Bourgeois contact structures: tightness, fillability and applications
BBourgeois contact structures:tightness, fillability and applications
Jonathan Bowden, Fabio Gironella, Agustin Moreno
Abstract
Given a contact structure on a manifold V together with a supportingopen book decomposition, Bourgeois gave an explicit construction of a contactstructure on V × T . We prove that all such structures are universally tightin dimension 5, independent on whether the original contact manifold is tightor overtwisted. In the planar case we give obstructions to the existence ofstrong symplectic fillings and we also obtain a broad class of new examples ofweakly but not strongly fillable contact 5–manifolds.The techniques developed in the 5–dimensional case also allow us to obtaintwo further results in arbitrary dimensions. Firstly, that the unit cotangentbundle of the n -torus has a unique symplectically aspherical strong filling upto diffeomorphism; secondly, that the Bourgeois contact manifold associatedto the open book with monodromy given by a single Dehn–Seidel twist onthe unit cotangent bundle of the n -sphere admits no strong filling. The latteranswers a question of Lisi–Marinkovi´c–Niederkr¨uger and provides the first ex-amples of weakly but not strongly fillable contact structures in all dimensions. Contents a r X i v : . [ m a t h . S G ] D ec Proof of tightness in dimension
226 Topology of the capped filling 26 T survives homologically in W cap . . . . . . . . . . . . . . . . . . . . 306.3 Parametric Wendl family of holomorphic cylinders . . . . . . . . . . . 326.4 Intersections with co-cores in the capped filling . . . . . . . . . . . . 39 S ∗ T n . . . . . . . . . . . . . . . . 558.2 Symplectic fillability of BO ( T ∗ S n , τ ) . . . . . . . . . . . . . . . . . . 59 In [Bou02b], Bourgeois showed that, whenever (
V, ξ ) is a contact manifold endowedwith a supporting open book decomposition (which always exist by work of Giroux[Gir02]), then the manifold V × T carries a natural contact structure. The mainmotivation behind such a construction was the problem of the existence of contactstructures on higher–dimensional manifolds. For instance, it showed that every odddimensional torus admits contact structures, a problem that has been open sinceLutz [Lut79] proved that T is contact, more than 20 years before.It was not until recently that Borman-Eliashberg-Murphy [BEM15] proved thatcontact structures in higher-dimensions actually exist in abundance (i.e. wheneverthe obvious topological obstructions disappear) by generalizing Eliashberg’s [Eli89]notion of overtwistedness , as well as the h –principle that comes with it, to higherdimensions. Overtwisted contact manifolds are topological/flexible in nature, andmost of the associated contact-topological invariants (e.g. those coming from holo-morphic curves) simply vanish. As a result, it has become relevant to find exam-ples of high–dimensional contact structures beyond the overtwisted ones, which aremore geometric/rigid and which potentially have rich associated invariants. Contactstructures which are not overtwisted are usually referred to as tight .2he construction in [Bou02b] actually fits very well in this setting, as it is bothvery explicit and yields a very broad class of contact manifolds in arbitrary odd di-mensions, with remarkable properties. For instance, [Pre07] used it to construct thefirst examples of high dimensional (closed) contact manifolds admitting a Plastik-stufe, as defined in [Nie06], which is equivalent to overtwistedness [CMP19, Hua17].In [BCS14], the authors also used it to construct contact structures on the prod-uct of a contact manifold with the 2–sphere. More recently, Lisi, Marinkovi´c andNiederkr¨uger [LMN18] started the systematic study of the Bourgeois construction,in particular studying its fillability properties, and in this paper we continue thisline of research.In what follows we will use the following notation. Given an abstract open book(Σ n , φ ) and the associated contact (2 n + 1)–manifold OBD (Σ , φ ), we denote by BO (Σ , φ ) the contact manifold obtained via the Bourgeois construction [Bou02b].Smoothly, BO (Σ , φ ) = OBD (Σ , φ ) × T and we refer to Section 2 for further details. Tightness.
We begin by addressing the natural question of whether a given Bour-geois contact structure is tight or overtwisted.In [LMN18], the authors give examples, in every odd dimension, of an overtwisted(
V, ξ ) such that the associated Bourgeois contact manifold is tight. Moreover, in[Gir19], it was shown that if V is a 3-manifold with non-zero first Betti number,then there exists a supporting open book such that the associated Bourgeois contactstructure is (hyper)tight. In this paper, we prove that, at least in dimension 5, theseare particular instances of a more general fact. Namely, 5-dimensional Bourgeoiscontact structures are rigid, inherently geometric objects, independently of the rigidor flexible nature of ( V, ξ ): Theorem A (Tightness) . For every abstract open book (Σ , φ ) , the contact –manifold BO (Σ , φ ) is (universally) tight. Recall that universally tight means that the universal cover is tight. In particular,universal tightness implies tightness. The fact that 5–dimensional Bourgeois contactstructures are universally tight is a simple consequence of the fact that they are tightand that finite covers on either factor of the product again yield Bourgeois contactstructures.While there are many ways of making contact/symplectic manifolds more flex-ible (e.g. by adding a Lutz twist, or by taking the flexibilization of a Weinsteinmanifold), Theorem A says that the Bourgeois construction can be interpreted asa “tightifying” procedure. To our knowledge, there is currently no other procedurewith an analogous property.Moreover, the above result is sharp with respect to taking branched covers.This is a consequence of an argument of Massot and Niederkr¨uger, based on ideas3rom [Pre07] (cf. [Nie14, Theorem I.5.1]). The interested reader can consult [Gir19,Observation 5.9] for details.
Symplectic fillability.
Another important problem in contact topology that isrelated to the flexible/rigid classification of contact structures, is that of charac-terizing which contact structures admit symplectic fillings. Indeed, one of the firstresults of the theory of holomorphic curves is that symplectically fillable contactmanifolds are tight. Given Theorem A, it is natural to wonder whether Bourgeoiscontact structures are (at least weakly) symplectically fillable. While there existpartial results in this direction [MNW13, LMN18], a complete answer is yet to befound. See also Remark 9.1 for subtleties in finding weak fillings.In this paper, we consider the planar case, i.e. the case where the page of theopen book of the original manifold has genus zero. In the 3-dimensional situation,strong symplectic fillings of contact structures supported by planar open books existin abundance, and in fact are in 1-1 correspondence with the factorizations of themonodromy into products of positive Dehn-twists [Wen10c]. However, the situationof planar 5-dimensional Bourgeois contact manifolds is surprisingly more rigid:
Theorem B (Fillability) . Let (Σ , φ ) be an abstract open book, where Σ has genuszero. If the contact –manifold BO (Σ , φ ) admits a strong symplectic filling, thenthe monodromy φ can be written as a product of commutators in the mapping classgroup of the page (rel. boundary). The case of φ isotopic to Id corresponds precisely to the case where V admitsa subcritical Stein filling [Cie02]. According to [LMN18, Theorem A.b], in this casethe associated Bourgeois contact manifold admits a Stein filling. In the particularcase of the annulus or the pair of pants the corresponding mapping class groups areabelian and we obtain the stronger statement: Theorem C (Stein vs. Strong fillability) . Let ( P, φ ) be an abstract open book whosepage P is an annulus or a pair of pants. The contact –manifold BO ( P, φ ) admitsa strong symplectic filling if and only if the monodromy φ is trivial in the mappingclass group of the page (rel. boundary).In particular, BO ( P, φ ) is strongly fillable if and only if it is Stein fillable. Concerning Theorem B, note that whilst mapping class groups of higher genussurfaces are perfect , meaning that any element can be written as a product of com-mutators, this is not the case in the planar case. In fact, any non-trivial product ofpositive Dehn twists will not lie in the commutator subgroup. For any product ofpositive Dehn twists gives a non-trivial positive braid after identifying all but oneappropriately chosen boundary component to (marked) points, and positive braidssurvive in the abelianization of the braid group of the disk, which is infinite cyclic(cf. [FM12]). 4ow, contact 3–manifolds admitting a supporting open book with planar pageand monodromy a product of positive Dehn twists are the convex boundaries ofsymplectic Lefschetz fibrations. In particular, according to [LMN18, Theorem A.a](cf. [MNW13, Example 1.1]), the associated Bourgeois contact manifold is weaklyfillable. Moreover, according to [LMN18, Theorem B], the same is true if the mon-odromy is a product of negative Dehn twists (cf. Remark 9.1 below). Theorem Bthen also implies the following:
Corollary D.
Let (Σ , φ ) be an abstract open book with Σ of genus zero and φ aproduct of Dehn twists, all of the same sign. Then, the contact -manifold BO (Σ , φ ) is weakly but not strongly fillable. Examples of weakly but not strongly fillable contact structures in dimension 3 arewell-known. The first examples of such contact manifolds in higher dimensions wereobtained in [MNW13, Theorem E], also in dimension 5 on manifolds diffeomorphicto a product of a 3-manifold with a torus. Those examples are associated to contact3-manifolds whose contact structure belongs to a
Liouville pair (see [MNW13, Def-inition 1]). These are in particular co–fillable, hence necessarily not planar by aresult of Etnyre [Etn04]. Corollary D then complements [MNW13, Theorem E], andprovides a broad class of new examples.Moreover, using Eliashberg’s classification of overtwisted contact structures indimension 3 [Eli90], and the fact that overtwisted contact structures are planar inthat dimension [Etn04], we have:
Corollary E.
For each almost contact structure ( M , η ) on a closed -manifold,there exists a universally tight but not strongly fillable contact structure on M × T which is homotopic to the product almost contact structure η ⊕ T T . Note that the open book given by Etnyre’s result [Etn04] may a priori decomposeas a product of commutators. However, this can be remedied by performing a singlestabilization, so that the resulting monodromy is not a commutator anymore, andwe can thus apply Theorem B to the stabilized open book. Recall that an almostcontact structure on V n +1 is a hyperplane field ξ equipped with a complex structure J : ξ → ξ . On a 3-manifold, this simply reduces to the data of a homotopy classof oriented 2-plane fields. We remark that the contact structure in Corollary E canoccasionaly be weakly fillable (e.g. in the case where the monodromy is a productof all negative Dehn twists, cf. Corollary D).
Symplectically aspherical fillings in higher dimensions.
Using similar tech-niques to those in the proof of Theorem B, we also study symplectically asphericalfillings of higher dimensional Bourgeois contact manifolds.First, we study the diffeomorphism type of the strong symplectically asphericalfillings of the unit cotangent bundle S ∗ T n of T n with its standard contact structure5 std , induced by the restriction of the standard Liouville form λ std on the unit diskcotangent bundle D ∗ T n to its boundary S ∗ T n . In fact, ( S ∗ T n , ξ std ) is none other than BO ( D ∗ T n − , Id). This follows from the following observation. The contact manifold
OBD ( D ∗ T n − , Id) is the convex boundary of the subcritical Stein manifold W = D ∗ S × · · · × D ∗ S (cid:124) (cid:123)(cid:122) (cid:125) n − × D .According to [LMN18, Theorem A.b], BO ( T ∗ T n − , id ) is then the convex boundaryof the Stein manifold (cid:0)(cid:81) n − i =1 D ∗ S (cid:1) × D ∗ T = D ∗ T n , with its split Stein structure,which is just the standard one. We then prove the following uniqueness result: Theorem F.
The contact manifold ( S ∗ T n , ξ std ) has a unique strong symplecticallyaspherical filling up to diffeomorphism. Theorem F is a smooth higher–dimensional version of a result by Wendl [Wen10c,Theorem 4] (cf. [Sti02, Theorem 1.6]). In dimension 5 (i.e. n = 3), Theorems C and Fgive a complete smooth characterization of symplectically aspherical strong fillingsfor the Bourgeois contact 5–manifolds associated to any open book with page D ∗ S .Note that, according to [LMN18], all such examples are weakly fillable. Remark 1.1.
Theorem F has also been independently obtained by Geiges–Kwon–Zehmisch [GKZ19].Lastly, we consider the case of BO ( D ∗ S n , τ ), where τ is the Dehn–Seidel twiston D ∗ S n , and we give a negative answer to [LMN18, Question 1.6]: Theorem G.
The Bourgeois contact manifold BO ( D ∗ S n , τ ) admits no strong sym-plectic filling. Remark 1.2.
The conclusion of Theorem G actually also holds for BO ( D ∗ S n , τ k ),for every k ∈ Z such that the 0-section S n inside the page of OBD ( D ∗ S n , τ k ) isnull-homologous in the resulting manifold, as is manifest in the case of a sphere S n +1 = OBD ( D ∗ S n , τ ). This follows by inspecting the proof. Observe that thiscondition is clearly not satisfied if k is such that τ k = smoothly, since then S n × S n +1 ∼ = OBD ( D ∗ S n , ) as smooth manifolds (the contact manifold BO ( D ∗ S n , )is actually Stein fillable). Moreover, it follows from Theorem 3.1 below that the setof powers k ∈ Z for which BO ( D ∗ S n , τ k ) is actually strongly fillable is a subgroupof Z , and so of the form k Z for some k ∈ Z . The generator satisfies k (cid:54) = 1 byTheorem G.It is well known that the contact manifolds BO ( D ∗ S n , τ ) admit weak fillings[LMN18, Theorem A.a]. As a consequence we obtain the first known examplesweakly but not strongly fillable contact structures in all dimensions (cf. [MNW13]). Corollary H.
There exist weakly fillable contact manifolds that admit no strongsymplectic fillings in all dimensions. O( , ) Σ Ψ Φ � � O( , ) Σ Ψ � O( , ) Σ Φ (C, ω ) C Figure 1: The pseudo-Liouville cobordism (
C, ω C ) given in Theorem 3.1. Outline of the proofs.
For convienence of the reader, we outline the main argu-ments of the proofs of Theorems A, B, F and G.
Tightness in dimension . The proof of Theorem A involves some geometricgroup theory and hyperbolic geometry as well as some holomorphic curve techniques.The first ingredient is the construction of a strong symplectic cobordism betweenBourgeois contact structures; this is done in Section 3.1. More precisely, Theorem 3.1is a “stabilized” version of the analogous result for open books, which was proven(independently) in [Avd12, Klu18]; see Figure 1. We point out that, while thesymplectic form on the strong cobordism of Theorem 3.1 is exact, the Liouville vectorfield associated to the global primitive is not inwards pointing along the negativeends. We shall refer to a strong symplectic cobordism with an exact symplectic formas pseudo-Liouville .For “most” cases of surfaces (the rest are dealt with case by case), standardresults from low-dimensional topology then allow to write any monodromy as acomposition, such that the contact structures on the negative ends of the cobordismin Figure 1 are hypertight; see Corollary 4.2. Then, a standard application ofthe holomorphic curve machinery `a-la [Hof93, Nie06, AH09] gives a holomorphicplane in the symplectization of one of the negative ends starting with a Bishopfamily associated to a Plastikstufe in the positive end. While bubbles are ruled outby exactness, holomorphic caps at the negative ends are excluded via the explicitproperties of the cobordism (
C, ω C ) (see Theorem 3.1 for a precise statement) andvia the specific Reeb dynamics at the negative ends; this is a subtle point. Now, theexistence of such holomorphic plane contradicts hypertightness of each connectedcomponent of the concave boundary ( C, ω C ), thus concluding the proof. Obstruction to fillability in dimension . The proof of Theorem B ismostly based on holomorphic curve techniques and spinal open book decomposi-tions (SOBDs). The latter notion was introduced in dimension 3 in [LVHMW18],7nd generalized to higher dimensions in [Mor17a]. This geometric decompositiongeneralizes the standard notion of an open book decomposition, and supports con-tact structures in a natural way, in a sense which mimics the notion for open booksdue to Giroux. Moreover, as in [Mor17a], given a contact manifold supported bya SOBD with 2–dimensional pages, an analogue of the construction of holomor-phic open book decomposition of [Wen10b] (see also [Abb11]) gives a foliation ofthe symplectization, whose leaves are holomorphic curves lifting the pages of theSOBD. In the case where the pages are genus zero and we are in the presence of astrong symplectic filling, we get an induced moduli space of punctured holomorphicspheres which “probes” the filling. Wendl [Wen10c] used such techniques in thecase of planar 3-dimensional open books and 4-dimensional symplectic fillings togive very strong classification results.In the low-dimensional situation considered by Wendl, positivity of intersectionsfor holomorphic curves is a very powerful tool and is essential at many points. In our6-dimensional situation, where semi-positivity holds, despite the absence of positivityof intersections, we are still able to recover analogous statements in the context ofplanar SOBDs.In the case of a planar Bourgeois contact structure, one can rule out every kindof nodal (and multiple–level) degeneration, except perhaps for bubbling of closedspheres. This is the main novelty in the current setup. One then chooses a suitable2–disk in the (unmarked) moduli space which intersects the nodal strata in a finitenumber of points, so that the monodromy of the original open book is precisely themonodromy of the forgetful map (defined by forgetting the marked point) along itsboundary. A careful analysis of the possible bubbling allows one to deduce thatthe monodromy around points corresponding to intersections with nodal curves istrivial, thus concluding the proof of Theorem B.Ruling out all unwanted nodal degenerations is quite technical. The main in-gredient is a version of spine removal surgery for Bourgeois SOBDs, adapted fromthe 3-dimensional notion in [LVHMW18] (see Section 3.2 below). This provides asymplectic cobordism with contact concave boundary the original Bourgeois contactstructure and stable convex boundary T × S . This leads to two moduli spaces ofholomorphic curves: one given by the spherical factor and the second given by cylin-ders coming from the symplectization of the first factor, which is an S -parametricversion of the foliation of the symplectization of T from [Wen10c]. One can thendeduce various topological properties of the filling before and after spine removalsurgery and these are enough to rule out unwanted nodal degenerations in the SOBDholomorphic foliation above. Classification of aspherical fillings of S ∗ T n . This proof also relies on theexistence of a an SOBD on S ∗ T n whose pages are cylinders, inducing a moduli spaceon the Liouville completion of any filling. The key point is to show that this moduli8pace admits no nodal degenerations, which is done as in the case of symplecticallyaspherical fillings of Bourgeois manifolds in dimension 5. The forgetful map frommarked moduli space to the unmarked one is then a fibration with cylindrical fibers,and the evaluation map then gives a map of degree 1. With some algebraic topology,this allows to prove that W is homotopy equivalent to D ∗ T n . Finally, using the s –cobordism theorem as in [BGZ16, Section 8], one concludes that W is diffeomorphicto D ∗ T n . Non–existence of symplectic fillings of BO ( D ∗ S n , τ ) . The Bourgeois con-struction naturally yields an S –equivariant contact structure on V × T with respectto the S -factors of the torus (cf. [DG12, Section 5.3]). These contact structures arenaturally supported by a SOBD with 2–dimensional cylindrical pages [Mor17a, Sec-tions 5.1, 5.2]. One then has a capping construction that yields a moduli space ofholomorphic spheres. Using the fact that these spheres intersect the cocores of thecapping handles transversely, under the assumption of symplectic asphericity thereis no bubbling and the moduli space is compact. One can then uses to yield a con-tradiction to the fact that certain homology classes in the handle become trivial inthe filling. In the general case, one can appeal an abstract perturbation as providedby Hofer-Wysocki-Zehnder and argue in analogy to the aspherical case to concludethe proof of Theorem G. Acknowledgments.
The authors wish to thank Chris Wendl, for his hospitalitywhen receiving the first and second authors in Berlin, and for helpful discussionswith all three authors throughout the duration of this research. We also thankSebastian Hensel for suggesting looking at [BF02, BF07], as well as Bahar Acu,Zhengyi Zhou, Ben Filippenko, Kai Cieliebak, Sam Lisi, Patrick Massot, KlausNiederkr¨uger, Fran Presas, Richard Siefring, Andr´as Stipsicz and Otto van Koertfor many helpful discussions and comments on preliminary versions of this paper.We are particularly grateful to Brett Parker and Urs Fuchs for pointing out a flawin an earlier version.The first and second authors greatly acknowledge the hospitality of the MATRIXcenter, where part of this research was conducted. Similarly, the second and thirdauthors are grateful to Monash University. The second author is supported by thegrant NKFIH KKP 126683.
Consider a closed, oriented, connected smooth manifold V n − and an open bookdecomposition ( B, θ ), together with a defining map Φ : V → D having each z ∈ int( D ) as regular value. Here, B ⊂ V is a closed codimension-2 submanifold, θ = Φ / | Φ | : V \ B → S is a fiber bundle, and Φ is such that Φ − (0) = B .9 1-form α on V is said to be adapted to Φ if it induces a contact structure on theregular fibers of Φ and if dα is symplectic on the fibers of θ = Φ / | Φ | . In particular,if ξ is a contact structure on V supported by ( B, θ ), in the sense of [Gir02], then(by definition) there is such a pair ( α, Φ) with α defining ξ . Theorem 2.1. [Bou02b]
Consider an open book decomposition ( B, θ ) of V n − ,represented by a map Φ = (Φ , Φ ) : V → R as above, and let α be a -formadapted to Φ . Then, β := α + Φ dq − Φ dq is a contact form on M := V × T ,where ( q , q ) are coordinates on T . The contact form β on M = V × T will be called Bourgeois form associated to( α, Φ) in the following. Note that, by Gray’s stability, if α and α are homotopicamong 1-forms adapted to Φ, then the associated β and β on M = V × T definethe same contact structure up to isotopy. Remark 2.2.
