spond to the non-glued hypersurfaces, in the very same way as we did before forthe glued ones. Their components look like N Σ ( H ) := H × [0 ,
1] = Y × S × [0 , H . There is a coordinate s ∈ [0 , ∂M = { s = 0 } , and such that the contact structure on the corresponding Girouxdomain is given by ker( β + sdθ ), for some contact form β for ( Y, ξ ). Denote by N Σ the disjoint union, over all unglued H ’s, of all of the N Σ ( H )’s. The Giroux form.
If we denote by λ ± = f ± ( β ± + dθ ), we can extend the ex-pression λ + = β + sdθ , a priori valid on N + ( H ), to N ( H ), by the same formula.Observe that, by choice of our coordinates, the resulting 1-form glues smoothly to λ − := f − ( β − − dθ ). Therefore, we may still think of the contact structure ξ − = ker λ − as a contactization contact structure over the region (Σ − × S ) \N − ( H ), with thecaveat that we need to switch the orientation in the S -direction.Denote the resulting contact form by λ := λ + ∪ Φ λ − . We may globally write it as λ = f ( β + dθ ) . Here, f is a function which is either strictly positive or strictly negativeover the interior of the Giroux domains, and vanishes precisely at the (glued andunglued) boundary components. The 1-form β coincides with the Liouville forms ± β ± where these are defined, and is undefined along said boundary components. Inthe case of blown-down components, the contact form λ also extends in a naturalway. 45 aper paper spine paper spine spine spine Figure 7: The isotoped function f in the extended Giroux torsion domain GT (cid:15) .From this construction, in M , each of the subdomains G ⊂ M used in the gluingprocedure carries a sign (cid:15) ( G ), which we define as the sign of the function f | ˚ G .Observe that f coincides with (cid:15) ( G ) along every H × { s ± = 1 } , the inner boundarycomponents of the collar neighbourhoods. Therefore, we may isotope it relative everycollar neighbourhood to a smooth function which is constant equal to (cid:15) ( G ) along G \ (cid:0) ( M δP ∩ G ) ∪ N δ∂ (cid:1) (see Figure 7). Here, M δP and N δ∂ denote small δ -extensions inthe interval direction of M P and N Σ , respectively, so that now | s | ∈ [0 , δ ]. Theregions M δP \ M P play the role of smoothened corners. Observe that isotoping f aswe just did does not change the isotopy class of λ , by Gray stability (version withboundary), and has the effect of transforming the Reeb vector field of λ into (cid:15) ( G ) ∂ θ along M Y . Observe also that along the paper, we have λ = β + γ , where γ = sdθ is a Liouville form for S × [ − , λ as a Giroux form, as we did with contact formconstructed in Section 2.1. (cid:15) -extension. It will be convenient to consider an (cid:15) -extension of our SOBD, whichwe call M (cid:15) , by gluing small collars to the boundary. To each component N Σ ( H ) for H lying in ∂G for a subdomain G of M , we glue a collar neighbourhood of the form H × [ − (cid:15),
0] for some small (cid:15) >
0. We extend our function f so that the extendedversion coincides with (cid:15) ( G ) id near s = 0, f (cid:48) ( − (cid:15) ) = 0, and sgn ( f (cid:48) ( s )) = (cid:15) ( G ) (cid:54) = 0for s > − (cid:15) (see Figure 7). We will define the paper M (cid:15)P to be the union of M P and the region H × [ − (cid:15)/ , δ ], and the spine M (cid:15) Σ to be the union of M Σ with H × [ − (cid:15), − (cid:15)/ λ coincides with − (cid:15) ( G ) ∂ θ along the boundary. Remark 3.4.