The contact structure determined by a Bourgeois contact form isstable up to contactomorphism under finite covers of the torus factor. Indeed, upto precomposing by an automorphism of T , any such cover is of the form( q , q ) (cid:55)−→ ( kq , q ) .Pulling back gives a contact form β k = α + k Φ dq − Φ dq and a straightforwardcalculation shows that linear interpolation gives a family of contact forms. Abstract open books and Bourgeois contact structures.
For the proof ofTheorem A, it is also useful to interpret the Bourgeois construction in abstractterms. We briefly recall here the construction in order to fix some notation. Thereader can consult for instance [Gei08, Section 7.3] for further details.Consider a Liouville domain (Σ n − , λ ), together with an exact symplectomor-phism ψ of (Σ , dλ ) (i.e. ψ ∗ λ = λ − dh , for some smooth h : Σ → R + ), fixing pointwisea neighborhood of the boundary B := ∂ Σ. One can then consider the mapping torusΣ ψ of (Σ , ψ ), and the abstract open book V Σ ,ψ := ( B × D (cid:116) Σ ψ ) / ∼ (1)where ∼ identifies ( p, θ ) ∈ ∂ ( B × D ) with [ p, θ ] ∈ ∂ Σ ψ . One can also construct afiberwise Liouville form λ ψ on the mapping torus π : Σ ψ → S of (Σ , ψ ). For large K (cid:29) α K = Kπ ∗ dθ + λ ψ is contact on Σ ψ . Moreover, it can be extendedto a contact form on all of V Σ ,ψ by h ( r ) λ B + h ( r ) dθ on B × D , for a well chosenpair of functions ( h , h ), and λ B = λ | B .We denote the resulting contact form on V Σ ,ψ by α Σ ,ψ . The contact manifold( V Σ ,ψ , ker( α Σ ,ψ )) will also be called an abstract contact open book , and denoted itsimply with OBD (Σ , ψ ). Sometimes, we will also use the contact form α Σ ,ψ .10e point out that there is a well defined map Φ Σ ,ψ : V Σ ,ψ → D given by extend-ing the projection to the circle on Σ ψ by settingΦ Σ ,ψ | B × D ( p, r, θ ) = ρ ( r ) e iθ ∈ D , for some non-decreasing function ρ satisfying ρ ( r ) = r near 0 and ρ ( r ) = 1 near r = 1. Notice also that α Σ ,ψ is naturally adapted to Φ Σ ,ψ (as defined above).We then denote by β Σ ,ψ the Bourgeois form on M Σ ,ψ := V Σ ,ψ × T associated to( α Σ ,ψ , Φ) as in Theorem 2.1, and by ξ Σ ,ψ the contact structure it defines. Finally welet BO (Σ , ψ ) := ( M Σ ,ψ , ξ Σ ,ψ ). Hypertightness for Bourgeois Contact Forms.
In the following sections, wewill need a hypertightness criterion for α Σ ,ψ . We first give a definition: Let ( V × T , ξ ) be a contact manifold, and B any subset of the set of closed Reeb orbits ofa contact form β . We say that β has T -trivial Reeb dynamics concentrated in B ifthe image of every closed Reeb orbit not in B under the projection V × T → T ishomotopically non-trivial. A straightforward computation gives: Observation 2.3. [Gir19, Corollary 6.3]
The Bourgeois contact form β Σ ,ψ for ξ Σ ,ψ has T -trivial Reeb dynamics concentrated in the set B consisting of the submanifolds γ B × { q } ⊂ V × T , for all q ∈ T and all γ B closed Reeb orbit of ( B, α Σ ,ψ | B ) . Ifthe binding ( B, α Σ ,ψ | B ) of the natural open book of V Σ ,ψ admits no contractible Reeborbits inside V Σ ,ψ , then the Bourgeois contact structure ξ Σ ,ψ is hypertight. Notice that, in the 3-dimensional case, Observation 2.3 implies that, if the bind-ing consists of a collection of loops each having infinite order in π ( V ), then theassociated Bourgeois contact structure is hypertight.We point out that we will not make use of Observation 2.3 in the proof ofTheorem A, as we will apply it directly on a another contact form, which stilldefines the Bourgeois contact structure up to isotopy (see Lemma 5.1 below). A supporting spinal open book decomposition.
We now present a geometricway of understanding the Bourgeois construction, via SOBDs (see also Section 8.2for an alternative SOBD).Consider Φ = Φ Σ ,ψ = ρe iθ = ( ρ cos( θ ) , ρ sin( θ )), a defining map for V = V Σ ,ψ = OBD (Σ , ψ ), together with the Giroux form α = α Σ ,ψ and the associated Bourgeoisform β = β Σ ,ψ . Let θ = Φ / | Φ | : V \ B → S be the open book coordinate. FromEquation (1), we obtain a decomposition M = V × T = B × D ∗ T ∪ Σ ψ × T ,where we identify D ∗ T ∼ −→ D × T via ( q , p , q , p ) (cid:55)→ ( p , − p , q , q ).11e denote by M S := B × D ∗ T , which we call the spine , and M P := Σ ψ × T ,the paper . We also have an interface region M I ∼ = B × [ − (cid:15), (cid:15) ] × T , correspondingto the region where M S and M P glue together. Observe that we have fibrations π S : M S → D ∗ T , π P : M P → S ∗ T = T , where the monodromy of π P coincides with ψ along the cotangent S -direction, andis trivial along T . The map π S has contact fibers and Liouville base, whereas π P has contact base and Liouville fibers. This is a spinal open book decomposition orSOBD for M (see [LVHMW18] for the 3-dimensional notion, and [Mor17b] for ahigher-dimensional one). Observe that the fibers of π P , the pages of the SOBD,coincide with the pages of the OBD for V . One may also view the SOBD as afibration (cid:98) π P : M \B → S × T , where we define the binding of the SOBD as B = B ×{ r = 0 }× T ⊂ B × D × T = M S . This fibration has fibers which symplecticallyare copies of the Liouville completion of the page Σ, and has monodromy ψ alongthe first factor, and trivial along the second one.The Bourgeois contact structure ξ = ξ Σ ,ψ is “supported” by the SOBD describedabove, in a sense which we now describe. Via the identification D ∗ T → D × T above, up to isotopy of contact forms, we have β | M S \ M I = λ B + λ std , where λ std = p dq + p dq is the standard Liouville form on D ∗ T . In other words, β | M S \ M I is a split contact form, having a Liouville and a contact summands. Notealso that on M S \ M I the Reeb vector field R β of β agrees with R B over the binding B , and is transverse to the pages away from it. Similarly, up to isotopy β | M P \ M I = λ ψ + α std , where α std = cos( θ ) dq + sin( θ ) dq is the standard contact form on T , and so splitsinto a Liouville summand and a contact summand. In particular, the restriction of dβ to the pages of the SOBD is a positive symplectic form, and the Reeb vectorfield is transverse to the pages, agreeing with that of α std and so tangent to the T factor. In other words, the contact structure, the contact form, as well as the Reebdynamics of β are “compatible” with the underlying geometric decomposition. Thisinterpretation also allows us to reobtain Observation 2.3. It then makes sense tomake the following: Definition 2.4.
A contact form is a
Giroux form for this SOBD if the contactstructure it induces is isotopic to the contact structure induced by the above β .12 Two cobordisms for Bourgeois manifolds
We present here two cobordisms involving Bourgeois contact structures. In Sec-tion 3.1 we describe a strong (actually, pseudo–Liouville) cobordism between Bour-geois contact manifolds with the same page. Its purpose is to relate the Bourgeoismanifold coming from two different monodromies to the one coming from theircomposition, and it will be used in the proof of Theorem A. Section 3.2 describesa cobordism from a Bourgeois contact manifold to a stable Hamiltonian manifold,which is obtained by attaching a symplectic handle over the spine of the former.This will be used in the proof of Theorem B.
Let (Σ n − , λ ) be a Liouville manifold, and let φ be an exact symplectomorphismrelative to the boundary. Notice that the boundary ( B, λ B ) := ( ∂ Σ , λ | ∂ Σ ) cannaturally be seen as the “binding” submanifold of the associated open book. Foreach q ∈ T , we also let B q be B × { q } ⊂ V Σ ,φ × T = M Σ ,φ .The aim of this section is to give a proof of the following result (recall Figure 1): Theorem 3.1.
There is a smooth cobordism C from M Σ ,ψ (cid:70) M Σ ,φ to M Σ ,ψ ◦ φ . Thiscobordism is smoothly a product C × T , where C is a smooth cobordism from V Σ ,ψ (cid:70) V Σ ,φ to V Σ ,ψ ◦ φ . Moreover, there is a symplectic form ω C on C which satisfiesthe following properties:1. ω C admits local Liouville forms λ + and λ − near M Σ ,ψ ◦ φ and M Σ ,ψ (cid:70) M Σ ,φ respectively, satisfying:(a) λ + induces a contact form on M Σ ,ψ ◦ φ which defines, up to isotopy, theBourgeois contact structure ker( β Σ ,ψ ◦ φ ) ,(b) λ − induces a contact form on M Σ ,ψ (cid:70) M Σ ,φ , which has (on each connectedcomponent) T -trivial Reeb dynamics concentrated in { B q } q ∈ T , and ishomotopic to β Σ ,φ , through contact forms whose restriction to each B q is λ B (up to a positive scalar multiple);2. ω C admits a global primitive ν which coincides with λ + at the convex boundaryand such that ν | B q = λ − | B q for each B q ⊂ M Σ ,ψ (cid:70) M Σ ,φ . Item 1 means in particular that, up to attaching cobordisms at its ends, (
C, ω C )is a strong symplectic cobordism with convex boundary BO (Σ , φ ◦ ψ ) and concaveboundary BO (Σ , φ ) (cid:116) BO (Σ , ψ ). Notice however that we do not claim that theglobal 1-form ν defines a contact structure at the concave boundary; in other words,the cobordism we give is not claimed to be Liouville, but just pseudo-Liouville (as13efined in the introduction). Lastly, we point out that Theorem 3.1 can be thoughtof as a “stabilized” version of [Avd12, Proposition 8.3] and [Klu18, Theorem 1]; infact, smoothly (but not symplectically), the cobordism C is just the product of thecobordism from [Avd12, Klu18] with T .We now proceed to give a proof of Theorem 3.1, following very closely, with someadaptations, the one given in [Klu18]. Some notation.
We start by completing the Liouville domain (Σ , ω = dλ ) andby defining an auxiliary function τ on the completion. Let Y be the Liouville vectorfield on (Σ , ω = dλ ) defined by ι Y ω = λ . By the definition of Liouville manifold, Y is positively transverse to B = ∂ Σ. Consider a collar neighborhood ( − δ, × B of B inside Σ, with coordinates ( t, q ) ∈ ( − δ, × B , where Y = ∂ t . We then extend(Σ , λ ) to a complete Liouville manifold ( (cid:98) Σ , (cid:98) λ ) given by setting (cid:98) Σ = Σ ∪ [0 , + ∞ ) × B and (cid:98) λ = (cid:26) λ on Σ e t λ B on [0 , + ∞ ) × B. Let (cid:98) ω = d (cid:98) λ . We denote by (cid:98) Y the natural extension of this Liouville vector field onΣ to (cid:98) Σ. Consider then a smooth function τ : (cid:98) Σ → R > such that:1. τ = − δ on (cid:98) Σ \ ( − δ, + ∞ ) × B ;2. ∂τ∂t > τ = τ ( t ) on ( − δ, + ∞ ) × B ;3. τ ( t, q ) = t on [0 , + ∞ ) × B .Notice in particular that dτ ( Y ) > − δ, + ∞ ) × B . A toroidal pair of pants cobordism in dimension . We need to utilise astrong 4-dimensional symplectic cobordism with concave end ( S ∗ T , ξ std ) (cid:116) ( S ∗ T , ξ std )and convex end ( S ∗ T , ξ std ). To this end, consider the unit disk cotangent bundle D ∗ T of T , together with its standard symplectic structure ω std = dλ std , wherein coordinates λ std = p dq + p dq is the standard Liouville form. To be precise,we need to work with scalar multiples Kω std and Kλ std , where K is a positive realconstant that will be determined later on in the proof. We also denote by X theLiouville vector field p ∂ p + p ∂ p .Consider the submanifold D (cid:15) ∗ T of D ∗ T made of those covectors of norm lessthan a certain (cid:15) < /
10, and denote by j ± : D ∗ (cid:15) T → D ∗ T the symplectomorphisms( p , q , p , q ) j ± (cid:55)−→ ( p ± / , q , p , q ).For ease of notation, we consider the inclusion j = j − (cid:116) j + : ( D ∗ (cid:15) T , Kω std ) (cid:116) ( D ∗ (cid:15) T , Kω std ) → ( D ∗ T , Kω std ) . X C t �� = X T � R Figure 2: Picture of the submanifold (with corners) C top ⊂ (cid:98) Σ × T ∗ T . Here, weidentify T ∗ T = R × T and (cid:98) Σ \ Σ = B × [0 , + ∞ ), with coordinate t ∈ [0 , + ∞ )which coincides with the function τ .Then, the desired cobordism is ( Q, ω Q ) := ( D ∗ T \ j ( D ∗ (cid:15) T ) , Kω std ). Topologicallythis is just a product of the torus with a pair of pants P ⊂ R in the ( p , p )-plane.The Liouville field on a neighbourhood of the convex boundary is just given by X , whereas the one near the concave boundary is given by j ∗ X , which is the vectorfield ( p ∓ / ∂ p + p ∂ p on the image of j ± respectively. Lastly, we consider anauxiliary smooth function f : T ∗ T → R , which depends only on p , p , and satisfies:1. f = p + p on T ∗ T \ D ∗ T ,2. f = ( p ∓ / + p on the image of j ± respectively,3. (cid:15) < f < Q .Notice that Items 1 and 2 imply in particular that df ( X ) > ∂ ( D ∗ T ), and df ( j ∗ X ) > j (and not only on a neighbourhoodof the concave boundary of the cobordism Q ). Description of the cobordism.
Consider the Σ-bundle E over the pair of pants P , where the monodromies along the two negative boundary components are givenby φ and ψ respectively, and by their composition along the remaining one. Sinceall monodromies are exact symplectomorphisms, there is a fiberwise Liouville form λ , and since all monodromies are the identity near the boundary of Σ, this formcan be assumed to agree with e t λ B near the horizontal boundary B × P .Without loss of generality assume that the form is invariant under (a lift) of thegradient vector field of f over a neighbourhood of ∂P . In particular, for any large K (cid:29) dλ + Kπ ∗ dλ std on the product E = E × T is an exactsymplectic form with primitive ν = λ + Kπ ∗ λ std .We then consider the manifolds with corners C bot := E and C top = (cid:110) ( x, q, p ) ∈ (cid:98) Σ × T ∗ T | τ ≥ , τ + f ≥ (cid:15) , τ + | p | ≤ (cid:111) .15 X C �� = X T Φ -1 Ψ T T Figure 3: Picture of C bot .Notice that C bot and C top can be identified along the subsets { τ = 0 } which arediffeomorphic to B × P × T . The symplectic form Ω on C bot also glues to therestriction of d (cid:98) λ + Kdλ std to C top ⊂ (cid:98) Σ × T ∗ T . We thus obtain a symplectic cobordism( C, ω C ), and we now proceed to check that it satisfies the required properties. Wewill only do this for the negative ends, as the other case is entirely analogous. Proof of Theorem 3.1
Notice first that by construction C = C × T is topo-logically a product. Furthermore, the primitives λ , (cid:98) λ described above glue togetherto give a global 1-form denoted (cid:98) λ which is a pullback of a form on C . Further-more, the intersection of each boundary component of C with the subsets C bot and C top determine T -stabilised open books, which we think of as supporting SOBD’sfor the Bourgeois contact structures. Moreover, these SOBD’s coincide, at leasttopologically, with those used to define the Bourgeois contact structure.Consider now the primitive j ∗ λ std of the standard symplectic form near the in-terior boundary of P , which in coordinates is just p dq + p dq ∓ / dq . TheLiouville vector field associated to the local primitive λ − = λ + Kj ∗ λ std is definedon a neighborhood of the negative boundary components C − of C , and is transverseto C − , since it projects to j ∗ X on P × T . Moreover, by construction, the restrictionof the associated Liouville form coincides with the Bourgeois contact form on thepaper region in C bot , and so is adapted to the Bourgeois SOBD in that region (in thesense of Definition 2.4). In the top part of the cobordism the Liouville vector fieldis just the linear combination ∂ t + r∂ r , where r is the radial parameter | p ∓ / | atthe respective boundary components. This is then also transverse to the boundary.By parametrising the interior hemispherical caps using the canonical coordinatescoming from the cotangent bundle of the torus, we obtain the following coordinatedescription on C top : λ − = e √ −| r | λ B + Kp dq + Kp dq ∓ / dq .Thus on each boundary component we have that α = λ − ± / dq is a Bourgeoiscontact form as in Theorem 2.1, for a suitable choice of open book map Φ. It is easyto check that α t = α + tdq gives a contact deformation, since α t ∧ ( dα t ) n = α ∧ ( dα ) n for all t . Moreover, an explicit computation using the description of λ − given abovethen shows that all periodic orbits either project non-trivially to the torus factor,16r are parallel to the binding (where r = 0). This is clear in the bottom region andfollows from the explicit form of λ − in the top region. This concludes the proof ofItem 1b.Finally, notice that there is a global Liouville primitive ν for ω C defined on all C , given by gluing λ + Kπ ∗ λ std on C bot = E and (cid:98) λ + Kλ std on C top ⊂ (cid:98) Σ × T ∗ T .Observing that ν coincides with λ − on the subsets of the form B q described inItem 2, we conclude the proof. (cid:3) We consider now an open book
OBD (Σ , ψ ) with surface page. The aim in thissection is to construct a symplectic cobordism having BO (Σ , ψ ) as strong concaveend, and a fiberwise-symplectic fibration over S ∗ T as stable convex end. Here,stable convex end means convex end of a strong symplectic cobordism betweenstable Hamiltonian structures (SHS), as defined in [CV15, Definition 6.2].This is a particular case of a more general construction for SOBDs, called spineremoval surgery , which has been defined in dimension 3 in [LVHMW18], and gen-eralizes a part of Eliashberg’s capping construction (namely, [Eli04, Theorem 1.1])that uses 3-dimensional open books. Similar higher-dimensional constructions havealso appeared in [AM18, DGZ14]. The proofs below will follow along similar lines asthose in Section 3.1. Theorem 3.2.
Let
OBD (Σ , ψ ) be an open book with surface page. Consider theclosed surface ( (cid:98) Σ , (cid:98) ω ) obtained from Σ by capping by disk s along B , and let (cid:98) ψ be thenatural extension of ψ and to (cid:98) Σ by the identity.Then there exists a symplectic cobordism ( C, ω C ) having BO (Σ , ψ ) as strong concaveend, and ( C + , H = ( λ + , Ω + )) as stable convex end, where:1. C + = (cid:98) Σ (cid:98) ψ × T π → S × T is the total space of a fiberwise-symplectic (cid:98) Σ -fibrationthat is trivial over the second factor and fiberwise symplectic form ω + ;2. λ + = π ∗ α std and Ω + = π ∗ dα std + ω + , where α std = cos θ dq − sin θ dq is thestandard contact form on T , with coordinates ( θ, q , q ) ∈ T = S × T .Moreover, ( C, ω C ) can be seen as attaching the symplectic handle H := (( (cid:98) Σ \ Σ) × D ∗ T , (cid:98) ω ⊕ ω std ) to the convex boundary BO (Σ , ψ ) × { } of the trivial cobordism BO (Σ , ψ ) × [0 , along its spine M S × { } = B × D ∗ T × { } . We will refer to the submanifolds of the form { } × D ∗ T as co-cores of thehandle H . It is easy to attach a cobordism that realises a deformation of the SHSon the positive end to that given by the product structure so that we obtain thefollowing: 17 X C �� = X T Ψ -1 Figure 4: The manifold with corners C bot . Corollary 3.3.
Let
OBD (Σ , ψ ) be an open book with planar page (i.e. a surfaceof genus zero). Then, there exists a symplectic cobordism ( C, ω C ) having BO (Σ , ψ ) as strong concave end, and ( T × S , H ) as stable convex end, where the stableHamiltonian structure H = ( λ, Ω) is given by λ = α std , Ω = dα std + ω S , with α std the standard contact form on T = S ∗ T and ω S an area form on S .Proof (Theorem 3.2). Without loss of generality, we suppose that ψ is the identityon a neighborhood of the boundary of Σ, and that ψ is an exact symplectomorphismfor ω = dλ . Consider then a collar neighborhood N := B × ( − δ, δ ) of B = ∂ Σ in (cid:98)
Σ, in such a way that
N ∩
Σ = B × ( − δ, ψ | N ∩ Σ = Id N ∩ Σ , λ | N ∩ Σ = e s λ B and ω | N = d ( e s λ B ), where s ∈ ( − δ, δ ) and λ B = λ | B . Denote also by (cid:98) ω a (non-exact)extension of dλ to (cid:98) Σ that equals d ( e s λ B ) on N .Fix also an auxiliary smooth function f : (cid:98) Σ → R satisfying the following: • f = − \ N , • f = 2 on (cid:98) Σ \ (Σ ∪ N ), • f depends only on the coordinate s ∈ ( − δ, δ ) on N = B × ( − δ, δ ), with respectto which is non-decreasing, • f = 0 on a neighborhood of B × { } ⊂ N .Let A ⊂ R denote the annulus with inner radius r = 1 and outer radius r = 3,and consider the Σ-bundle E over A , with monodromy ψ along the core circle. Thisbundle then inherits an exact and fiberwise symplectic form denoted ω E = dλ E . The2-form ω E + Kdλ std , for K (cid:29) E × T , consideredas a symplectic fiber bundle over A × T ⊂ T ∗ T .The fiberwise function f given above determines a well-defined function on thebundle that we again denote by f . We now set C bot = E and C top := { ( x, p, q ) ∈ (cid:98) Σ × T ∗ T | f ( x ) ≥ , | p | + f ( x ) ≥ , | p | ≤ } .The obvious symplectic forms piece together to give a symplectic manifold ( C, ω C )whose positive end C + corresponds to the union of { π ( y ) = 3 } ⊂ C bot and { p =3 } ⊂ C top , and whose negative end C − are those points with | p | + f ( x ) = 1 on C top , or π ( y ) = 1 on C bot . We now show that ( C, ω C ) satisfies the desired properties.18 he positive end. The positive end C + = (cid:98) Σ (cid:98) ψ × T is by construction just thesymplectic fiber bundle given by capping the open book, and then taking the productwith the torus. Notice that a neighborhood of C + in C has a naturally definedprojection onto a neighborhood of the component A + = {| p | = 3 } in A . TheLiouville vector field Y = p ∂ p + p ∂ p on A then naturally lifts, with respect tothis projection, to a vector field (cid:98) Y in the kernel of the radial collapse map A → A + ,implicitly used to define E . Contracting with (cid:98) Y then gives a SHS on the positiveend as described in Item 2 (after multiplying by suitable constants). The negative end.
On this part the set up is identical to that described in theproof of Theorem 3.1 above so that the obvious local Liouville vector field associatedto the primitive λ E + Kλ std gives the Bourgeois contact manifold BO (Σ , ψ ). Handle description of cobordism.
The last thing left to prove is that suchcobordism can be described as attaching a symplectic handle to BO (Σ , ψ ). For this,we subdivide C top in two subsets: C pr := { ( x, p, q ) ∈ C top ⊂ (cid:98) Σ × T ∗ T | | p | + f ( x ) ≤ } , H := { ( x, p, q ) ∈ C top ⊂ (cid:98) Σ × T ∗ T | | p | + f ( x ) ≥ , | p | ≤ } .Now, the union of C pr and C bot gives a trivial symplectic cobordism from BO (Σ , ψ )to itself. Indeed, there is a globally defined Liouville vector field transverse to thetwo boundary components: on C pr this is given by the sum of the standard Liouvillefield on T ∗ T and of the Liouville field ∂ s on N ⊂ (cid:98)
Σ and on C bot by the sum of thefiberwise-Liouville field associated to the Liouville form λ E on E and of the standardLiouville field on A ⊂ T ∗ T . Finally, H has a natural symplectic structure inducedby the ambient space (cid:98) Σ × T ∗ T and is symplectically deformation equivalent to aproduct ( (cid:98) Σ \ Σ) × D ∗ T . Let Σ denote a connected orientable surface with boundary. We will denote themapping class group as
M CG (Σ), which is defined to be the set of isotopy classesof orientation preserving homeomorphisms of Σ; note that we do not require thesehomeomorphisms to fix the boundary components. This group is naturally iso-morphic to the group of isotopy classes of homeomorphisms of the correspondingpunctured surface. One may also consider
M CG (Σ , ∂ Σ) of mapping classes fixingthe boundary, and there is a natural forgetful map
M CG (Σ , ∂ Σ) → M CG (Σ) whosekernel is generated by boundary parallel Dehn twists.19e will refer to a surface as sporadic if it is either a disk , an annulus or a pair ofpants. These cases correspond to the mapping class group being virtually abelian.The aim of this section is to prove the following:
Lemma 4.1 (Factorization Lemma) . Let φ ∈ M CG (Σ , ∂ Σ) for a non-sporadicsurface Σ . Then φ can be factored as φ = φ ◦ φ , where, for each i = 1 , , φ i issuch that each connected component of the binding of V i := OBD (Σ , φ i ) has infiniteorder in π ( V i ) . A direct consequence of Lemma 4.1 and Observation 2.3 is the following:
Corollary 4.2.
Let φ be a mapping class of a compact, orientable, non-sporadicsurface Σ with boundary. Then φ can be factored as φ = φ ◦ φ , with φ , φ suchthat the Bourgeois contact manifolds BO (Σ , φ ) and BO (Σ , φ ) are hypertight. In order to prove Lemma 4.1, we start by recalling some results from geometricgroup theory and 3-dimensional hyperbolic geometry, respectively, in Sections 4.1and 4.2. The proof is then given in Section 4.3.
We recall that, by the Nielsen-Thurston classification theorem (see for instance[FM12, Theorem 13.2]), every element in
M CG (Σ) or
M CG (Σ , ∂ Σ) is either pseudo-Anosov, reducible or of finite order.Recall also that a quasi-homomorphism on a group G is a map H : G → R with D ( H ) := sup g,h ∈ G | H ( gh ) − H ( g ) − H ( h ) | < ∞ .The quantity D ( H ) is called the defect of H . Observe that any bounded functionis trivially a quasi-homomorphism. A quasi-homomorphism is called homogeneous if H ( g k ) = kH ( g ) for all k ∈ Z and g ∈ G . It is a standard fact that any quasi-homomorphism can be made homogenized by an averaging process analagous to thedefintion of the Poincar´e translation number.The following lemma is direct consequence of (the proofs of) [BF02, Theorem 1]and [BF07, Proposition 5]. Lemma 4.3.
Let Σ be a connected orientable surface with boundary which is notsporadic. Then there exists a pseudo-Anosov map f on Σ fixing the boundary anda homogeneous quasi-homomorphism H on G = M CG (Σ , ∂ Σ) such that H is un-bounded on the cyclic subgroup of G generated by f , and H vanishes on the cyclicsubgroups generated by either finite order or reducible elements in G . In other words, if H is non-zero on (cid:104) ψ (cid:105) , then ψ must be pseudo-Anosov. Inparticular, one has the following straightforward consequence of Lemma 4.3:20 orollary 4.4. Let φ be an arbitrary mapping class on a non-sporadic compact,orientable, surface Σ with boundary, and let f be as in Lemma 4.3. Then, forsufficiently large k , the mapping class f k φ ∈ M CG (Σ , ∂ Σ) is pseudo-Anosov. We recall the following theorem on hyperbolic mapping tori due to Thurston [Thu98]:
Theorem 4.5. [Thu98]
Let Σ be a compact, orientable surface with boundary andnegative Euler characteristic. If φ is a pseudo-Anosov map on Σ , then the interiorof the associated mapping torus has a complete hyperbolic structure of finite volume. We will also need another result due to Thurston on Dehn fillings of hyperbolicmanifolds; an introductory account, as well as a detailed proof, can be found forinstance in [Mar16, Chapter 15]. For the reader’s ease, we give here a statement ofsuch a theorem which is adapted to the specific setting in which we will apply it.Let N be an orientable 3-manifold with boundary ∂N a finite union T (cid:116) · · · (cid:116) T c of 2-dimensional tori. For each i = 1 , . . . , c , let also m i , l i be generators of π ( T i ).For any c -tuple s = ( s , . . . , s c ) of Dehn filling parameters , i.e. of pairs s i = ( p i , q i )of coprime integers, one can consider the compact (boundary-less) 3-manifold N fill obtained by Dehn filling the boundary tori with parameters s = ( s , . . . , s c ); moreexplicitly, for each i = 1 , . . . , c , a solid torus P i := D × S is glued to N via the(unique up to isotopy) gluing map ∂P i → T i sending a meridian of ∂P i to a curvein the class p i m i + q i l i ∈ π ( T i ). Theorem 4.6. [Thu97]
In the setting described above, suppose moreover that theinterior of N admits a complete hyperbolic metric of finite volume. Then, there is acompact set K ⊂ R such that, if every Dehn filling parameter s i is in R \ K , theclosed -manifold N fill obtained by Dehn filling N with parameters s = ( s , . . . , s c ) admits a finite–volume complete hyperbolic structure g . Moreover, the cores of thefilling solid tori are closed geodesics of ( N fill , g hyp ) . Notice that since each s i being a pair of coprime integers, the theorem impliesthat the we can ensure that a Dehn filling is hyperbolic by excluding finitely manyvalues for each slope s i . Remark 4.7.
Since the fundamental group of a closed hyperbolic manifold istorsion-free and its closed geodesics are all non-contractible, the cores of the Dehnfilling tori will have infinite order in π ( N fill ).21 .3 Proof of the Factorization Lemma Let f be a pseudo Anosov map on Σ as in Lemma 4.3. According to Corollary 4.4, f k φ is pseudo Anosov on Σ for sufficiently large k . We then write φ = F ◦ G , where F = f − k , G = f k φ, where both are pseudo Anosov for k (cid:29)
0. By Theorem 4.5, the interiors of themapping tori associated to (Σ , F ) and (Σ , G ) carry complete hyperbolic structures.Let γ , . . . , γ n be the components of the boundary ∂ Σ. For each i = 1 , . . . , n ,we then denote by c i a curve in Σ which is parallel to γ i and contained in ˚Σ; wecan assume, up to isotopy, that they are pairwise disjoint. We also denote by τ , . . . , τ n the corresponding right-handed Dehn twists, and τ := τ . . . τ n . Observethat τ r = τ r . . . τ rn for every r ∈ Z , since the c i ’s are disjoint.Let φ := F τ r and φ := τ − r G . It is easy to check that the 3-manifolds OBD (Σ , φ ) and OBD (Σ , φ ) correspond to Dehn fillings of, respectively, the map-ping tori Σ F and Σ G with respect to Dehn filling parameters s ( r ) = ( s ( r ) , . . . , s n ( r ))and t ( r ) = ( t ( r ) , . . . , t n ( r )) such that | s i ( r ) | , | t i ( r ) | → + ∞ for each i = 1 , . . . , n as r → + ∞ Thus, for sufficiently large r , the hyperbolic Dehn filling Theorem4.6 implies that OBD (Σ , φ ) and OBD (Σ , φ ) carry hyperbolic structures and thatthe binding components (which coincide with the cores of the Dehn filling tori) aregeodesics. In particular the latter have infinite order in the fundamental group (seeRemark 4.7). In other words, we have found the desired decomposition φ = φ ◦ φ as posited in Lemma 4.1. The aim of this section is to prove Theorem A on tightness of the Bourgeois contactstructures in dimension 5. For this, we use the following lemma, which is an analogueof the well-known fact that the convex end of a Liouville cobordism with hypertightconcave end must be tight [Hof93, AH09]:
Lemma 5.1.
Suppose the connected components of the bindings of
OBD (Σ , φ ) and OBD (Σ , ψ ) have infinite order in the corresponding fundamental groups. Then, BO (Σ , φ ◦ ψ ) is tight.Proof. Let (
C, ω C ) be a symplectic cobordism as in Theorem 3.1. According toObservation 2.3, Item 1b of Theorem 3.1 and our hypothesis on OBD (Σ , φ ) and OBD (Σ , ψ ), the Reeb flow of λ − | ∂C − has no contractible periodic orbits. We nowshow that this implies that BO (Σ , φ ◦ ψ ) is tight.We assume by contradiction that its convex boundary BO (Σ , φ ◦ ψ ) is overtwisted.According to [BEM15], this implies the existence of an embedded Plastikstufe PS ,22s defined in [Nie06]. Up to attaching a topologically trivial Liouville cobordism to( C, ω C ) along its positive end, we may then assume that the induced contact format the positive end is (a positive multiple of) a contact form α P S which is “adapted”to PS , i.e. it has the normal form described in [Nie06, Proposition 4] near its core.Take a sequence of smooth functions f ( k ) on the negative boundary ∂ − C of C which C ∞ -converges to the constant function f ( ∞ ) ≡
1, such that the contact form λ ( k ) − := f ( k ) λ − is non-degenerate, and λ ( ∞ ) − = λ − . We modify the symplectic formnear ∂ − C to obtain a symplectic form ω ( k ) C → ω C so that the induced contact formon the boundary is λ ( k ) − for a fixed Liouville vector field. Since ω C = dν in C , wededuce that, for each k , there is a 1-form ν ( k ) on C such that ω ( k ) C = dν ( k ) on C , insuch a way that ν ( k ) → ν in the C ∞ -topology. The non-degeneracy of the forms λ ( k ) − will mean that we can apply SFT compactness theorem directly for these perturbedforms. Taking limits we will then deduce the general case.Attaching a cobordism at the negative ends using the local Liouville vector fieldsassociated to λ ( ∞ ) − , we obtain the negative Liouville completion (cid:98) C ( ∞ ) of λ ( ∞ ) − . Ob-serve that the negative Liouville completion (cid:98) C ( k ) of λ ( k ) − is smoothly the same as (cid:98) C ( ∞ ) , although with a different symplectic form at the negative ends, so that wework in a fixed smooth manifold. We denote by (cid:98) ω ( k ) C the negative completion of thesymplectic form ω ( k ) C in (cid:98) C ( k ) , so that it coincides with d ( e t λ ( k ) − ) at the negative ends.The negative ends of (cid:98) C ( k ) are negative symplectisations of nondegenerate contactforms. We can therefore apply the following standard argument. Take an (cid:98) ω ( k ) C -compatible almost complex structure J ( k ) , converging to a (cid:98) ω ( ∞ ) C -compatible J ( ∞ ) ,all of them extending the local model of [Nie06], and cylindrical in the cylindricalends. We have a Bishop family of Fredholm regular J ( k ) -holomorphic disks in (cid:98) C ( k ) with Lagrangian boundary, stemming from the core of the Plastikstufe. Analogouslyto [Nie06, Proposition 10], one can check that the exactness of the symplectic formnear the positive end, and hence near the Plastikstufe, provides uniform boundson the Hofer energy, defined as in [Wen16, Page 115]. By SFT compactness (usingthe nondegeneracy condition at the negative ends), we thus obtain a non-trivial J ( k ) -holomorphic building configuration, with potentially multiple levels. Since thesymplectic form on (cid:98) C ( k ) is exact, there are no bubbles in the building. Also, there isno boundary bubbling, as shown in [Nie06]. We conclude, as in [Hof93, AH09], thatit must contain non-trivial components in the negative ends.For sufficiently large k , one can rule out holomorphic caps as follows: this is notautomatic from standard arguments, since ( (cid:98) C ( k ) , ω ( k ) C ) is only pseudo-Liouville. Aftertaking a subsequence of k ’s, assume the existence of a sequence of J ( k ) -holomorphiccaps c ( k ) for k → + ∞ , considered as maps to the compactification C ( k ) , and havingboundary in the negative boundary of C ( k ) . The Hofer energy bounds on the Bishopfamiliy provide universal bounds on the action of the boundary orbits γ k of each23 ( k ) . So, after passing to a further subsequence, by the Arzel`a–Ascoli theorem, wehave that γ k → γ ∞ converges to a periodic Reeb orbit of the Reeb flow of λ − in the C ∞ -topology.Now as each γ k is nullhomotopic in the cobordism C ( k ) ∼ = C , the same is true of γ ∞ . Projecting to T , using the globally defined projection, we see that the imageof γ ∞ in T is also nullhomotopic. We conclude by Theorem 3.1 that the Reeb orbit γ ∞ must be a binding component B q .Let π C : (cid:98) C ( k ) = (cid:98) C ( ∞ ) → C denote the collapsing map of the negative Liouvilecompletion onto the compact cobordism. Given their explicit forms on the ends,the 2-forms π ∗ C ω ( k ) C naturally extend to forms ω ( k ) C on C ( k ) . We let ν ( k ) denote thecorresponding primitives. As the almost complex structures are cylindrical on theends, the forms ω ( k ) C integrate non-negatively on all J ( k ) -holomorphic curves andpositively on those which are not completely contained in the cylindrical end. Inparticular, this is the case for any holomorphic cap given the assumption that allReeb orbits have infinite order in the fundamental group of the end.By Item 2 of Theorem 3.1 the primitive ν is positive along binding components,and hence ν ( ∞ ) is positive on γ ∞ . The same is true by continuity for ν ( k ) restricted to γ k , for k (cid:29)
0. Then integrating the exact form Ω ( k ) = dν ( k ) along the holomorphiccap c ( k ) , and using Stokes’s theorem, we obtain the following contradiction: (cid:90) c ( k ) Ω ( k ) = (cid:90) − γ k ν ( k ) < k (cid:29) J ( k ) -holomorphic building contains a J ( k ) -holomorphicplane P ( k ) in the bottom level. By passing to a subsequence and using Arzel`a–Ascoli,we obtain a contractible Reeb orbit in the negative symplectization of λ − = λ ( ∞ ) − .But there are no such orbits at the negative ends, and this finishes the proof.We can now proceed to the proof of Theorem A: Proof (Theorem A).
We start by proving the result in the “generic” case of non-sporadic page Σ . We then deal with sporadic pages on a case by case basis. Case 1: non-sporadic Σ . By Lemma 4.1 we may factorise the monodromy φ = φ ◦ φ , where the components of the bindings in OBD (Σ , φ ) and OBD (Σ , φ ) haveinfinite order. Then, according to Lemma 5.1, we conclude that BO (Σ , φ ) is tight. Case 2: Σ is a disk. In this case, the monodromy φ is necessarily isotopic to theidentity. In other words, the resulting contact 3-manifold is ( S , ξ std ) and the openbook structure is the one induced by the subcritical Stein-filling D . Accordingto [LMN18, Theorem A.(b)], the associated Bourgeois contact structure is Steinfillable, and hence tight. 24 ase 3: Σ is an annulus. The mapping class group of the annulus is generatedby a single positive Dehn twist around the core circle. If the monodromy is anon-negative power of such generator, then the resulting contact 3-manifold is Steinfillable; then, according to [MNW13, Example 1.1], the associated Bourgeois contactstructure is weakly fillable, and hence tight. If the power is negative, according to[LMN18, Theorem B], the Bourgeois contact structure associated to
OBD (Σ , φ ) iscontactomorphic to that associated to OBD (Σ , φ − ), so we obtain tightness for thiscase. Case 4: Σ is a pair of pants. For simplicity, enumerate from 1 to 3 the connectedcomponents of ∂ Σ. For i = 1 , ,
3, let τ i be a positive Dehn twist along the i -thconnected component of ∂ Σ; Notice that the monodromy φ is necessarily of the form τ a ◦ τ a ◦ τ a . We then define τ := τ ◦ τ ◦ τ and, for any N ∈ N > , we can decompose φ as φ = F ◦ G , with F := φ ◦ τ N = (cid:81) i =1 τ N + a i i and G := τ − N = (cid:81) i =1 τ − Ni . Wethen use the following result, whose proof is postponed: Lemma 5.2. If N > is big enough, each binding component of OBD (Σ , F ) is ofinfinite order in π ( OBD (Σ , F )) . The same is true for OBD (Σ , G ) . Combining Lemmas 5.1 and 5.2, we conclude that BO (Σ , φ ) is tight, as desired.Lastly, we prove the lemma used above in the case of a pair of pants: Proof (Lemma 5.2).
We deal only with the case of
OBD (Σ , F ); the proof for themanifold OBD (Σ , G ) is completely analogous.We first point out that, as explained in detail for instance in [Ozb07, Section3], the manifold OBD (Σ , F ) can be seen as obtained by Dehn surgery on the totalspace of S × S → S along three S -fibers, with coefficients r i := − N + a i , for each i = 1 , ,
3. In other words,
OBD (Σ , F ) is the Seifert manifold { , ( o , N + a , − , ( N + a , − , ( N + a , − } .Moreover, the orbit space O of the Seifert fibration of OBD (Σ , F ) is a 2-dimensionalorbifold, with underlying topological surface S , and the binding B of OBD (Σ , F )consists of a union of fibers of the Seifert fibration.We recall that there is a notion of orbifold Euler characteristic χ orb for orbifoldsthat behaves multiplicatively under finite covers of orbifolds. In our special case ofthe base orbifold B of the Seifert fibered space OBD (Σ , F ), we have χ orb ( O ) = χ ( S ) − (cid:88) i =1 (cid:18) N + a i (cid:19) = − − (cid:88) i =1 N + a i .25rom now on, let N > χ orb ( O ) <
0. In particular,
OBD (Σ , F ) isfinitely covered by a circle bundle X over a hyperbolic surface S , in such a way thatfibers of X → S are mapped to fibers of OBD (Σ , F ) → O (see [Sco83] for instance).Now, S being hyperbolic, the fibers of X are of infinite order in π ( X ). As X covers OBD (Σ , F ) in a compatible way with their Seifert bundle structures, it follows thefibers of OBD (Σ , F ), hence its binding too, are of infinite order in its fundamentalgroup, as desired.We finally note that all these arguments remain valid when we pull-back underany finite cover of V × T of the first factor. Since finite covers over the secondfactor do not change the contact structure up to contactomorphism (cf. Remark2.2), such covers also preserve tightness. Now any finite cover is itself covered by acomposition of covers of the respective factors. Consequently, the contact structureremains tight under any finite cover on the first factor. Since the fundamental groupof any closed 3-manifold is residually finite (cf. [Hem87]) so is π ( V × T ) and hencetightness on finite covers is equivalent to tightness on the universal cover of V × T and universal tightness follows. This concludes the proof of Theorem A. (cid:3) We now assume that BO (Σ , ψ ) admits a strong symplectic filling W . The aim ofthe section is to obtain some control on the topology of the capped filling W cap givenby attaching the spine removal handle described in Theorem 3.2.More precisely, in Section 6.1 we study a moduli space of spheres, naturally aris-ing from the foliation by spheres near the boundary. This is then used in Section 6.2to show that the T -factor of the Bourgeois construction survives homologically inthe capped filling. In Section 6.3, we describe a moduli space of cylinders, aris-ing from a parametric version of the foliation for the symplectization of T from[Wen10c]. In Section 6.4 we study intersection properties of the capped filling,proving in particular that 2–cycles intersect the co-cores in a controlled way. Let (
W, ω W ) be a strong symplectic filling of BO (Σ , ψ ). After attaching thecobordism ( C, ω C ) of Corollary 3.3 along BO (Σ , ψ ) using the Liouville flow, weobtain a symplectic manifold ( W cap , ω cap ) = ( W, ω W ) ∪ ( C, ω C ) with stable boundary( T × S , H ). We will construct a moduli space of holomorphic spheres in ( W cap , ω cap ),and show that it is a 4-dimensional manifold with boundary T for suitable choicesof data. A good reference for general backround on moduli spaces of holomorphicspheres, as we shall use them here, is [MS12]. What follows is inspired by theconstructions in [MNW13]. 26 ocal symplectic models. From Corollary 3.3, we have a symplectic model fora collar neighbourhood of the positive boundary of C , given by C = (( − (cid:15), × T × S , ω = d ( e t λ ) + Ω) , (2)where λ = α std and Ω = dα std + ω S , and t ∈ ( − (cid:15), t -coordinate can in factbe identified with the Liouville coordinate for the radial vector field in D ∗ T , nearits boundary.Therefore ker λ = ξ std ⊕ T S , the Reeb vector field is R λ = R std , the Reeb vectorfield of α std , and the stabilizing vector field is ∂ t . We also have a model symplectichandle A = ( D ∗ T × D , ω std ⊕ dσ ) , (3)where dσ is a symplectic form on D . We attach k symplectic handles A , . . . , A k to each component of the spine, all modelled on A .The two models A and C are compatible with each other, provided we view( − (cid:15), × T as the standard symplectic neighborhood of T = ∂D ∗ T in D ∗ T ,Σ ∪ (cid:83) kj =1 D j as a 2–sphere S , where D j is the core of the handle A j , which we call a capping disk , and we take ω S so that it coincides with dσ along each of the cappingdisks (after suitably smoothening the corners). Almost complex structure in the local models.