Because of the above discussion on orientations, from which we gath-ered that the S -orientation that we need depends on the sign of the Giroux domain,46e rule out the case where we have a sequence ( G , . . . , G N − ) of Giroux subdomainsof M , where G i has been glued to G i +1 (modulo N ) along some collection of bound-ary components, and N >
We can generalize the previous construction to the case where the S -bundles arenot necessarily globally trivial, but trivial on the boundary components which areglued together. Definition 3.5. [DiGe12] Let Σ be a compact 2 n -manifold with boundary, ω int asymplectic form on ˚Σ, ξ a contact structure on ∂ Σ, and c a cohomology class in H (Σ; Z ). The tuple (Σ , ω int , ξ , c ) is called an ideal Liouville domain if for some(and hence any) closed 2-form ω on Σ with − [ ω/ π ] = c ⊗ R ∈ H (Σ; R ) there existsa 1-form β on ˚Σ such that • dβ = ω int − ω on ˚Σ. • For any smooth function f : Σ → [0 , ∞ ) with regular level set ∂ Σ = f − (0),the 1-form β = f β extends smoothly to ∂ Σ such that its restriction to ∂ Σ isa contact form for ξ .An ideal Liouville domain (Σ , ω int , ξ ,
0) is an ideal Liouville domain in the senseof Giroux, where we may take ω = 0. In the case where c | ∂ Σ = 0, so that we maytake ω = 0 in any collar neighbourhood [0 , × ∂ Σ of the boundary, we will call(Σ , ω int , ξ , c ) an ideal strong symplectic filling .Now let π : M → Σ be the principal S -bundle over Σ of (integral) Euler class c . Choose a connection 1-form ψ with curvature form ω on this bundle. As in thetrivial case c = 0 (where we may take ψ = dθ ), the 1-form f ( ψ + β ) = f ψ + β definesa contact structure ξ on M , and we call ( M, ξ ) the contactization of (Σ , ω int , ξ , c ).Observe that d ( ψ + β ) = π ∗ ( ω + ω int − ω ) = π ∗ ω int , where we identify β with π ∗ β , sothat on ˚Σ, ξ may be regarded as the prequantization contact structure correspondingto ω int . Definition 3.6. [DiGe12] Let B be a closed, oriented manifold of dimension 2 n .An ideal Liouville splitting of class c ∈ H ( B ; Z ) is a decomposition B = B + (cid:83) Γ B − along a two-sided (but not necessarily connected) hypersurface Γ, oriented as theboundary of B + , together with a contact structure ξ on Γ and symplectic forms ω ± on ± ˚ B ± , such that ( ± B ± , ω ± , ξ , ± c | B ± ) are ideal Liouville domains.47roposition 3.6 in [DiGe12] tells us that an S -invariant contact structure ξ =ker( β + f ψ ) on the principal S -bundle π : M → B defined by c ∈ H ( B ; Z ) leadsto an ideal Liouville splitting of B of class c , along the dividing setΓ = { b ∈ B : ∂ θ | b ∈ ξ } = f − (0)If we require that c ∈ H ( B, Γ; Z ), so that c | Γ = 0, then the S -principal bundle π is trivial in any collar neighbourhoods [ − , × Γ of Γ, along which we may take ψ = dθ , ω = 0. In this situation, we may regard the total space M as carrying whatwe will call a prequantization SOBD , which is obtained in the same way as we defineda Giroux SOBD. That is, we declare suitable collar neighbourhoods of Γ to be papercomponents, which we can do by the triviality assumption on c along Γ. The onlydifference now is that the spine is no longer globally a trivial S -bundle. Reciprocally,one can glue prequantization spaces over ideal strong symplectic fillings to obtain amanifold M (possibly with non-empty boundary) carrying an S -invariant contactstructure ξ = ker( β + f ψ ), and a prequantization SOBD. The gluing constructionis the same as for Giroux SOBDs, and is a particular case of the gluing constructionof Thm. 4.1 in [DiGe12].Observe that, using the coordinates of the previous section, the contact 1-form α = β + f ψ defining ξ satisfies that dα = ds ∧ dθ > S × [ − , f in the same way as for Giroux SOBDs, itsReeb vector field is tangent to the S -fibers of π Σ along M Σ . Moreover, along theboundary, it induces the distribution ker α ∩ T ( ∂M ) = ξ ⊕ T S , with characteristicfoliation given by the S -direciton, and its Reeb vector field coincides with R , theReeb vector field of β . From this, and having the definition in [L-VHM-W], andthe standard one due to Giroux, both in mind, one can define Definition 3.7. A Giroux form for a prequantization SOBD (in particular, for aGiroux SOBD) is any contact form inducing a contact structure which is isotopic tothe S -invariant contact structure ξ = ker( β + f ψ ).With this definition, any S -invariant contact form α inducing an S -invariantcontact structure in a principal bundle defined by a cohomology class c satisfying c | Γ = 0, where Γ is a set of dividing hypersurfaces for α , is a Giroux form for theinduced prequantization SOBD. The proof of this theorem is a reinterpretation of what we did in the constructionof our contact manifold models of Section 2.1.
Thm. (3.2).