Along the collar neighbour-hood C , we take an H -compatible almost complex structure J which maps ∂ t to R λ ,and such that J | ker λ = J std | ξ std ⊕ j S , where J std is the standard complex structure in D ∗ T , and j S is any compatiblecomplex structure in S .We extend J to the cobordism ( C, ω ) is such a way that it coincides, alongthe symplectic handle A , with J std ⊕ j σ , where j σ is the restriction of j S to eachcapping disk and so dσ -compatible. Near the concave end BO (Σ , ψ ), we take it tobe cylindrical and generic. Local moduli space of spheres.
With these choices, the co-cores D ∗ T × { } of the handles are J -holomorphic and complex codimension 1. The cores K q = { ( q, } × D , for q ∈ T , are also holomorphic disks. By construction, for each( t, q, θ ) ∈ ( − (cid:15), × T × S , we have a holomorphic sphere u = u ( t,q,θ ) : S → ( − (cid:15), × T × S × S , z (cid:55)→ ( t, q, θ, z ) Lemma 6.1.
The spheres u ( t,q,θ ) are Fredholm regular and have Fredholm index . C, ω ). The shaded region is the model collarneighbourhood C . The model symplectic handle A consists of everything that liesabove the horizontal dotted line. Proof.
Fix u = u ( t,q,θ ) . The complex normal bundle N u of u splits into a sum of J -invariant integrable and trivial line bundles, which along the collar neighbourhood,can be described by N u = T ( t,q,θ ) D ∗ T = (cid:104) ∂ q , ∂ q (cid:105) ⊕ (cid:104) ∂ q , ∂ p (cid:105) . This implies that thenormal component of the linearized Cauchy-Riemann operator also splits as D Nu = ∂ ⊕ ∂ , where ∂ is the standard Cauchy-Riemann operator acting on W , -sectionsof the corresponding line bundle. By automatic transversality, each summand issurjective, and we deduce surjectivity of D Nu : W , ( N u ) → L (Ω , ( N u )) so that u is indeed Fredholm regular. Since u is immersed, its index is the Fredholm index of D Nu , and from the Riemann-Roch formula we obtainind( u ) = ind( D Nu ) = 2 χ ( u ) + 2 c ( N u ) = 2 χ ( S ) = 4 . Moduli of spheres in ( W cap , ω cap ) . We extend J to an ω cap -compatible almostcomplex structure in W cap , so that it is generic along the original filling W . We thenobtain a (connected) moduli space M of holomorphic spheres in W cap , containingthe curves u ( t,q,θ ) . By Lemma 6.1, our choice of J and [Wen16, Thm. 7.2], it is a4-dimensional smooth manifold with non-empty boundary. We consider its Gromovcompactification M , which is obtained from M by adding strata of nodal spheres.Every element in M intersects the holomorphic co-cores of the handles precisely onceby positivity of intersections, and therefore, M consists of simply covered spheres. M is a 4-dimensional stratified space (see below), with non-empty boundary, whichwe now identify via the following: Lemma 6.2 (Local Uniqueness) . For sufficiently small (cid:15) > , any curve u in themoduli space M , that intersects the collar ( − (cid:15), × T × S , is a reparametrizationof one of the u ( t,q,θ ) . Proof.
The proof is the same as in [MNW13, AM18]. Assume first that u is acomponent of an element v in M that intersects { } × T × S , namely, at t = 0.28e define the open set U = u − (( − (cid:15), × T × S ) . Then u | U = ( u , u ) , where u : U → ( − (cid:15), × T and u : U → S are both holomorphic. But the boundaryof ( − (cid:15), × T is strictly pseudoconvex, and so u is constant. This implies that U = S , and u is a reparametrization of a sphere u (0 ,q,θ ) . All the adjacent spherecomponents of v to this particular one intersect u (0 ,q,θ ) , and therefore they intersect { } × T × S , which completely contains the image of u (0 ,q,θ ) . So, it inductivelyfollows that all sphere components of v are reparametrizations of the same u (0 ,q,θ ) (since different such spheres never intersect), which implies that v ∈ M is non-nodaland satisfies the claim.In general, assume that (cid:15) k → u k which is acomponent of a curve in M , intersects ( − (cid:15) k , × T × S and is not a reparametriza-tion of a sphere u ( t,q,θ ) . Since the moduli space M is compact, after passing to asubsequence, we obtain a limiting nodal curve u ∞ in M which intersects the bound-ary at t = 0. As above, u ∞ is a reparametrization of u (0 ,q,θ ) . But Fredholm regularityimplies that every sphere in M near it is a reparametrization of a sphere u ( t (cid:48) ,q (cid:48) ,θ (cid:48) ) , which is absurd.From now on, we assume (cid:15) > C .Consider a co-core D ∗ T , and recall that we arranged it to be J –holomorphic.Notice that the J –holomorphic spheres u ( t,q,θ ) intersect D ∗ T in exactly one point.Moreover, positivity of intersections and the fact that intersections can’t escape atthe boundary (due to Lemma 6.2) tell us that every J –holomorphic sphere in thesame homology class as the u ( t,q,θ ) ’s must intersect the considered co-core (positivelyand) exactly once. Thus, the map I : M → D ∗ T given by sending each curve u ∈ M to the intersection point of u with the co-core D ∗ T is well defined.We also consider the moduli space M ∗ obtained by adding a marked point tothe domain of each curve in M , which is a 6-dimensional smooth manifold withboundary, and its Gromov compactification M ∗ . We have a forgetful map π : M ∗ → M which forgets the marked point. Therefore the fiber over u ∈ M is the domain of u itself. We also have an evaluation map ev : M ∗ → W cap which evaluates at the marked point. As a corollary of Lemma 6.2, we obtain: Corollary 6.3.
The intersection map I and the evaluation map ev are diffeomor-phisms near the boundary. In particular, ∂ M = T , and ∂ M ∗ = T × S . odal stratification. The moduli space M is a stratified space ∅ = M ⊂ M ⊂ M ⊂ M = M . Here, M i consists of nodal configurations of spheres which have at least i nodes, andthe interior part int( M i ), consists of nodal configurations with precisely i nodes. Forour choice of J , the top open stratum M = int( M ) is a smooth 4-dimensional man-ifold consisting of simply covered spheres. The elements of the i -th strata int( M i )contain main sphere components, which intersect holomorphic co-cores preciselyonce and so are again simply covered, while the rest are possibly multiply coveredbubbles which intersect no co-cores. By the Uniqueness Lemma 6.2, no element in M i touches the boundary of W cap , for i ≥ ∅ = M ∗ ⊂ M ∗ ⊂ M ∗ ⊂ M ∗ = M ∗ , and the forgetful map respects the stratification.Since every 6-dimensional symplectic manifold is semi-positive, it follows stan-dard dimension computations in Gromov-Witten theory that the dimension of theimage of M i ∗ under the evaluation map is at most − i , in the sense of [MS12, Sec.6.5]. More precisely, the image under the evaluation map of M i ∗ is covered by theimages of underlying moduli spaces of simple stable maps, each of which is actu-ally a smooth manifold for generic choice of almost complex structure. We shalloccasionally say that the corresponding unmarked piece M i has dimension at most4 − i . Remark 6.4.
In the case where the filling W is symplectically aspherical, a priorithere can be only bubbles that go through the symplectic handles. However, afterTheorem 6.12 below is established, this actually implies that there are no bubblesat all and M = M is a smooth 4-manifold. T survives homologically in W cap The natural constructions of symplectic fillings of Bourgeois contact manifolds allhave a global product structure with T . This seems to be a manifestation of somedeeper mechanism: in the planar case, will show that any filling “remembers” this T -factor. More precisely, we shall show that the inclusion of any torus fiber isinjective on homology. In this paper, by stratifed space , we will mean a filtration ∅ = M m +1 ⊂ M m ⊂ · · · ⊂ M = M of a compact topological space M , where the interior int( M i ) of each M i is called the i -th strata,and the closure of each strata satisfies M i = (cid:83) j ≥ i M j . W be a strong symplectic filling of the contact manifold BO (Σ , ψ ) associatedto an open book OBD (Σ , ψ ) with surface page, and let W cap be the symplectic man-ifold with stable boundary S × T obtained by attaching the handle from Section 3.2on W . Consider the following commutative diagram T S × T ∂ M ∗ M ∗ W cap M D ∗ T i i (cid:48) fev − ∂ ∼ j evπ I ∗ I Here, j is the natural inclusion ∂ M ∗ ⊂ M ∗ , π is the map forgetting the markedpoint and i is a section of the ( S × S )–bundle S × T → T induced by the naturalprojection T = S ∗ T → T . Moreover, the map f , defined by the commutativityof the diagram, is actually homotopic to the zero section of D ∗ T → T . Finally, i (cid:48) denotes the composition of all the horizontal arrows and I ∗ the composition of thevertical ones. Proposition 6.5.
The map i (cid:48) is injective in integer homology. The proof uses the fact that the moduli space gives a relative pseudo-cycle on thefilling which induces pseudo–cycles on generic submanifolds – in our case surfaces.We refer to [MS12, Section 6.5] for further background.
Proof (Proposition 6.5).
The result is clearly true for H . We also claim that thestatement for H follows from the statement for H . Indeed, suppose i (cid:48)∗ ( a ) , i (cid:48)∗ ( b ) (cid:54) = 0in H ( W cap ), for a, b generators of H ( T ). Then i (cid:48)∗ ( H ( T )) is torsion-free, andso there are α, β ∈ H ( W cap ) such that α ( a ) and β ( b ) are both non–zero integers.Then, α (cid:94) β is non–zero on the image under i (cid:48)∗ of the generator a ⊗ b of H ( T )given by the K¨unneth splitting H ( T ) = H ( S ) ⊗ H ( S ). Hence it suffices to provethe statement for H .Let γ be a curve in T and consider its image γ (cid:48) := i (cid:48) ( γ ) in W cap . We can supposewithout loss of generality that it is simple. Notice also that this lies (by definitionof i (cid:48) ) in the boundary ∂W cap , so that ev − γ (cid:48) = j ◦ ev − ∂ ◦ i ( γ ) . (4)Suppose by contradiction that γ (cid:48) bounds a 2–chain δ (cid:48) in W cap . By general positionwe can assume that this chain is an embedded surface with boundary, which we willconsider as a relative cycle. 31e claim that the map ( ev, I ∗ ) : M ∗ → W cap × D ∗ T is a 6–dimensional pseudo–cycle. Indeed, the closure of the image of ( ev, I ∗ ) | M ∗ is the image of the compactifiedmoduli space M ∗ , on which the map ( ev, I ∗ ) is continuous. Moreover, the image ofthe nodal set M ∗ for both maps agrees with the image of the space M s ∗ of underlyingsimple curves. As M s ∗ consists of components of dimension ≤
4, this gives the claim.What’s more, one can perturb δ (cid:48) so that δ (cid:48) × D ∗ T becomes weakly transverse tothe pseudo–cycle ( ev, I ∗ ). According to (a relative version of) [MS12, Proposition6.5.17], the restriction of I ∗ to ev − ( δ (cid:48) ) gives a relative pseudo–cycle in D ∗ T . Now,such a relative pseudo–cycle is a weak representative of a relative homology classmodulo torsion (see [MS12, Discussion after Lemma 6.5.6]), and it moreover givesthe original homology class under the boundary map from relative homology to theboundary. We conclude that γ (cid:48) must be trivial in H ( D ∗ T ). This contradicts thefact that the map f in the diagram above induces a homotopy equivalence. Remark 6.6.
Note that the proof of Proposition 6.5 is much more straight forwardin the absence of bubbles, e.g. when W is assumed symplectically aspherical. In thiscase, according to Remark 6.4, the moduli space itself is a smooth manifold and onecan simply apply standard general position arguments without needing to considerpseudo–cycles. In this section, we construct a different moduli space of holomorphic curves in thecompletion of the capped symplectic filling ( W cap , ω cap ), using an S -parametrizedversion of the construction in [Wen10c]. Wendl family.
From [Wen10c], we obtain a suitable stable Hamiltonian struc-ture H = ( λ , Ω ) on T , together with a symplectic filling ( D ∗ T , ω ) for suitable ω , coinciding with ω std away from a small collar of the boundary. The completion( T ∗ T , ω ) is endowed with an ω -compatible almost complex structure J , givingrise to a suitable smooth 2-dimensional moduli space M . The latter consists ofsomewhere injective holomorphic cylinders in T ∗ T with two positive ends, eachasymptotic to a distinct Morse-Bott family of Reeb orbits parametrized by a cir-cle. There are precisely two such Morse-Bott families, and we denote their disjointunion inside T by B . We call B the binding of T , a disjoint union of two 2-tori. The moduli space M gives a foliation of T ∗ T , as follows from standard4-dimensional techniques, and M is itself diffeomorphic to R × S . We can write M = M triv ∪ M ntriv , where M triv is the family of cylinders whose cylindricalends in T ∗ T \ D ∗ T are trivial cylinders over orbits in B (they are precisely thosecylinders which intersect the zero section T ⊂ T ∗ T ).32 + ∞ � , 0 θ T S , 0 S , 1 S , 0 P , 1 P D * T * T TT T TTT V V V Figure 6: The Wendl holomorphic foliation of D ∗ T . We will use this S -parametrically. The curves in the non-shaded region are a cylindrical end of M ∼ = R × S over S × S . Remark 6.7.
The above construction of [Wen10c] can be understood in termsof holomorphic spinal open book decompositions: T has a natural planar SOBDstructure (cf. [LVHMW18, Mor17a, Wen10c]). We write T = T S ∪ T P , where the spine is T S = B × [ − ,
1] = T S, (cid:70) T S, . Here, the [ − , S -factor of T = S × T , centered around the Morse-Bottorbits, and we split T S into its two connected components. We have trivial fibrations π S,i : T S,i → {∗}× [ − , × S , where we call V i := π S,i ( T S,i ) the i - vertebrae ( i = 0 , paper T P . See Figure 6.We denote by ( C = [0 , × T , ω ) a small collar neighbourhood of the boundaryof ( D ∗ T , ω ). We view it as a topologically trivial exact symplectic cobordism withconcave boundary the contact manifold ( T , α std ), and positive stable boundary thestable Hamiltonian manifold ( T , H ). Parametric Wendl family.
In our situation, by attaching a symplectic cobor-dism of the form ( C × S , ω + ω S ) to the boundary of W cap , we may modify themodel symplectic collar C = (( − (cid:15), × T × S , ω = d ( e t λ ) + Ω) of Section 6.1by replacing λ with λ , and Ω with Ω + ω S , without changing notation, so that33ow W cap has stable boundary ( M cap := ∂W cap , H = ( λ, Ω)). We then consider thecompletion ( (cid:99) W cap , (cid:98) ω cap ) = ( W cap , ω cap ) ∪ (cid:98) C , where (cid:98) C = (( − (cid:15), + ∞ ) × T × S , ω = d ( e t λ ) + Ω)is the symplectization of the stable Hamiltonian structure H at M cap . For a ∈ ( − (cid:15), + ∞ ), we denote W acap := (cid:99) W cap \ (( a, + ∞ ) × ∂W cap ), its truncation at level a ,and M acap = ∂W acap . We will also write M acap = M acap,P ∪ M acap,S , where M acap,P = { a } × T P × S is the paper region, and M acap,S = { a } × T S × S is the spine region(recall Remark 6.7).We modify the H -compatible almost complex structure J so that, with respectto the splitting T (cid:0) ( − (cid:15), + ∞ ) × T × S (cid:1) = (cid:104) ∂ t , R H (cid:105) ⊕ ξ H ⊕ T S , we have J = i ⊕ J ⊕ j S , Here, j S is, as before, a compatible complex structure in S , which we take to be thestandard one. Similarly as in Section 6.1, we specify J along the model symplectichandle A = ( D ∗ T × D , ω std ⊕ dσ ) as J = J ⊕ j σ , where j σ is j S restricted to D when viewed as a capping disk, and is compatible with dσ = ω S | D . We denotethe k handles by A , . . . , A k , which are all modelled on A , and A = (cid:83) nj =1 A j . Let D , . . . , D k be the corresponding cores, which are the capping disks for the pages.We extend J to all of (cid:99) W cap , generically away from (cid:98) C ∪ A .Observe that, for each z ∈ S , the hypersurface H z := ( − (cid:15), + ∞ ) × T × { z } is holomorphic in the cylindrical end (cid:98) C . Moreover, for z ∈ D j ⊂ S , H z gluesto D ∗ T × { z } in the symplectic handle A j . We still denote the resulting gluedhypersurface by H z , which is a copy of T ∗ T inside (cid:99) W cap for every j and z ∈ D j . Set (cid:98) A j = (cid:83) z ∈ D j H z and (cid:98) A = (cid:83) kj =1 (cid:98) A j . Observe that (cid:98) C ∪ A = (cid:83) z ∈ S H z .We now obtain a moduli space of cylinders of the form u z : ( R t × S ϕ , i ) → ( (cid:98) C ∪ A , J ) , u z ( t, ϕ ) = ( u ( t, ϕ ) , z ) , where u ∈ M and z ∈ S , so that im( u z ) ⊂ H z . Here, for z ∈ Σ ∼ = S \ (cid:16)(cid:83) j D j (cid:17) away from the capping disks, we assume u is taken to vary in the portion of M consisting of curves lying in the cylindrical ends.The cylindrical moduli space M W + := { u z : u ∈ M , z ∈ S } extends to a modulispace in (cid:99) W cap , which we denote by M W .We consider the moduli spaces M W ∗ and M ∗ obtained from elements M W and M , respectively, by adding a marked point to the domains, and the resulting eval-uation map ev : M W ∗ → (cid:99) W cap , and forgetful map π : M W ∗ → M W . We denote by34 W the Gromov compactification of M W . We have M W = M Wntriv ∪ M
Wtriv , where M Wtriv consists of those curves in M W whose cylindrical ends are trivial cylindersover Reeb orbits in S × B . The maps ev and π extend to M W , and we denote theirextensions with the same notation. Siefring intersection theory.
In what follows, we will make use of Siefring in-tersection theory for holomorphic curves and hypersurfaces, to appear in [Sie] (seealso [MS19]). The setup for this is as follows. Consider an asymptotically cylindricalholomorphic curve u in the completion (cid:99) W of a symplectic manifold W with stableHamiltonian boundary, and H a holomorphic hypersurface in (cid:99) W which is asymp-totically cylindrical (in a well-defined sense [MS19]) to trivial cylinders over strongstable hypersurfaces in ∂W , and its intersection H ∩ W with the compact piece W has stable boundary satisfying ∂ ( H ∩ W ) ⊂ ∂W . Here, a strong stable hypersurfaceis defined as a Reeb-flow invariant codimension 2 submanifold of ∂W for which therestriction of the ambient SHS is again a SHS. We assume moreover that the ambientSHS is Morse-Bott. Then there is a well-defined holomorphic intersection pairing u ∗ H , which is homotopy invariant under homotopies through asympotically cylin-drical curves or hypersurfaces, and non-negative whenever im( u ) is not containedin H . Intuitively, the pairing considers contributions from the standard intersectionpairing together with contributions coming from “infinity”.In our setup, observe that for z = 0 ∈ D j , the origin in a capping disk, thecompletion of the corresponding co-core (cid:98) C j = H z ∼ = T ∗ T is a holomorphic hy-persurface in (cid:99) W cap which is R -invariant in the cylindrical end of (cid:99) W cap , and is thecompletion of a codimension 2 symplectic filling of the strong stable hypersurface M z := T × { z } ⊂ T × S = ∂W cap . It is therefore asymptotically cylindricalover its own cylindrical end. This also holds for any hypersurface H z with z ∈ D j .While the hypersurfaces H z for z away from the capping disks are also asymptoti-cally cylindrical, the corresponding strong stable hypersurface M z = T × { z } is nota priori necessarily symplectically filled inside the unknown filling W , and thereforeintersections with a holomorphic curve might escape towards the interior of W . Properties of the compactified moduli space.
We are ready for the following:
Theorem 6.8 (Properties of M W ) . There exists a (cid:29) , such that the followingstatements hold:(i) (compactness) We have a stratification ∅ = M W, ⊂ M W, ⊂ M W, ⊂ M W, = M W ere, the top strata int ( M W, ) is a -dimensional smooth manifold consistingof somewhere injective cylinders. The i -th strata int ( M W,i ) has dimension atmost − i , and consists of nodal curves having precisely one component whichis a somewhere injective cylinder, and precisely i (possibly multiply covered)closed sphere bubbles, for i = 1 , . In particular, there are no multiple-floorbuilding degenerations from curves in M W . The bubbles lie completely in theinterior of the filling W , and thus have empty intersection with the symplectichandles of W cap .(ii) (boundary behaviour) The evaluation map is (weakly) transverse to the bound-ary ∂W acap .(iii) (relative pseudocycle) For a ≥ a , consider M W ∗ ,a = ev − ( W acap ) ⊂ M W ∗ , andlet ev a denote the corresponding restriction of the evaluation map. Then ev a : M W ∗ ,a → W acap is a relative pseudocycle representing a fundamental class. Inparticular, the evaluation map on the compactification ev a : M W ∗ ,a → W acap issurjective.(iv) (truncated fibers) Let M Wa be the set of umarked curves in M W with non-emptyintersection with W acap . Then the fibers of the forgetful map π : M W ∗ ,a → M Wa are compact truncations of the corresponding punctured curves.(v) (diffeomorphism on vertical boundary) For a ≥ a , we consider the verticalboundary ∂ v M W ∗ ,a = ev − ( M acap,P ) ⊂ M W ∗ corresponding to the paper region M acap,P = { a } × T P × S . Let ∂ v ev a denote the corresponding restriction of theevaluation map. Then ∂ v ev a : ∂ v M W ∗ ,a → M acap,P is a diffeomorphism.(vi) (cylindrical end) For a ≥ a , M W has a cylindrical end over ∂ M Wa ∼ = S × S × S . Remark 6.9.
Note that we are not claiming that the moduli space M W providesa foliation of (cid:99) W cap , in contrast to the 3-dimensional situation. Remark 6.10.
It follows from the proof (Step 5 below) that (cid:98) A , the completion ofthe handles and collar, is foliated by non–nodal curves. Remark 6.11 (Symplectically aspherical case) . In case that W is symplecticallyaspherical (e.g. exact), there are no bubbles. In particular, Item (i) says that M W = M W . Moreover, the forgetful map is a fibration with typical fiber T ∗ S . Proof.
We split the proof in several steps.36 tep 1: Regularity for the cylindrical moduli space M W + . We show surjec-tivity of the normal linearized Cauchy-Riemann operator for u z = ( u, z ), which isequivalent to regularity [Wen10a, Cor.3.13] (note that these curves are immersed).This operator splits as D Nu z = D Nu ⊕ ∂, where D Nu is the normal operator of u inside H z , and ∂ is the standard Cauchy-Riemann operator acting on the holomorphic nor-mal bundle to H z , the trivial line bundle over R × S with fiber T z S . From [Wen10c],we know that M is regular, and so the first summand is surjective. After addinga small weight to the Sobolev spaces in the domain and target of ∂ so that sectionsdecay exponentially (cf. [Wen10a, p. 14-15]), and the Conley-Zehnder index at eachpuncture becomes 1, the Fredholm index of the resulting Fredholm operator is 2.Since its kernel consists of holomorphic sections which decay at infinity, it consistsof the 2-dimensional space of constant sections. It follows that its index is the sameas the dimension of its kernel, and so the second summand is also surjective, andregularity follows. It also follows from the Riemann-Roch formula that the Fredholmindex of u z is 4. Step 2: Intersection with holomorphic hypersurfaces.
We use Siefring in-tersection theory to restrict the behaviour of curves in M W . Indeed, let z = 0 ∈ D j be the origin in a capping disk. We have that u z ∗ H z = 0 , for every z ∈ S . This is obvious for z (cid:54) = z , and for z = z we can use homotopyinvariance of the pairing. This implies that u ∗ H z = 0 for every u ∈ M W . Step 3: Uniqueness in the upper levels.
Let v be a (possibly nodal) holomor-phic curve in the Gromov compactification M W , consisting only of upper levels, i.e.its components lie completely in the symplectization of H . Then the intersectionpairing v ∗ H z is defined, for every z ∈ S (we consider only homotopies throughcurves in upper levels), and vanishes identically. Since the holomorphic hypersur-faces H z foliate the upper levels, we conclude that each component of v has imagein H z for some z , which is independent of the component. Alternatively, withoutappealing to intersection theory, one can observe that the ω S -energy of the (closedbut potentially nodal) holomorphic map obtained as the projection of v to S van-ishes. Either way, we obtain that v = ( v , z ) for v either a trivial cylinder over anorbit in B , or an element in M . From [Wen10c], we know that there are no nodaldegenerations from curves in M , and so v is in both cases non-nodal and consistsof a single level, and we conclude the same for v .This implies that every curve in M W with components completely lying in theupper levels is of the form u z for some u ∈ M and z ∈ S , and in particular thereare no nodal degenerations in the upper levels.37 tep 4: There are no multiple level degenerations. This follows from unique-ness in the upper levels, since curves in M have no negative ends, and the stabilitycondition for holomorphic buildings implies that there are no floors consisting solelyof trivial cylinders. Step 5: Uniqueness in (cid:98) A . As for uniqueness in the upper levels, Siefring in-tersection theory implies that if u ∈ M W has non-empty intersection with (cid:98) A , thenit lies completely in a holomorphic hypersurface H z for z ∈ D j in some cappingdisk. Uniqueness in H z ∼ = T ∗ T (from [Wen10c]) implies u is of the form u z , and inparticular is a non-nodal curve. Step 6: Nodal stratification.
The asymptotics of elements in the moduli M W are of the form { z } × { γ } × { pt } ⊂ S × T × S , for γ ⊂ B , and in particularcorrespond to 1-cycles in { pt } × T . We have shown in Section 6.2 that no 2-chain in W cap can bound 1-cycles coming from these tori, and so this rules out thepossibility of nodal degenerations consisting of two holomorphic planes intersectingat a point. We have already ruled out bubbles completely contained in (cid:98) C , andthere are no bubbles having non-trivial intersection with A by uniqueness in (cid:98) A .Therefore bubbles can only appear in the interior of W , and we obtain Item (i) (thedimension counts follow from semi-positivity). Step 7: M W is a smooth -manifold. This follows by Fredholm regularity ofeach u z , uniqueness in (cid:98) C ∪ A , genericity of J away from this region, and [Wen16,Thm. 7.2]. Step 8: Boundary behaviour.
Consider M W the family of curves in M W having non-empty intersection with W cap = W cap ⊂ (cid:99) W cap . Since W cap is compact,this is a compact family. Therefore, for every (cid:15) > a (cid:29) a , + ∞ ) × ∂W cap under every curve in M W consists of smallneighbourhoods of their punctures whose diameter (in a fixed, suitable metric in thedomain) is bounded from below; and their image under the corresponding curve in M W lies in [ a , + ∞ ) × N (cid:15) ( B ) × S , where N (cid:15) ( B ) ⊂ T is an (cid:15) -neighbourhood of the T -binding B , so that all are C -close to trivial cylinders determined by their Reebasymptotics (cf. [DRGI16, Lemma 5.14]). In particular, if a curve in M W intersects[ a , + ∞ ) × ∂W cap \ ( N (cid:15) ( B ) × S ), then it is a non-trivial leaf of the foliation M W + . Byour choice of a , all curves in M W intersect { a } × ∂W cap transversely in a collectionof two circles.We next observe that the evaluation map is weakly transverse to M acap = ∂W acap .Indeed, by our choice of a , the cylindrical ends of any curve in M W meets the38oundary of W acap transversely, along the spine region M acap,S . The same is true for thenodal curves in M W \M W . Moreover, along the paper region M acap,P , transversalityfollows by observing that we may translate curves in M W + (the only elements of M W near M acap,P ) in the Liouville direction, which is transverse to M acap . Thisproves Item (ii). Step 9: Diffeomorphism on vertical boundary.
Since the only curves thatmeet { a } × T P × S are curves in M W + , we obtain that the evaluation map is adiffeomorphism of ∂ v M W ∗ ,a onto M acap,P . This proves Item (v). Step 10: cylindrical end.
Item (vi) follows immediately from Item (v) (recallFigure 6).
Step 11: The truncated evaluation map is a relative pseudocycle.
Recall(cf. Step 8) that the evaluation map is weakly transverse to M acap = ∂W acap , and itfollows in particular that ev a indeed gives a relative pseudocycle by restricting to M W ∗ ,a = ev − ( W acap ) ⊂ M W ∗ . Furthermore, since the map is a diffeomorphism on apart of the boundary it follows that the pseudo-cycle represents a fundamental classin relative homology. Step 12: Truncated fibers.
By the choice of a and uniqueness in the end ofthe completion, we have that the fibers of the forgetful map M W ∗ ,a → M Wa aretruncations of the cylindrical components of the corresponding curve in M W ∗ . Thisproves Item (iv), and finishes the proof of Theorem 6.8. Let W be any symplectic filling of the Bourgeois contact manifold BO (Σ , ψ ), whereΣ is a genus zero surface with k boundary components. Consider the symplecticmanifold W cap with stable convex boundary M cap := S × T obtained from W asdescribed in Section 3.2, i.e. by attaching k model symplectic handles A , . . . , A k (see Equation (3)) to the convex boundary of W . For each i = 1 , . . . , k , denote C i the co-core D ∗ T × { } of the i -th symplectic handle A i . Theorem 6.12.
For all [ σ ] ∈ H ( W cap ) and ≤ i, j ≤ k , we have [ σ ] · [ C i ] = [ σ ] · [ C j ] .In particular, if a -cycle intersects a co-core, it intersects all of them. For the proof of Theorem 6.12 we will need the following:39 emma 6.13.
Let δ be an embedded surface in W cap . Then, there is a decompositionof the associated homology class [ δ ] = [ δ W ] + [ γ ] such that:1. δ W is disjoint from the co-cores;2. γ is contained in the boundary of W cap .Proof (Theorem 6.12). Let [ σ ] ∈ H ( W cap ). Apply Lemma 6.13 to decompose thecycle as [ δ ] = [ δ W ] + [ γ ]. The first summand is disjoint from the co-cores so onlythe second can intersect them non-trivially. However, observe that a cycle in theboundary has non-trivial intersection number with a co-core if and only if it hasa non-trivial component coming from the S -factor with respect to the K¨unnethdecomposition H ( M cap ) = H ( S × T ) = H ( S ) ⊕ H ( T ), in which case it mustintersect all co-cores with the same algebraic count of intersections. Proof (Lemma 6.13).
Up to perturbing δ , we can assume that it doesn’t intersectthe nodal stratum ev ( M W, ∗ ,a ) and that it intersects transversely ev ( M W, ∗ ,a \ M W, ∗ ,a ).In particular, this intersection is 0-dimensional and so the surface meets finitelymany curves with one node. Let ∆ ⊂ δ denote a small neighbourhood of theseintersection points. By perturbing the evaluation map near the boundary we canassume that these do not lie on the co-core. One can do so at the expense of losing J -holomorphicity in that region, but this will not affect the topological argumentsthat follow. Alternatively, one can prove using Siefring intersection theory that theco-cores are actually foliated by curves without nodes, so that nodal curves muststay away from the co-cores; see Remark 6.10.Now delete the interior of the discs δ reg = δ \ ∆ from the surface and note thatthis surface is disjoint from the nodal locus. Hence by perturbing the evaluationmap on the smooth part of the moduli space, we can pull it back to obtain a compactsurface with boundary ¯ δ reg that maps onto δ reg .We then consider a retraction given by “pushing up” along the cylindrical fibers ofthe forgetful map π : M W ∗ ,a → M Wa . More precisely, define the horizontal boundaryof M W ∗ ,a to be the region ∂ M W ∗ ,a \ π − ( ∂ M Wa ). Then, there is a deformation retract H + t : M W ∗ ,a → M W ∗ ,a , for t ∈ [0 , H +0 = Id ,onto a connected component ∂ + h M W ∗ ,a (which we call positive) of ∂ h M W ∗ ,a with respectto π . Such a deformation retraction is just given by pushing vertically up cylindersin the domains of the holomorphic curves, in such a way that H + t is the identity on ∂ + h M W ∗ ,a for all t ∈ [0 ,
1] and the image of H +1 lies in ∂ + h M W ∗ ,a . Note that this mapdoes not extend continuously over the nodal substratum, but it maps points near agiven nodal curve C to points near the positive boundary ∂ + C ⊂ ∂ + h M W ∗ ,a of C .Let ¯ δ be the image of ¯ δ reg under H +1 , and δ = ev (¯ δ ). One can then form anew surface homologous to δ as follows. 40irst flow the preimage of the boundary of ∂ ∆ to the positive boundary ∂ + h M W ∗ ,a via H + t to form a tube T = { H + t ( p ) | p ∈ ∂ ¯ δ reg , t ∈ [0 , } .Notice that each connected component c of ∂δ lies in a small neighbourhood N of ev ( ∂ + C ), for a nodal curve C . In particular, inside such neighbourhood, itis either contractible or homotopic to a non–trivial multiple of ev ( ∂ + C ). We claimthat the latter cannot occur.Indeed, by the construction in Section 6.3, ev ( ∂ + C ) is homotopic to a curve inthe T -factor of the spine in M cap = S × T . Now, according to Proposition 6.5,this T -factor survives in homology in W cap . On the other hand, c is contractible in W cap , as it bounds a connected component of ∆ T = ev ( T ) ∪ ∆, which is a union ofdisks.Let then D be the union of the disks capping the boundary components of ∂δ ,each contained in a small neighbourhood of a nodal curve. As the latter do notintersect the co-cores, the same is true for D .Denote then (cid:98) ∆ T = ∆ T ∪ D and (cid:98) δ = δ ∪ D . We thus obtain a decomposition[ δ ] = [ (cid:98) ∆ T ] + [ (cid:98) δ ] = [ δ W ] + [ γ ],where the first surface does not intersect the co-cores and the second one is com-pletely contained in (a neighborhood of) the boundary, as desired. Remark 6.14.
Analogously to what pointed out in Remark 6.6 for the proof ofProposition 6.5, the proof of Theorem 6.12 becomes much more straightforward inthe absence of bubbles, e.g. if W is symplectically aspherical. Indeed, accordingto Remark 6.11, in this case the moduli space has no nodal degenerations, henceit is a smooth manifold. The forgetful map is also a fibration, and the markedmoduli space retracts onto a connected component of its horizontal boundary. Inparticular, one can simply apply standard general position arguments to pull backany class in H ( W acap ) to a class in H ( M W ∗ ,a ) and homotope it to the boundary viathe retraction. The image of this homotopy under the evaluation map shows thatthe class H ( W acap ) actually comes from the boundary, i.e. from H ( S × T ), thusproving Theorem 6.12. The aim of this section is to prove Theorem B from the Introduction. The sectionis organized as follows. In Section 7.1 we describe a codimension–4 holomorphicfoliation on the symplectization of (a stable Hamiltonian structure obtained as a41 S M P M I � =0 � =1 Figure 7: The codimension-4 holomorphic foliation.deformation of) BO (Σ , φ ). This is then used in Section 7.2 to construct a modulispace of punctured spheres in the filling, which are, in the cylindrical end, lifts ofthe pages of the SOBD of BO (Σ , φ ). Finally, in Section 7.3 we use this moduli spaceto give a proof of Theorem B above. In this section, we make use of the supporting SOBD discussed in Section 2. Thedetails for this section are mostly deferred to Appendix A.We can adapt the construction of “model A” in [Mor17a], which is inspired bythe construction in [Wen10a] for open book decompositions. Here, we provide ageometric description, and defer further details of the construction to the appendix.We point out that this construction can be carried out in all dimensions.
Qualitative description.
One deforms the Bourgeois contact structure alongthe paper region M P to a stable Hamiltonian structure H A which is tangent to thepages, such that arbitrary small perturbations are contact and isotopic to the originalBourgeois contact structure. Then one constructs a H A -compatible almost complexstructure J in the symplectization of the stable Hamiltonian manifold ( M, H A ), andan R -invariant foliation F of R × M by codimension-4 J -holomorphic submanifolds,whose leaves are classified into two types:1. (Trivial cylinders) Trivial cylinders over each component of the binding. Sym-plectically, these are symplectizations of the contact manifold ( B, λ B ). They42an be parametrized by {F q } q ∈ T , where F q = R × B × { q } × { } ⊂ R × M S ,having as many connected components as the binding B = ∂ Σ.2. (Holomorphic pages) Lifts of the pages Σ with cylindrical ends attached. Sym-plectically these are Liouville completions of (Σ , λ ), parametrized by a family {F ( a,θ,q ) } ( a,θ,q ) ∈ R × S × T . The leaf F ( a,θ,q ) projects to q ∈ T under the naturalprojection R × M → T , and consists of a lift of the θ -page at R -level a along R × M P , together with a cylindrical end along R × ( M S ∪ M I ) which projects tothe flow line of the Morse-Bott function H ( q, p ) = | p | in D ∗ T correspondingto constant angle θ .The holomorphic foliation is then as follows (see Figure 7): F = {F ( a,θ,q ) } ( a,θ,q ) ∈ R × S × T ∪ {F q } q ∈ T .In any dimension, the holomorphic page F ( a,θ,q ) has k positive cylindrical ends,where k is the number of components of B , each of the form B × { q } × { } ⊂ M S ,for B a component of B . Namely, it is asymptotically cylindrical (in the senseof [MS19]) to the symplectization of B . Translating the a -parameter correspondsprecisely to a -translation of F ( a,θ,q ) in the R -direction.Moreover, the “double completion” construction of [LVHMW18], as used in[Mor17a], provides the following: Lemma 7.1.
There exists a topologically trivial symplectic cobordism ( Y, ω Y ) , havingthe original Bourgeois contact structure ( M, ξ BO ) as concave boundary component,and the stable Hamiltonian manifold ( M, H A ) as positive stable boundary. A proof of the above lemma is given in Appendix A. Therefore, one may at-tach (
Y, ω Y ) to any strong filling of ( M, ξ BO ) and obtain a diffeomorphic symplecticmanifold with stable boundary ( M, H A ).In the 5-dimensional case, the holomorphic pages are holomorphic curves, whichin the genus zero case are Fredholm regular, and have Fredholm index 4, as followsfrom adapting the Fredholm analysis of [Mor17a] to the current SOBD. Observethat in our setup there is possibly non-trivial monodromy, but this does not affectthe proof of regularity, which is localized around a page. By our choice of function H , the asymptotics of the holomorphic pages are Morse-Bott, each arranged in a T -family.Moreover, as explained in Remark A.3, given any T (cid:29)
0, one can arrange thatevery Reeb orbit that is distinct from the binding components has action which isgreater than T , and that the action of every binding component is the same (in theMorse-Bott case). The 2-form of H A is by construction exact, and the integral of itsprimitive along the binding is precisely their action as Reeb orbits. Taking T largeenough, Stokes theorem provides the following:43 emma 7.2. Fix k ∈ N , and an H A -compatible J in the symplectization R × M of H A . Let u : ˙ S → R × M be a connected J -holomorphic curve with simply coveredpositive asymptotics Γ + ( u ) all of the form B × { q } × { } , such that + ( u ) ≤ k .Then its negative asymptotics Γ − ( u ) are also simply covered, of the same form, and − ( u ) ≤ + ( u ) , with equality if and only if u is a trivial cylinder. (cid:3) We shall assume from now on that W is a strong symplectic filling of M = BO (Σ , φ ).Denote by H A = (Λ , Ω) the SOBD stable Hamiltonian structure on M described inSection 7.1, such that its symplectization admits a finite energy foliation F inducedby the holomorphic spinal open book decomposition, for a suitable compatible al-most complex structure J . We attach the symplectic cobordism of Lemma 7.1 to W ,obtaining a symplectic manifold with stable boundary ( M, H A ) which we still call W . We take the Liouville completion (cid:99) W = W ∪ [0 , + ∞ ) × M by adding the sym-plectization, and extend J to a compatible almost complex structure in (cid:99) W which isgeneric in the interior of W . For a ∈ [0 , + ∞ ) , we denote W a = (cid:99) W \ (( a, + ∞ ) × ∂W )the truncation at level a , and M aP := { a } × M P .We obtain a moduli space M A , consisting of k -punctured holomorphic spheres in (cid:99) W , where k is the number of connected components of B = ∂ Σ. We denote by M A the Gromov compactification of M A . We have M A = M Antriv ∪ M
Atriv , where M Atriv consists of those curves in M A which have cylindrical ends lying in the union of thetrivial cylinders over the SOBD binding B = B × T × { r = 0 } . We consider M A ∗ , obtained by adding a marked point to curves in M A , and the resulting evaluationmap ev : M A ∗ → (cid:99) W , and forgetful map π : M A ∗ → M A . Properties of the compactified moduli space.