Let N be a subdomain as in the hypothesis, carrying a Giroux form λ = f ( β + dθ ) obtained by gluing. Since the contact embedding condition is open,48 Σ ε S x M Σ ε M Σ ε M Σ ε M Σ ε M Σ ε M Σ ε M P ε M P ε M P ε M P ε M P ε M P ε M P ε M P ε Figure 8: An (cid:15) -extension of a domain N consisting of two Giroux domains gluedalong three boundary components. We draw the Morse–Bott submanifolds of H in green. These are a “barrier” for page-like holomorphic curves, needed to adaptTheorem 2.10. We also draw an index 1 holomorphic cylinder (of the first kind).and we are assuming that there are boundary components of Σ + not touching Σ − ,we can find a small (cid:15) > M admits a contact embedding of the (cid:15) -extension N (cid:15) . Endow this extension with a Giroux SOBD N (cid:15) = M (cid:15)P ∪ M (cid:15) Σ as in the previoussection. On this decomposition, add corners where spine and paper glue togetheras we explained before. Also choose a small Morse/Morse–Bott function H in M (cid:15) Σ ,which lies in the isotopy class of f , and vanishes as we get close to the paper. Withthis data, we then may construct a Morse/Morse–Bott contact form Λ on N whichlies in the isotopy class of λ , along with a SHS deforming it, in the analogous wayas done in Section 2.1.Since in our situation we are allowing Σ ± to be more general than a semi-filling Y × I , we need to specify what we mean by the Morse–Bott situation. We willtake our Morse–Bott function H so that it depends only on the interval parameteralong the collar neighbourhoods N Σ close to the boundary and along a slightlybigger copy of M (cid:15)P , and matches the function f close to ∂N . In particular, ∂N is a Morse–Bott submanifold. We also impose that H is Morse in the interior ofthe components of M Σ which are away from the boundary. For simplicity, we willassume that, besides the boundary, H only has exactly one Morse–Bott submanifoldclose to each boundary component of M Σ which is glued to a paper component, ofthe form Y × { t } for some t (see Figure 8). The Morse situation is then obtainedby a perturbation of this situation obtained by choosing Morse functions along theMorse–Bott submanifolds. Observe that we may always choose the interior Morse-Bott submanifolds so that they lie in the cylindrical ends of the Liouville domainsΣ ± .We have an almost complex structure J (cid:48) compatible with the SHS, for whichwe get a stable finite energy foliation by holomorphic cylinders, which come in twotypes: either they are obtained by gluing constant lifts of the cylindrical pagesand flow-line cylinders over M (cid:15) Σ along the corners (first kind); or they are flow-linecylinders completely contained in M (cid:15) Σ (second kind). Both kinds have as asymptotics49 aper paper corner spine corner corner cornerspine paper spine corner cornerspine Figure 9: The foliation by holomorphic cylinders (with respect to SHS data) of thesymplectization of GT (cid:15) .simply covered Reeb orbits γ p corresponding to critical points p of H , either along theboundary, or at interior points of Σ (see Figure 9). These cylinders have two positiveends if its corresponding critical points lie in Giroux subdomains with different sign,and one positive and one negative end if these signs agree. The Fredholm indexformula is exactly as before.One can adapt the proof of Fredholm regularity in the Morse-Bott case (Section2.5) to this particular situation, to obtain regularity for cylinders of both kinds.There is no change in the proof for cylinders of the second kind. For regularity forcylinders of the first kind, we adapt the proof of Section 2.5.1. For this, we observethat our choices of interior Morse-Bott submanifolds imply that the cylindrical endsof the cylinders of the first kind lie completely over cylindrical ends of the Liouvilledomains Σ ± (see Figure 8). This means that we may assume that the obviousanalogous version of Assumptions 2.7 hold, and the rest of the proof is a routineadaptation. Fredholm regularity in the sufficiently nearby Morse case follows fromthe implicit function theorem.We also have a version of the uniqueness Theorem 2.10, adapted to this situation.We have a few small changes, as follows. In the hypothesis we require that thepositive asymptotics are simply covered; correspond to critical points all of which,in the Morse–Bott case, belong to a Morse–Bott submanifold of the type alreadydescribed; and any two lie in distinct components of M Σ , both of which are awayfrom the boundary, separated by