We now analyse the structureof the compactified moduli space M A . Crucially for Theorem B, we show that theonly possible degenerations consists of bubbling of closed spheres. The following isthe analogue of Theorem 6.8 in this setting: Theorem 7.3 (Properties of M A ) . There exists sufficiently large (cid:28) a ∈ [0 , + ∞ ) ,so that the following statements hold (see Figure 8):(i) (compactness) We have a stratification ∅ = M A, ⊂ M A, ⊂ M A, ⊂ M A, = M A Here, the top strata M A = int ( M A, ) is a smooth -manifold consisting ofsomewhere injective k -punctured spheres. For i = 1 , , the i -th strata int ( M A,i )44 S M P M I � = 𝜑 ( � , � ) � � WW 𝜑 ( � , � ) + ∞ � =0 � =1 � Figure 8: The moduli space M A . The unmarked moduli M A has a cylindrical end[ a , + ∞ ) × T which corresponds to curves lying away from the shaded region. Wehave ev ( a, m ) = ( ϕ ( a, m ) , m ), for a smooth function ϕ : [ a , + ∞ ) × M \B → [0 , + ∞ )satisfying ϕ ( a, m ) = a for m ∈ M P , lim r → ϕ ( a, r ) = + ∞ along R × M S . has dimension at most − i , and consists of nodal curves having precisely onecomponent which is a somewhere injective k -punctured sphere, and precisely i (possibly multiply covered) closed spheres bubbles. In particular, there are nomultiple-floor building degenerations from curves in M A , and no degenerationswith components consisting of l -punctured spheres for l < k .(ii) (boundary behaviour) The evaluation map is (weakly) transverse to the bound-ary ∂W acap .(iii) (diffeomorphism on vertical boundary) For a ≥ a , denote ∂ v M A ∗ ,a = ev − ( M aP ) ⊂M A ∗ the vertical boundary, and ∂ v ev a the corresponding restriction of the eval-uation map. Then ∂ v ev a : ∂ v M A ∗ ,a → M aP is a diffeomorphism. iv) (relative pseudocycle) For a ≥ a , consider M A ∗ ,a = ev − ( W a ) ⊂ M A ∗ and let ev a denote the corresponding restriction of the evaluation map. Then ev a : M A ∗ ,a → W a is a relative pseudocycle representing the fundamental class.In particular, the evaluation map on the compactification ev a : M A ∗ ,a → W a issurjective.(v) (truncated fibers) Let M Aa be the set of umarked curves in M Aa with non-emptyintersection with W a .Then the fibers of the forgetful map π : M A ∗ ,a → M Aa are compact truncationsof the corresponding punctured curves in M A .(vi) (cylindrical end) For a ≥ a , M A has a cylindrical end over ∂ M Aa ∼ = T .Proof. (Theorem 7.3) For the purposes of exposition, we divide the argument inseveral steps. Step 1: Curves in the top levels.Lemma 7.4.
Let u : ˙ S → R × M be a connected holomorphic curve with precisely k = π ( B ) positive asymptotics all of the form B × { q } × { } , such that any twolie in different components of the SOBD binding B . Then u is a leaf in F .Proof. (Lemma 7.4.) The proof is a direct and straightforward adaptation of [Mor17a,Thm. 3.9, case B], and therefore we will provide only the key argument. While a pri-ori there is non-trivial monodromy in our setup, we carry out the relevant changesin the proof. We point out that it can be carried out for any spinal open bookdecomposition supporting a contact structure.Consider u as in the statement. We will show that u is a reparametrization of aleaf in F . By Lemma 7.2, its negative ends also correspond to binding components.Let U = (cid:70) j U j ⊂ Σ be the union of the connected components of Σ \ supp( φ ) whichcontain ∂ Σ. Define V = U × S × T = (cid:70) j V j , where V j = U j × S × T , whichwe view as a subset of M P . The J described in Section 7.1 can be constructed insuch a way that the submanifolds R × { z } × S × T ⊂ R × V , for z ∈ U , areactually J -holomorphic hypersurfaces. Moreover, the projection p : R × V → U isholomorphic.If the spine is disconnected so that k >
1, then, since u is connected and ap-proaches every boundary component of Σ, necessarily it intersects all connectedcomponents of R × V . In the case where k = 1, either u is a trivial cylinder, or it isa plane by Lemma 7.2. In the latter case, it also necessarily intersects all connectedcomponents of R × V , because the binding is non-contractible in R × M S .46o we may consider S := u − ( R × V ) = (cid:70) j S j , where each S j = u − ( R × V j )is assumed non-empty. Up to generically perturbing the boundary of V , we mayassume that S is a disjoint union of genus zero surfaces with non-empty boundarymapping to R × ∂ V under u , and that the portion of ∂ V near ∂ Σ lies in some { t = t } , where t ∈ ( − δ,
0] is the collar parameter in ( − δ, × B × S × T ⊂ M P .We then have that F = ( p ◦ u ) | S : S → U is a holomorphic branched cover, such that the degree d j ≥ F j = F | S j is thealgebraic intersection number of u with R × V j .One then estimates the Ω-energy of u , where Ω = Ω ν is the (exact) 2-form in theSHS H A constructed in Appendix A. This form looks like Ω = dλ + Kdλ std along M P for some large K (cid:29)
0, where λ is the Liouville form in Σ, so that λ = e t dϕ inthe collar, and λ std is the standard contact form in T . Denote S P = u − ( R × M t P ),where M t P is the part of the paper whose boundary corresponds to { t = t } , and by A = a/ π , where a is the action of any binding component (which is the same foreach, and can be taken to be as close to 1 as desired; see Remark A.3). Then,2 πAk ≥ πA ( k − − ( u )) = (cid:90) ˙ S u ∗ Ω ≥ (cid:90) S P u ∗ Ω ≥ (cid:90) S P u ∗ dλ ≥ πe t k ≥ πe − δ k, where we have used that (cid:82) S P u ∗ dλ std ≥ (cid:90) S P u ∗ dλ = e t (cid:90) ∂S P u ∗ dϕ = 2 πe t (cid:88) j d j . Since A and e − δ can be chosen arbitrarily close to 1 independent of u , the aboveestimate implies that d j = 1 for every j , and Γ − ( u ) = ∅ . In other words, each F j isa biholomorphism, and u has no negative ends. Moreover, we obtain that (cid:90) S P u ∗ dλ std = 0Using the Morse-Bott condition of the orbits and unique continuation, the restof the proof follows almost word by word as in [Mor17a, Thm. 3.9, case B], and weomit further details. Step 2: Degenerations.
We now study the possible degenerations. Let u ∈ M A be a stable and potentially nodal building with multiple components distributedamongst levels (a unique main level, and perhaps several upper levels).47 tep 2.A: Only bubbles as nodal degenerations. We show that none of thecomponents of u consists of a sphere with l punctures, for l < k .Indeed, let u be such a component. By adding + ∞ to the cylindrical end of (cid:99) W and choosing a suitable smooth structure on [0 , + ∞ ], we obtain a compact manifold W ∞ with ∂W ∞ = { + ∞} × M =: M ∞ . We then do (topologically) spine removalsurgery to M ∞ by attaching handles as described in Section 3.2, obtaining a smoothcopy of W cap . By Lemma 7.2, all the asymptotics of every component in u is asimply covered binding component. So we may attach the cores of the handles ateach of the asymptotics of u , obtaining a sphere in W cap which intersects precisely l < k co-cores. But, by Theorem 6.12, there are no such spherical classes in W cap .It follows that u can only have closed sphere bubble components. Their imagecannot be completely contained in (cid:99) W \ W , by the exactness of the 2-form of H A .Therefore there are no nodal degenerations in the upper levels. Step 2.B: There are no multiple levels.
By the previous step, any componentof the lower-most level of u is necessarily a sphere with precisely k punctures, andperhaps some bubbles. When combined with Lemma 7.2, we conclude that the levelsabove the lower-most can only consist of trivial cylinders. The stability conditionimplies that u coincides with its lower-most level.By Step 1, if such level is an upper level, then u is a non-trivial leaf in F . If not,it is a element in M Atriv . Step 3: M A is a stratified space. From genericity of J along W , Fredholmregularity of curves in F , and Lemma 7.4, we obtain smoothness of the top stratum.The dimension counts follow from semi-positivity of symplectic 6-manifolds (cf. theproof of Lemma 7.5 below). This proves Item (i).The proof of the remaining items is a word by word adaptation of the corre-sponding proof in Theorem 6.8 above, and is left as an exercise for the reader. Nodal substrata.
In this section, we describe the nodal configurations that canappear in M A \M A . Lemma 7.5 (The
Lemma) . The set M A, ⊂ M A of nodal configurationswith at least node consists of the following configurations (see Figure 9):A) A single somewhere injective k -punctured sphere component of index , and asingle simply covered bubble of index . The substratum of int ( M A, ) consistingof these configurations is -dimensional. u A B1 u u � ( � )=1 B2 u � :1 u � ( � )=0 C D dim=2 dim=2 dim=0dim=0dim=2 E u u e.g. u u u u u u u Figure 9: The possible degenerations in M A \M A . Case E, i.e. corresponding to con-figurations with non-contractible vanishing cycles, is ruled out using Theorem 6.12,as in Step 2.A in the proof of Theorem 7.3.49 ) A single somewhere injective k -punctured sphere component of index , andeither:B1) A simply covered bubble of index . The corresponding nodal substratumof int ( M A, ) is -dimensional; orB2) A doubly covered bubble of index , whose underlying simple sphere has in-dex . The corresponding nodal substratum of int ( M A, ) is -dimensional.C) A single somewhere injective k -punctured sphere component of index , and anindex multiple cover of some degree, whose underlying simple sphere has alsoindex . The corresponding nodal substratum of int ( M A, ) is -dimensional.D) A single somewhere injective k -punctured sphere component, together with twobubbles (i.e. an element in M A, ⊂ M A, ). Independently of the indices ofeach component, the corresponding nodal substratum is always -dimensional.In all cases A)–D), configurations with multiply covered bubbles either come in -dimensional substrata or the multiple cover is rigid.Proof. Let u be a nodal configuration in M W , with u its k -punctured sphere com-ponent, and (cid:101) u , (cid:101) u the (possibly constant or multiply covered) bubbles. Let u and u be the underlying simple curves, and d , d be the corresponding coveringdegrees. We have:4 = ind( u ) = 2 c τ ( u ) + µ τCZ ( u ) = 2 c τ ( u ) + µ τCZ ( u ) + 2 d c ( u ) + 2 d c ( u ) , where τ is a trivialization of the contact structure along the asymptotics of u . Wehave µ τCZ ( u ) = µ τCZ ( u ) = 4 with respect to a natural such trivialization, and so c τ ( u ) = 0. This implies c τ ( u ) + d c ( u ) + d c ( u ) = 0 (5)Since u is somewhere injective and intersects in an open set the region where J isgeneric, we have 0 ≤ ind( u ) = 2 c τ ( u ) + µ CZ ( u ) = 2 c τ ( u ) + 4 , and so c τ ( u ) ≥ − ≤ d c ( u ) + d c ( u ) ≤ ≤ ind( u ) + ind( u ) + ind( u ) − i = 2 c τ ( u ) + µ τCZ ( u ) + 2 c ( u ) + 2 c ( u ) − i = 2(1 − d ) c ( u ) + 2(1 − d ) c ( u ) + 4 − i (7)where i is the number of non-constant bubbles, and where we used (5). The lemmafollows by combining (6) and (7). We consider M Aa , for a ≥ a , as in the statement of Theorem 7.3. Consider theclosed disk ∆ = { a } × {∗} × { q } × D ⊂ { a } × M S , which intersects each curve in M Aa at most once (since they are uniformly C -close to trivial cylinders, cf. Step 8in the proof of Theorem 6.8). Here, recall that M S = ∂ Σ × T × D is the spine ofthe SOBD from Section 2.Note that the curves that intersect ∂ ∆ can be identified with the truncatedpages of the open book used in the Bourgeois construction. Pushing the interiorof the disc slightly into the filling, we can assume that the boundary of the disc istransverse to the evaluation map. Note also that this part of the disc lies near thevertical boundary where the evaluation map is a diffeomorphim. After a furtherperturbation, relative to its boundary, we can assume that it intersects the nodalstrata – more precisely its image under the evaluation map – transversely in itsinterior, and hence in a finite number of points that lie near the boundary of somecurves in the marked moduli space M A ∗ ,a .In view of Lemma 7.5, we can assume by general position that we only meetcurves in the strata corresponding to Cases A, B1, and C. Notice also that in caseB1 the main component has index 0, i.e. (each connected component of) the cor-responding nodal substratum is made of nodal curves with a fixed main puncturedcomponent, and a bubble component varying in a 2-dimensional family. In partic-ular, for dimensional reasons, one can perturb ∆ so that it avoids such rigid mainpunctured component.We can hence assume that ∆ intersects only nodal curves of type A and C, in afinite number of points, which we denote y j . Let also N an arbitrary neighborhood(which will be taken small in the following) of the nodal locus in the unmarkedmoduli space and π : M A ∗ ,a → M Aa be the forgetful map; by general position, we canthen assume that the restriction of the evaluation map to M Aa \ π − ( N ) is transverseto the disc ∆. Note that although there may be a large subset of the moduli spacepassing through a point of the disc, given for example by reparametrising a multiply-covered component, there will only be finitely many underlying simple nodal curvesmeeting the disc. By general position we can assume that these simple nodal curves51eet the disc meet the disc at distinct points, hence we denote u j the underlyingsimple nodal curve meeting ∆ at y j , and we let C j be its images in W a . We alsonote that there may be multiple non-nodal curves going through a given point in theinterior of the disc, and perhaps also points through which both nodal and smoothcurves pass.Given that the dimension of the ambient manifold is greater than four, we canassume that, for every curve u j , the underlying simple curve of its spherical com-ponent as well as its punctured component are both embedded (by a puncturedversion of [Wen18, Ex. 2.27]). Moreover, the image of the nodal curve itself is alsoembedded for similar reasons.Let N j = N ( C j ) be a regular neighbourhood of C j inside W a . Notice that N j retracts onto C j . We can easily arrange that the resulting collapse map is compatiblewith the natural projection to the binding on boundary components. One canalso blow down the bubble component of C j . This gives a blow down of C j to its(truncated) punctured component C j ∼ = Σ.Choose small discs D j = D ε j ( y j ) ⊂ ∆ about each intersection point y j ∈ ∆.Now, since the evaluation map on the uncompactified moduli space ev : M A ∗ ,a → W a is transverse to the disc, taking ε j generic (and small), we can assume that thethe evaluation map is also transverse to the small circle ∂D j . Furthermore, up toshrinking the neighborhood N of the nodal locus in the unmarked moduli spaceconsidered above, we can assume that its preimage π − ( N ) in the marked modulispace is mapped in ∪ j N j inside W a via the evaluation map and that ev − ( ∂D j ) ⊂M ∗ ,a \ π − ( N ). We can then perturb the evaluation map itself relative to a slightlysmaller neighborhood N (cid:48) ⊂ N so that the y j are regular values for the restriction tothe complement of N in the moduli space, which is entirely contained in the smooth(i.e. uncompactified) moduli space. In particular, for each j , the intersection ofthe preimage ev − ( y j ) with M A ∗ ,a \ π − ( N ) (where π is the forgetful map) is a 0-dimensional submanifold in M A ∗ ,a \ π − ( N ), and ev ( π − ( N )) is entirely contained in ∪ j N j ⊂ W a .Denote by M A, ∆ a ⊂ M Aa the set of unmarked curves which intersect ∆, andconsider the map I ∆ : M A, ∆ a → ∆ sending a curve to its intersection point with ∆(recall that we arranged that any curve intersects the disc in at most one point, usingthe fact that punctured curves are asymptotically trivial). These intersection points,considered as marked points, then give a natural lift M A, ∆ ∗ ,a = ev − (∆) of M A, ∆ a tothe marked moduli space. Moreover, the map I ∆ factors through the space M A, ∆ qu defined as follows: M A, ∆ qu is obtained from M A, ∆ a by identifying two (necessarilynodal) curves in M A, ∆ a if they have the same image in W a under the evaluationmap. The quotient topology is Hausdorff and compact, since preimages of pointsunder the quotient map M A, ∆ a −→ M A, ∆ qu are compact and M A, ∆ a is Hausdorff and52ompact (here we use that ev is proper, so that M A, ∆ ∗ ,a is compact). Note that M A, ∆ qu contains a smooth part M A, ∆ qu ∼ = M A, ∆ a which is identified with the non-nodal partof M A, ∆ a . We let I ∆ qu be the induced map on M A, ∆ qu , which is continuous.Our discussion above implies that point preimages under M A, ∆ qu only have finitelymany accumulation points, all of which consists of (images of) nodal curves (thisfollows by transversality to ∆ of the evaluation map along the uncompactified modulispace). In particular, since ( I ∆ qu ) − ( y j ) intersects M A, ∆ a \ N in a 0-dimensionalsubmanifold, the curves in ( I ∆ qu ) − ( y j ) whose images are not entirely contained insome N j consists of a finite collection of non-nodal curves. Therefore we may cover( I ∆ qu ) − ( y j ) via a finite collection of disjoint open sets V jk ’s in the quotient space M A, ∆ qu , each of which is either contained in a local smooth chart for M A, ∆ a (and wemay take it to be a disk) or such that the images of any curve in it is containedin some N j . By taking ε j small (and shrinking the N accordingly), and usingcompactness of M A, ∆ qu , we can assume that ( I ∆ qu ) − ( ∂D j ) ⊂ (cid:70) k V jk . (Notice that I ∆ qu ) − ( ∂D j ) is a disjoint union of embedded circles, because ev is transverse to each ∂D j .)Let ∆ (cid:48) := ∆ \ (cid:83) j D j , and consider ∆ := ev − (∆ (cid:48) ) ⊂ M A, ∆ ∗ ,a and B := π (∆) ⊂M A, ∆ a . Define π | B : E = π − ( B ) → B , where π is the forgetful map. Then E is a fiber bundle with fiber Σ (and ∆ is a section), and note that B is a smoothsurface with boundary ∂B = ( I ∆ qu ) − ( ∂D j ) ∪ ( I ∆ qu ) − ( ∂ ∆). By construction, themonodromy of E along ( I ∆ qu ) − ( ∂ ∆) ⊂ ∂B (which is a single circle), is precisely ψ , the monodromy of the original open book used in the Bourgeois construction.Moreover, we claim that the monodromy of this bundle is trivial along all boundarycomponents contained in (the possibly disconnected) subset ( I ∆ qu ) − ( ∂D j ) ⊂ ∂B .Note first that the monodromy on the boundary of the page Σ is trivial over theentire bundle as there is a natural trivialisation given by projecting to the bindingof the original open book, which we chose compatible with the retraction of N j to C j . Next, for components contained in neighbourhoods of smooth curves, thebundle extends trivially over the corresponding disk V jk , and thus we can cap ofsuch boundary components with disks and extend the bundle to the capped surface,which we denote by (cid:98) B .We now consider the remaining case. In view of the previous paragraph, theimage of all the curves in every boundary component L j ⊂ ∂ (cid:98) B lies in a regularneighbourhood N j of some nodal curve C j . Thus composing with the collapse ontothe blown down nodal curve C j , we obtain a fiberwise map between the bundle over E | L j and the trivial bundle: 53 | L j L j × C j L j L j . f j Id We note that the map is a diffeomorphism near the boundary with respect to thenatural trivialisation coming from the projection to the binding of the open book.In particular, it induces degree one maps on each fiber. It is well known that anydegree one map of a surface that is a homeomorphism on the boundary is homotopicrel boundary to a homeomorphism by a result of Kneser cf. [Sko87]. In particular,after a homotopy we can assume that the map is a homeomorphism on a referencefiber E | L j ⊇ Σ j ∼ = C j .Then let ψ t be the flow on E | L j of a vector field transverse to the fibers, whichmoreover sends fibers to fibers and it induces the given trivialisation on the bound-ary. Let ψ j : Σ → Σ denote the monodromy given by the time one map of this flow.Pushing forward via f j we obtain a homotopy of maps ψ t = f j ◦ ψ t ◦ f − j | Σ j from ψ = ψ j to the identity relative to the boundary.In particular, the monodromy ψ j is homotopic to the identity rel. ∂ Σ. Kneser’sTheorem then implies that these are in fact isotopic to the identity. As the openbook monodromy ψ : Σ → Σ is just the composition of all the ψ j , plus an elementinduced by the commutator relation in the fundamental group of the base surface (cid:98) B , we see that ψ is isotopic to a product of commutators rel. ∂ Σ, as desired. Thisconcludes the proof of Theorem B. (cid:3)
Remark 7.6.
In the aspherical case, Item (i) of Theorem 7.3 symplifies in thefollowing way: M A = M A , i.e. there are no multiple–level or nodal degenerationsat all. In particular, the above strategy of intersecting with ∆ directly yields a fiberbundle with fiber Σ over a base surface B with one boundary component. Thisimmediately tells that the monodromy ψ of the original open book decompositionfactors as a product of commutators. In this section we study strong symplectically aspherical fillings of high dimensionalBourgeois contact manifolds. More precisely, Section 8.1 contains the proof of The-orem F, i.e. the classification of such fillings in the case of S ∗ T n = BO ( D ∗ T n − , Id).In Section 8.2 we prove Theorem G, i.e. that no symplectically aspherical fillingsexist in the case of BO ( D ∗ S n , τ ), with τ the Dehn–Seidel twist on D ∗ S n .54 .1 Symplectically aspherical fillings of S ∗ T n In this section, we consider ( S ∗ T n , ξ std ), the unit cotangent bundle of T n , with itsstandard Stein fillable contact structure, and we prove Theorem F from the Intro-duction. The proof follows the same lines as those of previous sections in a higher-dimensional setting and many arguments simplify greatly due to the asphericitycondition, which ensures that all relevant moduli spaces are automatically compactas bubbling can be excluded by assumption. A family of SOBDs and holomorphic foliations.