a paper component. The number k is replaced by2 in the proof. We have more than just one Morse–Bott submanifold now, but, forcase B where u lies over some paper component P = Y × I × S , we can still get asimilar upper bound on the energy: 50 ( u ) ≤ π (cid:32) (cid:88) z ∈ Γ + e (cid:15)H ( p z ) − (cid:88) z ∈ Γ − T z (cid:33) ≤ π (cid:88) z ∈ Γ + e (cid:15)H ( p z ) ≤ π || e (cid:15)H || C , (23)where T z > z ∈ Γ − , which a priorimight even lie in M \ N (but not a posteriori). Here, the energy is computed withrespect to a 2-form which is the derivative of a contact form for ξ outside of N ,and which restricts to the 2-form of the fixed SHS on N . Here we use that we havechosen the SHS so that it is contact near ∂N .Using that the number of boundary components of each paper component is 2,as before we get E ( u ) ≥ πe − δ deg P ( u ) , for some small δ >
0. Here, we denote by deg P ( u ) the covering degree of therestriction to u of the projection to I × S , over the component P . The rest ofthe proof is the same. Observe that since we are assuming that the critical pointsappearing in the positive asymptotics are away from the boundary of N , and theonly way of venturing into M \ N is to escape through the latter, any holomorphiccurve with these positive asymptotics which leaves N would necessarily need to gothrough a paper component, so is dealt with by case B in the proof of 2.10 (whichdeliberately does not need to assume that the whole curve stays over N ).Make all the necessary choices to have an SFT differential D SF T : A (Λ)[[ (cid:126) ]] → A (Λ)[[ (cid:126) ]] , which computes H SF T ∗ ( N, ker Λ; Ω | N ), where Ω ∈ O ( N ). Extend these choices to M so as to be able to compute H SF T ∗ ( M, ξ ; Ω).Let e be a maximum (index 2 n ) of H in M Σ ∩ (Σ − × S ), and let h be an index1 critical point of H in M Σ ∩ (Σ + × S ). We take both to lie in the Morse–Bottmanifolds of the Morse–Bott case, before the Morse perturbation. Denote by q e , q h the corresponding SFT generators. Define Q = q h q e , an element of A (Λ). There is a unique (perturbed) cylinder u of the first kind whichhas γ e and γ h as positive asymptotics, with ind( u ) = 1. If we choose H so thatit does not have any minimum, by uniqueness any other holomorphic curve over51 which may contribute to the differential of Q is a flow-line cylinder completelycontained in M Σ , connecting an index 2 n − p with e .Denote by d the element in H ( M ; R ) / ker Ω defined by any d ∈ H ( M ; R ), by M ( H ; p, e ) the space of positive flow-lines connecting p with e , u γ the flow-linecylinder corresponding to a flow-line γ , and (cid:15) ( u ) the sign of the holomorphic curve u assigned by a choice of coherent orientations. Since H n (Σ ± ) = 0, elements of M ( H ; p, e ) come in evil twins pairs γ ↔ γ such that (cid:15) ( u γ ) = − (cid:15) ( u γ ). As in Section2.9, one can choose suitable spanning surfaces such that [ u γ ] = [ u γ ].Then, D SF T Q = z [ u ] (cid:126) + (cid:88) γ ∈M ( H ; p,e ) ind p ( H )=2 n − (cid:15) ( u γ ) z [ u γ ] q p q h = z [ u ] (cid:126) + 12 (cid:88) γ ∈M ( H ; p,e ) ind p ( H )=2 n − ( (cid:15) ( u γ ) + (cid:15) ( u γ )) z [ u γ ] q p q h = z [ u ] (cid:126) , (24)which proves that ( N, ker Λ) has Ω | N -twisted 1-torsion. This implies that ( M, ξ )has Ω-twisted 1-torsion, since our uniqueness theorem gives that there are no holo-morphic curves with asymptotics in N which venture into M \ N , and therefore H SF T ∗ ( N, ker Λ; Ω | N ) embeds into H SF T ∗ ( M, ξ ; Ω). Here, we use that ker Λ is iso-topic to ker λ = ξ | N .For the second statement, assume that we have a blown-down boundary compo-nent, so that the corresponding D × Y is a paper component with disk pages. Asin Section 2.3, we have that the disk pages lift as finite-energy holomorphic planeswith a single positive asymptotic γ p , corresponding to a critical point p in M Σ . If u is such a plane, its index is ind( u ) = ind p ( H ). Take p so that ind p ( H ) = 1, and let P = q γ p . Since there is no minima for H , by our uniqueness theorem we know that D SF T (cid:16) z − [ u ] P (cid:17) = 1Since the choice of coefficients is arbitrary, ( N, ker Λ) is algebraically overtwisted,which implies that ( M, ξ ) is.In view of the definition of a prequantization SOBD, and the fact that we maystill construct a foliation by flow-line holomorphic cylinders over a non-trivial pre-quantization space [Mo2, Sie16], one has the following:52 heorem 3.8.
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