As explained in the In-troduction, we can identify (
M, ξ ) := ( S ∗ T n , ξ std ) ∼ = ( BO ( D ∗ T n − , id ) , ξ BO ) , up tocontactomorphism. This manifold is the convex boundary of the Stein manifold W := D ∗ S × · · · × D ∗ S , with its product Stein structure. We may write M = ∂W = D ∗ T j × ∂D ∗ T n − j (cid:91) ∂D ∗ T j × D ∗ T n − j , for j = 1 , . . . , n − . These decompositions all yield supporting SOBDs for (
M, ξ ),besides the SOBD which we have so far considered for Bourgeois contact manifolds,which corresponds to j = 2 (see Section 2). Observe that they are symmetric underthe substitution j ↔ n − j . From now on, we shall only consider the j = n − D ∗ S are 2-dimensional.As in Section 7, one can then adapt the construction in Appendix A for the j = n − D ∗ S , using as auxiliary Morse–Bott function on thevertebra D ∗ T n − the square of the distance from the zero section. This gives a(2-dimensional) holomorphic foliation F of the symplectization of a suitable stableHamiltonian structure H in M , for which there exists a symplectic cobordism withstrong concave boundary ( M, ξ std ) and positive stable boundary ( M, H ). Moreover,because of the particular choice of Morse–Bott function, which has no gradienttrajectory flowing from a critical point to another, the leaves of F are either trivialcylinders or Liouville completions of the pages D ∗ S , and the latter approach theformer asymptotically at infinity.As in Section 7 (cf. [Mor17a, Section 3.5]), one shows that F is Fredholm regular.Moreover, Riemann-Roch gives that each of its elements has Fredholm index 2 n − A moduli space.
Consider W a symplectically aspherical strong symplectic fillingof ( M, ξ ) = ( S ∗ T n , ξ std ). We now proceed as in Section 7.After attaching a cobordism to its boundary to obtain ( M, H ), and completingto a manifold (cid:99) W = W ∪ [0 , + ∞ ) × M by attaching the symplectization of ( M, H ),we obtain a moduli space M consisting of holomorphic cylinders inside (cid:99) W . We alsohave M ∗ , obtained by adding a marked point to the domains, an evaluation map ev :55 ∗ → (cid:99) W , and a forgetful map π : M ∗ → M . Denote by W a = (cid:99) W \ (( a, + ∞ ) × ∂W ))the truncation at level a .The following is the analogue of Theorem 7.3 for the current moduli space M ,but in the symplectically aspherical case. Note that in higher dimension one doesnot immediately have semi-positivity, but the asphericity assumption as well as thegeometric set-up (positivity of intersections with cocores) of handles means that allnodal curves are simple, and semi-positivity indeed holds in our situation. Thus theproof is an easy adaptation and is hence omitted: Theorem 8.1.
There exists sufficiently large (cid:28) a ∈ [0 , + ∞ ) , so that the followingstatements hold:(i) (compactness) The moduli space M coincides with its Gromov compactification M . Namely, there are no nodal nor multiple-floor building degenerations, norbubbling.(ii) (fibration) Let M a, ∗ = ev − ( W a ) . The forgetful map π : M a, ∗ → M a is afibration with fiber an annulus D ∗ S . (iii) (vertical boundary diffeomorphism) For a ≥ a , the evaluation map restrictsto a diffeomorphism ev a : ∂ v M ∗ ,a → M aP = between the vertical boundary ∂ v M ∗ ,a = π − ( ∂ M a ) and the paper M aP = { a } × S ∗ T n − × D ∗ S ⊂ W a . Symplectically aspherical fillings of S ∗ T n : homotopy type. Let W be asymplectically aspherical filling of S ∗ T n = BO ( D ∗ T n − , Id). As a first step towardsTheorem F, we will prove that W is homotopy equivalent to D ∗ T n .Consider S ∗ T n with its SOBD given by S ∗ T n = D ∗ T n − × S ∗ S (cid:91) S ∗ T n − × D ∗ S , where we view D ∗ S as page. In particular, as S ∗ S = {± } × S , the spine D ∗ T n − × S ∗ S has two connected components; we call D ∗ T n − × { +1 } × S the positive component , and the other one, the negative one. Notice that there isa natural map j : T n → S ∗ T n which factorizes through the inclusion of T n = { T ∗ T n − } × T n − × { +1 } × S into the positive component of the spine of S ∗ T n . Proposition 8.2.
Let j : T n (cid:44) → W be given by the composition of j defined aboveand the natural inclusion S ∗ T n = ∂W into W . Consider also a lift (cid:101) j of j to theuniversal covers R n and (cid:102) W . Then:1. H ( (cid:102) W ; Z ) = Z and H k ( (cid:102) W ; Z ) = { } for k > ,2. j ∗ : π ( T n ) → π ( W ) is an isomorphism. j , and hence theinclusion of its (trivial) normal bundle D ∗ T n , induces a homotopy equivalence, asdesired. It is hence enough to prove Proposition 8.2; we first fix some notation.Let a > a > W a := (cid:99) W \ (( a, + ∞ ) × ∂W ) and M ∗ ,a = ev − ( W a ) be asin Theorem 8.1 (cf. Theorem 6.8). Since W a deformation retracts onto W , it isequivalent to prove the statement with W a instead of W (and similarly with theiruniversal coverings). We then factor j : T n (cid:44) → W a as composition of a map i : T n →M ∗ ,a and the evaluation map ev : M ∗ ,a → W a , where i is defined as follows.According to Theorem 8.1 the forgetful map π a : M ∗ ,a → M a is a fibrationwith fiber D ∗ a S = [ − a, a ] × S . Moreover, the evaluation map ev : M ∗ ,a → W a induces a diffeomorphism on the vertical part of the boundary ∂ v M ∗ ,a . We let ∂ + h M ∗ ,a ⊂ ∂ h M ∗ ,a denote the subset of the horizontal boundary given by posi-tive boundary components of the fibers of the forgetful map. Then, as the subset ∂ + h M ∗ ,a ∩ π − ( ∂ M a ) also lies in the vertical boundary ∂ v M ∗ ,a on which the evalua-tion map is a diffeomorphism, we can factor the map j as claimed by first homotopingthe zero section into the boundary S ∗ T n − and then composing with the evaluationmap. Proof (Proposition 8.2).
Since ev is a diffeomorphism on the vertical boundary, ithas degree 1. In particular, it induces surjections in π and H ∗ . Moreover, as themoduli space retracts onto its positive boundary, which is mapped onto a neighbor-hood of the spine D ∗ T n − × S , it follows that j : T n (cid:44) → W a also induces surjectionson homology and fundamental groups. In particular, we deduce that the fundamen-tal group of the filling is abelian.We now prove that j is also H ∗ –injective; as π ( T n ) and j ∗ ( π ( T n )) = π ( W a )are abelian, this immediately implies that j is also π –injective. Consider a class x ∈ H ∗ ( T n ), and denote y := i ∗ ( x ) in H ∗ ( M ∗ ,a ). Suppose that ev ∗ ( y ) is zero in H ∗ ( W a ); this means that there is z ∈ H ∗ ( W a ) with ev ∗ ( y ) = ∂z . As M ∗ ,a and ev are both smooth, there is a well defined ev ! ( z ) in H ∗ ( M ∗ ,a ) such that ev ∗ ev ! ( z ) = z , where ev ! is given, geometrically, by perturbing ev to be transverse to a cyclerepresenting z and taking its preimage. Moreover, as y is supported near ∂ + h M ∗ ,a ,where ev restricts to a diffeomorphism, ∂ev ! ( z ) = y , so that y = 0 in H ∗ ( M ∗ ,a ).Moreover, as M ∗ ,a retracts onto ∂ + h M ∗ ,a , y = 0 in H ∗ ( ∂ + h M ∗ ,a ) too. Now, because ev ( ∂ + h M ∗ ,a ) = D ∗ T n − × S is the spine of S ∗ T n = ∂W a , this means j ∗ ( x ) = ev ∗ ( y ) =0 in H ∗ ( D ∗ T n − × S ), i.e. x = 0 in H ∗ ( T n ), as j factorizes as the natural inclusion T n = T n − × S (cid:44) → D ∗ T n − × S (cid:44) → ∂W (cid:44) → W .Lastly, we prove that the map (cid:101) j on the universal cover also induces an isomor-phism in H ∗ . Injectivity follows trivially from the fact that the universal cover of T n is contractible; we then prove surjectivity.Observe that all holomorphic curves in M naturally lift to maps from the planeto the universal cover (cid:102) W a of W a . This gives a corresponding moduli space (cid:102) M of57olomorphic planes in (cid:102) W a , together with its marked version (cid:102) M ∗ equipped with anevaluation map (cid:101) ev : (cid:102) M ∗ ,a → (cid:102) W a . One can check that the (cid:101) ev is proper and that hasdegree 1 (using cohomology with compact supports), since the diffeomorphism onthe vertical boundary lifts to a diffeomorphism of the appropriate cover consideredas a subset of the universal cover of W . As in the case of cylinders, (cid:102) M ∗ ,a can bepushed to its positive horizontal boundary ∂ h, + (cid:102) M ∗ ,a made of the lifts of the positiveboundaries of the holomorphic cylinders in M ∗ ,a . As j is an isomorphism in π ,using standard covering space arguments, one can check that this set is mapped,via the lifted evaluation map, to a lift D ∗ R n − × R of the spine D ∗ T n − × S of ∂W a ⊂ W a to (cid:102) W a . An argument analogous to the one of the previous paragraphthen allows to deduce that any homology class in (cid:102) W comes from D ∗ R n − × R sothat the relative homology H ∗ ( (cid:102) W , ∂ (cid:102) W ) is trivial in all degrees. Symplectically aspherical fillings of S ∗ T n : diffeomorphism type. Once thehomotopy type is established, understanding the diffeomorphism type can be doneby using the s–cobordism theorem. The argument below is just an adaptation of[BGZ16, Sections 5 and 8] to our setting. We thus give a sketch of the proof, referringto the proofs of the technical statements in [BGZ16]; for the readers’ ease, we alsoadopt their notations.We start by describing the spaces involved in the argument. Let W be theresult of attaching a topologically trivial cobordism [0 , × S ∗ T n to W along itsboundary M := S ∗ T n = { } × S ∗ T n . Consider also on S ∗ T n the SOBD givenby spine D ∗ T n − × S ∗ S and paper S ∗ T n − × D ∗ S , and recall that the spine hasa “positive” component D ∗ T n − × { +1 } × S given by the natural identification S ∗ S = {± } × S ⊂ R × S . The cobordism [0 , × S ∗ T n then contains a collar,diffeomorphic to D ∗ T n − × D ∗ S = D ∗ T n , of the positive component of the spineof { / } × S ∗ T n . In other words, there is a (smooth) copy of W := D ∗ T n whichis entirely contained in the cobordism [0 , × S ∗ T n . Let then X := W \ W and M := ∂W ; notice that ∂X = M ∪ ( − M ). The aim is now to prove that X isdiffeomorphic to a cylinder [0 , × S ∗ T n , so that W is actually diffeomorphic to W ,as desired.As explained in [BGZ16, Lemmas 5.1 and 5.2], the fact that W is homotopyequivalent to D ∗ T n implies that the inclusions M , M (cid:44) → X induce isomorphismson π and on H ∗ . Moreover, as S ∗ T n is a simple space (i.e. the action of its π onevery homotopy group is trivial), arguing exactly as in [BGZ16, Lemmas 8.1 and8.2] one can show that M , M (cid:44) → X actually induce isomorphisms on all homotopygroups. This proves that X is an h–cobordism between M and M .Now, as M = S ∗ T n , the Whitehead group Wh( π ( M )) vanishes, so that theWhitehead torsion of the inclusion M (cid:44) → X is necessarily zero. The s–cobordismtheorem then tells that X is diffeomorphic to [0 , × S ∗ T n , as desired.58 .2 Symplectic fillability of BO ( T ∗ S n , τ ) We denote by τ the Dehn–Seidel twist on D ∗ S n , and consider the Bourgeois contactmanifold BO ( D ∗ S n , τ ). The aim of the section is to prove Theorem G, i.e. that BO ( D ∗ S n , τ ) does not admit any strong symplectically aspherical filling. S –equivariant SOBDs. Bourgeois contact structures are T –equivariant con-tact structures on products of contact (2 n + 1)–manifolds with T , and it can bein particular reinterpreted using the construction from [GS10, Theorem 1] of S –equivariant contact structures on products with S as follows (cf. [DG12, Section5.3]).Consider the Weinstein manifold ( T ∗ ( S n × S x ) , λ st = λ T ∗ S n + sdx ), where s isthe cotangent direction of T ∗ S x and S x denotes the factor corresponding to the firstcoordinate of ( x, y ) ∈ T . The Dehn–Seidel twist τ on D ∗ S n gives the compactlysupported map ψ = τ × Id on the contactization ( D ∗ S n × S x , λ T ∗ S n + dx ), such that ψ ∗ ( λ T ∗ S n + dx ) = λ T ∗ S n − dH + dx, where H : D ∗ S n → R is a smooth function which is 0 outside the support of τ ,which we may extend to T ∗ ( S n × S x ) = T ∗ S n × T ∗ S x in the obvious way. As ( D ∗ S n × S x , ker( λ T ∗ S n + dx )) is a contact submanifold of the contact boundary S ∗ ( S n × S x )of D ∗ ( S n × S x ), one can then use the ψ as gluing map in the recipe given in [GS10]in order to obtain an explicit contact form β (cid:48) on (cid:16) T ∗ ( S n × S x ) ∪ ψ T ∗ ( S n × S x ) (cid:17) × S y . Remark 8.3.
To be precise, as ψ is not a strict contactomorphism, one needs tointerpolate between the contact forms λ T ∗ S n + dx and ψ ∗ ( λ T ∗ S n + dx ) = λ T ∗ S n − dH + dx on the subset D ∗ S n × S x of the hypersurface S ∗ ( S n × S x ) along which the gluingin the left factor happens. This can be done by interpolating in a linear way, alongthe normal coordinate to the hypersurface, which is equivalent to attaching a smallcobordism on one of the two Weinstein manifolds used in the gluing (as explainedin [GS10]).Now, the underlying smooth manifold is just S n +1 × T x,y ) , and β (cid:48) is of theform α (cid:48) + φ (cid:48) dx − φ (cid:48) dy , where (by choice of the gluing ψ ) α (cid:48) ∈ Ω ( S n +1 ) and φ (cid:48) , φ (cid:48) : S n +1 → R , i.e. they are T –equivariant. One then also explicitly check that φ (cid:48) = ( φ (cid:48) , φ (cid:48) ) : S n +1 → R actually defines an open book on S n +1 , with page D ∗ S n and monodromy τ , and, moreover, that α (cid:48) is adapted to such open book. In otherwords, up to isotopy, ( S n +1 × T , ker β (cid:48) ) is just BO ( D ∗ S n , τ ).59he advantage of this point of view is that ( S n +1 × T , ker β (cid:48) ) is clearly supportedby another SOBD, called here the S y –equivariant SOBD , which already appearedin [Mor17a, Section 5.1] (see also [Mor17a, Section 5.2] dealing with the non–trivialbundle setting described in [DG12, Section 5.3]). Remark 8.4.
The word ”supported” is used here as in Section 2; more precisely, itmeans that ker β (cid:48) is isotopic to the kernel of the explicit contact form on S n +1 × T constructed via the corner rounding procedure as in Appendix A (namely, the λ E | M − on M − there), but using this S y –equivariant SOBD instead of the Bourgeois SOBDdescribed in Section 2.More precisely, such an S y –equivariant SOBD is composed by the following twopieces. The paper is as follows:( D ∗ ( S n × S x ) , λ st ) M P = (cid:83) i =1 , ( D ∗ ( S n × S x ) × S y , λ st + dy ) . (cid:83) i =1 , ( S y , dy ) π P where S y is the S -factor of T corresponding to the second coordinate of ( x, y ) ∈ T and t is the cotangent coordinate of D ∗ S y .The spine is instead given by the gluing M S = S ∗ ( S n × S x ) × D ∗≤ S y (cid:91) ψ S ∗ ( S n × S x ) × D ∗≥ S y , where D ∗≤ and D ∗≥ denote respectively the non–positive and non–negative halfcotangent directions. Moreover, the contact form is given by α := ( λ T ∗ S n + sdx ) | S ∗ ( S n × S x ) + tdy on the “left piece” D ∗ S n × S x × D ∗≤ S y and of the form ( λ T ∗ S n + sdx ) | S ∗ ( S n × S x ) + f ( t ) dH + tdy on the “right piece” D ∗ S n × S x × D ∗≥ S y , where f : [0 , → [0 ,
1] issmooth, strictly decreasing, equal to 1 near 0 and equal to 0 near 1. In particular,one has a fibration structure S ∗ ( S n × S x ) M S = S ∗ ( S n × S x ) × D ∗≤ S y (cid:83) ψ S ∗ ( S n × S x ) × D ∗≥ S y ,D ∗ S yπ S with explicit contact forms on the fibers and Liouville form on the base as describedabove.Notice that we have chosen the S y ⊂ T factor in the above construction. How-ever, everything goes through exactly the same by exchanging the roles of x and y ,thus obtaining also an S x –equivariant SOBD supporting BO ( D ∗ S n , τ ).60 emark 8.5. We discussed here the case of BO ( D ∗ S n , τ ), but the same argu-ment shows that every Bourgeois contact structure admits analogous S –equivariantSOBDs. A capping cobordism.
Let now W be a hypothetical strong symplectic filling of BO ( D ∗ S n , τ ). Notice that the paper in the S y –equivariant SOBD described above isa trivial product. We can then attach a handle of the form D ∗ ( S n × S y ) × D , where D is a 2-disk, to each connected component D ∗ ( S n × S y ) × S y of the paper M P . Thiscan be done in a way which is completely analogous to the spine removal cobordismdescribed in Theorem 3.2; we omit here the details. (Notice however that we areremoving the paper instead of the spine, using the fact that it is a trivial product.)We denote W cap the result of attaching such a handle to W .The symplectic form on W cap admits a natural stabilizing vector field definedon a neighborhood of the boundary ∂W cap and transverse to it. Moreover, ∂W cap admits a natural decomposition as follows: ∂W cap = S ∗ ( S n × S x ) × D + (cid:91) ψ S ∗ ( S n × S x ) × D − , where D − and D + are seen as south and north hemispheres of a sphere ( S , ω S ), S ∗ ( S n × S x ) × D − is equipped with the stable Hamiltonian structure ( α, dα + ω S )and S ∗ ( S n × S x ) × D + is equipped with the stable Hamiltonian structure ( α + f ( r ) dH, d ( α + f ( r ) dH ) + ω S ), where r is the normal coordinate to ∂D + ⊂ D + and f is as in the description of the SOBD. In particular, W cap has a natural structureof fibration over ( S , ω S ) and fibers given by the smooth manifold S ∗ ( S n × S x ) witha varying contact form.In order to construct a foliation by holomorphic spheres near ∂W cap it is ac-tually useful to change the identification of its boundary. More precisely, up toreparametrizing via the (compactly supported) diffeomorphism D ∗ S n × S x × D − → D ∗ S n × S x × D − ( p, x, y ) (cid:55)→ ( τ ( p ) , x, y )on the subset D ∗ S n × S x × D − of S ∗ ( S n × S x ) × D − , the boundary ∂W cap can actuallybe identified with the stable Hamiltonian manifold obtained by gluing( S ∗ ( S n × S x ) × D + , α + f ( r ) dH, d ( α + f ( r ) dH ) + ω S )to ( S ∗ ( S n × S x ) × D − , α + dH, dα + ω S )via the identity gluing map. In other words, as a stable Hamiltonian manifold, ∂W cap is just (cid:0) S ∗ ( S n × S x ) × S , α + f ( r ) dH, d ( α + f ( r ) dH ) + ω S (cid:1) ,61here f is here to be interpreted as the extension of the f defined above on D + to S by 1 on D − .Let now χ : [0 , → [0 ,
1] be a smooth non-increasing function, equal to 1 near0 and 0 near 1. We can then stack the (topologically trivial) symplectic cobordism (cid:0) S ∗ ( S n × S x ) × S × [0 , t , d [ e t ( α + χ ( t ) f ( r ) dH )] + ω S (cid:1) on top of W cap , and denote the resulting symplectic manifold by W (cid:48) cap . Notice that ∂W (cid:48) cap is just the stable Hamiltonian manifold (cid:0) S ∗ ( S n × S x ) × S , α, dα + ω S (cid:1) . A moduli space of spheres in the capped filling.
Note that ∂W (cid:48) cap is just asplit stable Hamiltonian manifold, given by the product of ( S ∗ ( S n × S x ) , α ) with the2-sphere. Hence one can find a J compatible with the symplectic structure which,near a neighborhood of the boundary ∂W (cid:48) cap , splits as a direct sum of the standardalmost complex structure J S on the 2-sphere and some J α on the symplectizationof the contact factor. In particular, for any points ( t, q ) ∈ ( − (cid:15), × S ∗ ( S n × S x ), thespheres u t,q : S → ( − (cid:15), × S ∗ ( S n × S x ) × S y (cid:55)→ ( t, q, y )are all J –holomorphic.We also need some control on the J in the cobordism W (cid:48) cap \ W , as we now ex-plain. Notice that the handle D ∗ ( S n × S x ) × ( D + (cid:116) D − ) used to obtain W cap from W has a product symplectic structure, with the cocores C ± = D ∗ ( S n × S x ) × { D ± } being symplectic submanifolds. Moreover, these cocores naturally extend to sym-plectic cocores C (cid:48) + , C (cid:48)− in W (cid:48) cap thanks to the t –direction of the topologically trivialcobordism W (cid:48) cap \ W cap = S ∗ ( S n × S x ) × S × [0 , t . In the choice of J in the para-graph above, we could have moreover arranged that C (cid:48) + , C (cid:48)− are J -holomorphic in( W (cid:48) cap , J ); we hence assume this is the case. In the complement of a neighborhood of ∂W (cid:48) cap ∪ C (cid:48) + ∪ C (cid:48)− , the J can then be chosen generic, so that every simple holomorphicsphere intersecting said region is regular.Then, exactly as in Section 6.1, one can prove the following properties. The J -holomorphic spheres u t,q are regular, and each J -holomorphic sphere which issufficiently near to ∂W (cid:48) cap has to be of the form u t,q . The resulting (unmarked)moduli space has dimension 2 n + 2.One can then consider the moduli space M of J -holomorphic spheres in W (cid:48) cap in the same homology class as the S -factors coming from the boundary ∂W (cid:48) cap . Inparticular, each of these curve has to be somewhere injective, as they intersect thecocores geometrically once by positivity of intersections. Moreover, by uniqueness62ear the boundary of W (cid:48) cap , the moduli space M extends the family of u t,q ’s de-fined on the cylindrical end over ∂W (cid:48) cap . Lastly, we denote by M , M ∗ the Gromovcompactification of M , M ∗ respectively. Remark 8.6.
In the general case where W is an arbitrary strong filling, theremight be elements in M\M with multiply covered sphere components. However,since every element in M intersects the cocores once and positively, those multiplycovered components are necessarily disjoint from the cocores. Aspherical case.
By construction we have that each element in the moduli space M intersects each cocore C (cid:48)± transversely in a single point. This then gives a map I + : M → C (cid:48) + , which, by uniqueness of the u t,q ’s, is a diffeomorphism near theboundary. We further note that in the aspherical case M ∗ = M ∗ and hence the(marked) moduli space has a relative fundamental class that we denote [ M ∗ ].Now let S ∼ = S n ⊂ S ∗ ( S n × S x ) ⊂ C (cid:48) + denote a section over the sphere factor,which exists since the (co)tangent bundle of the product S n × S is trivial. Since thissphere can be pushed via an isotopy into the S n +1 -factor of the uncapped boundary ∂W = S n +1 × T , it is zero in the homology of the filling. We let b be a boundingcycle so that ∂b = S in the sphere S n +1 . We now recall the (relative) shriek map ev ! : H ∗ ( W (cid:48) cap , ∂W (cid:48) cap ) → H ∗ ( M ∗ , ∂ M ∗ ), defined as the composition of Poincar´eduality and capping with the fundamental class: ev ! ( a ) = (cid:2) M ∗ (cid:3) ∩ ev ∗ P D ( a ) . Using the fact that the evaluation map is a diffeomorphim near the boundary, weconclude that the diffeomorphic preimage of S in the (marked) moduli space alsobounds ev ! ( b ). Then composing the forgetful map π : M ∗ → M with the inter-section map I + we see that the sphere S ⊂ D ∗ ( S n × S x ), which is homotopic toa factor of the 0-section, bounds ( I + ◦ π ) ∗ ev ! ( b ) in C (cid:48) + , i.e. is null-homologous inthe cotangent bundle of S n × S x . This is obviously absurd, hence the contradictionproves Theorem G in the aspherical case. The general case: polyfold perturbations.
As above we have that each el-ement in the moduli space M intersects each cocore C (cid:48)± transversely in a singlepoint. This then gives a map I + : M → C (cid:48) + , which, by uniqueness of the u t,q ’s,is a diffeomorphism near the boundary. Notice that, as we are in the higher ( ≥ M ∗ as lying inside a Gromov-Witten polyfold B ∗ [HWZ17a,Section 2.2, Definition 2.29, Section 3.5] consisting of (not necessarily holomorphic)63table nodal configurations of spheres with one marked point and possibly multiplecomponents. It comes with a natural evaluation map ev : B ∗ → W (cid:48) cap , which extendsthe one of M ∗ . We view the nonlinear Cauchy-Riemann operator ∂ J as a Fredholmsection [HWZ09, Definition 4.1] of a strong polyfold bundle E ∗ → B ∗ [HWZ17a, Def-inition 2.37, 2.38, Section 3.6] with zero set ∂ − J (0) = M ∗ . We also have a forgetfulmap π : B ∗ → B , where similarly B is a polyfold containing M , given by forgettingthe marked point, and a strong polyfold bundle E → B satisfying π ∗ E = E ∗ , with acorresponding Fredholm section ∂ J making the obvious diagram commute.Since M ∂ := { u t,q } ⊂ M consisting of the spheres in ( − (cid:15), × S ∗ ( S n × S x ) × S is transversely cut out, the section ∂ J is in good position [HWZ09, Definition 4.12]at the boundary.According to [HWZ09, Theorem 4.22], we may introduce an abstract perturba-tion p , which is a multivalued section of E ∗ [HWZ09, Definition 3.35, Definition3.43], so that: • ∂ J + p is transverse to the zero section of E ∗ ; • The perturbed moduli space M p ∗ = ( ∂ J + p ) − (0) ⊂ B ∗ is a (2 n + 4)–dimensional compact, oriented, weighted branched orbifold withboundary and corners [HWZ09, Section 3.2, Definition 3.22]; and • The perturbation p is supported away from a closed neighbourhood (in B ∗ ) of M ∂ ∗ = π − ( M ∂ ).We then define M p ⊂ B to be the image of M p ∗ under the forgetful map π p := π | M p ∗ ,which is a perturbation of M . Observe that the last condition implies that a collarneighbourhood of the boundary of M p still consists of the holomorphic spheres M ∂ ,and local uniqueness still holds.Moreover, for p sufficiently small, every element in M p will be close to some element in M , in the Gromov-Hausdorff topology. Hence we can assume that allelements in M p are transverse to the cocores and have precisely one intersectionpoint, since this intersection is purely topological because of codimension reasons.Thus we obtain a well defined and smooth map I p + : M p → C (cid:48) + , that agrees with I + on M ∂ . Obstruction to the existence of W . Fix now any sufficiently small abstractperturbation p as described above, and consider I p + : M p → C (cid:48) + . Recall also thatin this setting one has a well defined notion of integrating over the moduli space: we64efer to [HWZ10, HWZ17a] for further details. For our purposes it suffices to use thefact that there is a notion of sc-smooth differential forms [HWZ10, Def. 1.8] in M p ∗ as well as a de–Rham cohomology group H ∗ dR ( M p ∗ ) [HWZ10, p. 10], such that saiddifferential forms can be pulled back under the evaluation map, and that Stokes’theorem holds [HWZ10, Thm. 1.11]. In our setting, one then has a well-definednotion of degree of a map, which agrees with the usual notion at the boundary ofthe moduli space, where smoothness is built into the construction. We then formallydefine (relative) homology H ∗ ( M p ∗ , ∂ M p ∗ ) := (cid:0) H ∗ dR ( M p ∗ ) (cid:1) ∗ , as the dual (over R ) of the de–Rham cohomology group, with the obvious notionsof push-forwards, i.e. as duals of pull-backs.This allows us to define a shriek map on homology (with real coefficients). For-mally one does this by defining maps on the duals of the associated de Rham coho-mologies via the integration map, as a map ev ! : H ∗ ( W (cid:48) cap , ∂W (cid:48) cap ) → H ∗ ( M p ∗ , ∂ M p ∗ )given via the equation: ev ! ( a )( β ) = (cid:90) M p ∗ ev ∗ P D ( a ) ∧ β, where we have first used Poincar´e-Lefschetz duality in the target.The remainder of the argument then goes through exactly as in the asphericalsetting yielding Theorem G in the general case. Remark 8.7 (Fundamental classes in the polyfold setting) . According to [HWZ17b,Remark 15.8] (cf. also [McD06]) any compact oriented weighted branched orbifoldadmits a rational (relative) fundamental class. Using this one could then avoid thediscussion of smoothness of the maps used above, as in this case continuity, whichis obvious, suffices to replicate the argument from the aspherical case.
Our results are, together with [LMN18], among the first steps in understanding thenature of the contact structures given by Bourgeois’ construction, and several openquestions remain. Firstly:
Question 1.
Are Bourgeois contact structures tight in all odd dimensions? More-over, is every Bourgeois contact structure weakly fillable, at least in dimension 5?It is an important problem to understand more precisely the dependence of theBourgeois structure on the starting open book decomposition. By a direct conse-quence of their definition, all Bourgeois contact structures are contact deformations65f the almost contact structure ξ V ⊕ T T (i.e. the endpoint η of a path ( η t ) t ∈ [0 , ofhyperplane fields starting at η = ξ V ⊕ T T and such that η t is contact for t > BO (Σ , φ ) and BO (Σ (cid:48) , φ (cid:48) ) for any OBD (Σ (cid:48) , φ (cid:48) ) and OBD (Σ , φ ) supporting the samecontact structure. Besides sharing the formal homotopy class, Theorem A in thispaper shows in particular that the tight vs overtwisted classification type of any5-dimensional BO (Σ , φ ) is independent of the open book.On the other hand, in [Bou02a, Corollaries 10.6 and 10.8], Bourgeois used cylin-drical contact homology with respect to noncontractible homotopy classes of Reeborbits, in order to distinguish infinitely many Bourgeois contact manifolds arisingfrom open books supporting the standard contact structure on S ; and similarly for T . Further instances of different open books supporting the same contact structurethat induce non-contactomorphic Bourgeois contact manifolds can also be found in[LMN18, Example 1.5]. Question 2.
Can we find further contactomorphisms of Bourgeois contact mani-folds, beyond the inversion of the monodromy from [LMN18]? More ambitiously,can we classify the contactomorphism type of all the Bourgeois contact manifoldsarising from some fixed contact structure, especially via rigid holomorphic curvesinvariants?A further interesting question relates to
Giroux torsion . This is a standardnotion in dimension 3, and was generalized to higher dimensions in [MNW13]. Itwas shown in [MNW13] to be an obstruction to strong fillability, and it was provenin [Mor17b, Theorem 1.7] that it can be detected by an SFT-type contact invariant, algebraic torsion (defined in [LW11]). Theorem A in this paper can be summarizedas: the Bourgeois construction “kills” overtwistedness. One can wonder if it also“kills” Giroux torsion.
Question 3.
Does there exist a Bourgeois contact manifold with Giroux torsion?An affirmative answer to Question 3 would provide an obstruction to strongfillability, but it would not immediately provide an obstruction to weak fillability,since Giroux torsion only provides obstructions to weak fillability under suitablecohomological conditions (see [MNW13, Corollary 8.2] or [Mor17b, Corollary 1.8]).We can also approach the question for fillability via subgroups of the mappingclass group. Let Σ be a surface with boundary, and denote
F ill (Σ) the set of elements ψ of M CG (Σ) such that BO (Σ , ψ ) is strongly fillable. From Theorem 3.1 and[LMN18, Theorem B], we know that F ill (Σ) is a subgroup of
M CG (Σ). Moreover,since monodromies are defined up to conjugacy, it is a normal subgroup.
Question 4.
What can we say about
F ill (Σ)?66 emark 9.1. (Obstructions to gluing weak cobordisms) Observe that in order toshow that Bourgeois contact structures are tight – at least in dimension 5 –, one couldbe tempted to argue as follows. By decomposing the monodromy into powers ofpositive and negative Dehn twists, and using the cobordism (
C, ω C ) of Theorem 3.1,we end up with having to find fillings of BO (Σ , τ ± ) where τ is a single Dehn twist.Notice that BO (Σ , τ ) is indeed weakly fillable, according to [LMN18, TheoremA.(a)]. More precisely, one can arrange the cohomology class of the symplectic forminduced at the boundary of the filling to be (cid:15) [ ω T ], where ω T is an area form on T and (cid:15) >
0. Since (
C, ω C ) is exact at each end, one can then consider the perturbedversion ( C, ω C + (cid:15)ω T ), which has weakly dominated boundary. Then, [MNW13,Lemma 1.10] guarantees that the weak filling of BO (Σ , τ ) glues to this perturbedversion. Now, in order to find an analogous filling for the case of BO (Σ , τ − ), onewould hope to appeal to [LMN18, Theorems A.(a) and B]. However, by carefullytracing the signs from the contactomorphism provided in [LMN18], one can seethat the corresponding weak filling does not glue correctly to ( Q, Ω + (cid:15)ω T ). Moreprecisely, the contactomorphism swaps the orientation of the T -factors. Of course,in the case where the monodromy is a product of powers of Dehn twists of the samesign, then indeed we do find weak fillings. A Construction of finite energy foliations
In this appendix, we fill in the details of Section 7.1. We first recall some notation.Consider (Σ n − , λ ) a Liouville domain, ψ a symplectomorphism of Σ restrictingto Id near B = ∂ Σ and satisfying ψ ∗ λ = λ − dh for h : Σ → [0 , ∞ ) some smoothfunction which vanishes near B . Denote the corresponding mapping torus by Σ ψ , and let V n − = OBD (Σ , ψ ) be the manifold with the corresponding open book.Let Φ = (Φ , Φ ) = ( ρ cos θ, ρ sin θ ) = ( p , − p ) be the defining map, so that θ =Φ / | Φ | : V \ B → S is the open book fibration. Let α be a Giroux form forthe contact structure ξ = ker α supported by the open book in V . Denote by λ B the restriction of λ to the binding B = ∂ Σ. Consider M = V × T , andlet β = α + Φ dq − Φ dq be the Bourgeois form associated to ( α, Φ) inducingthe Bourgeois contact structure ξ BO = ker β , where ( q , q ) ∈ T . We consider thespinal open book decomposition (SOBD) for M as described at the end of Section 2,for which β is a Giroux form (Definition 2.4). In particular, we have the spine M S = B × D ∗ T , the paper M P = Σ ψ × T , and an interface region between themof the form M I = B × [ − (cid:15), (cid:15) ] × T . We also have the structural SOBD fibrations π S : M S → D ∗ T and π P : M P → S ∗ T = T .67 � ν H( � ) M P M S M I M I (M, ξ ) � O H (M, ) A M S M P (Y, ω ) Y h ������ Li ������� fl�� 𝛿 � - 𝛿 � XE X SH Figure 10: The symplectic cobordism (
Y, ω Y ), corresponding to the shaded region,a bounded domain inside the double completion E ∞ , ∞ . Double completion and symplectic cobordism
In this section, we prove Lemma 7.1, which we now restate.
Lemma A.1.
There exists a topologically trivial symplectic cobordism ( Y, ω Y ) , hav-ing the Bourgeois contact structure ( M, ξ BO ) as concave boundary component, anda suitable stable Hamiltonian manifold ( M, H A ) as positive stable boundary. The stable Hamiltonian structure H A is the one used in Section 7.1 to obtain aholomorphic foliation, which we construct in the following section. The proof followsby adapting a construction in [Mor17a] (see also [LVHMW18, AM18]). The pictorialreference is Figure 10.Consider a cylindrical end N ( B ) = ( − δ, + ∞ ) × B ⊂ (cid:98) Σ, with coordinates ( t, b ) ∈N ( B ), so that λ = e t λ B . Similarly, consider N ( T ) = (1 − δ, + ∞ ) × S × T ⊂ T ∗ T ,with coordinates ( r, θ, q ) ∈ N ( T ), so that λ std = e r λ T , where λ T denotes thestandard contact form in T . This gives completions of the spine M S and paper M P ,which we denote by (cid:99) M S = B × T ∗ T and (cid:99) M P (a (cid:98) Σ-fibration over T ) respectively.Consider the completed mapping torus abstractly given by (cid:98) Σ ψ = { ( x, ϕ ) ∈ (cid:98) Σ × R : 0 ≤ ϕ ≤ h ( x ) } / ( ψ ( x ) , ∼ ( x, h ( x )) , ψ to (cid:98) Σ. We denote points in (cid:98) Σ ψ by[ x, ϕ ], and we have θ ([ x, ϕ ]) = ϕ/h ( x ) ∈ S . Let λ ψ be a fiberwise Liouville form on (cid:98) Σ ψ which coincides with λ on each fiber, and consider the fiberwise Liouville vectorfield X λ associated to λ ψ . We also have a corresponding completed paper piece (cid:99) M P = (cid:98) Σ ψ × T , and the subset N P ( B ) = N ( B ) × S × T ⊂ (cid:99) M P with coordinates( x = ( t, b ) , θ, q ). We consider the “double completion”, defined as the following open2 n -manifold: E ∞ , ∞ = (1 − δ, + ∞ ) × (cid:99) M P (cid:71) ( − δ, + ∞ ) × (cid:99) M S (cid:46) ∼ , where we identify ( r, t, b, θ, q ) ∈ (1 − δ, + ∞ ) × N P ( B ) ⊂ (1 − δ, + ∞ ) × (cid:99) M P with( t, b, r, θ, q ) ∈ ( − δ, + ∞ ) × B × N ∞ ( T ) ⊂ ( − δ, + ∞ ) × (cid:99) M S . See Figure 10.We may view E ∞ , ∞ as a “fibration” over ( T ∗ T , λ std ), but where the fibers changetopological type: they are copies of (cid:98) Σ over the cylindrical end (1 − δ, + ∞ ) × T ⊂ T ∗ T , and cylindrical ends ( − δ, + ∞ ) × B over { r ≤ − δ } ⊂ T ∗ T . We denote thismap by π E .For K (cid:29) σ = σ K : [0 , + ∞ ) → [0 , + ∞ ) satisfying: • σ ≡ , − δ ]. • σ ≡ K on [2 , + ∞ ). • σ (cid:48) > − δ, σ as a function on E ∞ , ∞ by identifying it with σ ◦ r , where the coordinate r is also viewed as a function.We view the 1-form λ ψ as a 1-form on E ∞ , ∞ , and we define λ E = λ ψ + σπ ∗ E λ std ∈ Ω ( E ∞ , ∞ ) . By construction, as is straightforward to check, this is a Liouville form on E ∞ , ∞ ,i.e. the 2-form ω E = dλ E is symplectic. The associated Liouville vector field is X E = X λ + V σ , where V σ = f ( r ) ∂ r is the Liouville vector field associated to theLiouville form σλ std , for f = σσ + σ (cid:48) >
0. In particular, it agrees with ∂ t + V σ in thesubregion of E ∞ , ∞ where the variable t is defined.We have the following preferred copies of M = V × T lying in E ∞ , ∞ as hyper-surfaces with corners: M − := ( { t ≤ } ∩ { r = 1 } ) ∪ ( { t = 0 } ∩ { r ≤ } ) ⊂ E ∞ , ∞ M + := ( { t ≤ } ∩ { r = 2 } ) ∪ ( { t = 1 } ∩ { r ≤ } ) ⊂ E ∞ , ∞ . F, G) - - δ - r � ρ - δ Figure 11: Our choice of corner smoothing.Here, by convention, whenever we write { t ≤ c } , we also include the region where t is not defined.We now smooth all corners. Choose smoothing functions F, G : ( − δ, δ ) → ( − δ, ( F ( ρ ) , G ( ρ )) = ( ρ, , for ρ ≤ − δ/ G (cid:48) ( ρ ) < , for ρ > − δ/ F (cid:48) ( ρ ) > , for ρ < δ/ F ( ρ ) , G ( ρ )) = (0 , − ρ ) , for ρ ≥ δ/ . See Figure 11. We replace the region M − ∩ ( { t ≥ − δ } ∪ { r ≥ − δ } ) ⊂ E ∞ , ∞ , whichcontains the corner M − ∩ { t = 0 , r = 1 } , with the region M − I := { ( r, t ) = ( F ( ρ ) + 1 , G ( ρ )) : ρ ∈ ( − δ, δ ) } × B × S × T ⊂ E ∞ , ∞ Similarly, we replace M + ∩ ( { t ≥ − δ } ∪ { r ≥ − δ } ) ⊂ E ∞ , ∞ with M + I := { ( r, t ) = ( F ( ρ ) + 2 , G ( ρ ) + 1) : ρ ∈ ( − δ, δ ) } × B × S × T ⊂ E ∞ , ∞ Then M − = M − S (cid:91) M − I (cid:91) M − P , where M − P = M − ∩ ( { t ≤ − δ } ∩ { r = 1 } ), and M − S = M − ∩ ( { t = 0 } ∩ { r ≤ − δ } ).We have a similar decomposition for M + . Observe that these two decompositionsare nothing else than the Bourgeois SOBD associated to M .Consider the Morse-Bott Hamiltonian function H : D ∗ T → R given by H ( q, p ) = −| p | + 12 . All of its critical points lie in the zero section T , which is a Morse-Bott submanifold,and its flow lines are the radial lines (where we take the standard flat metric on T ).70 |p| Figure 12: The smoothened Morse-Bott function H .The function H is naturally defined on M S , where it is B -independent, and aftersmoothening (see Figure 12) we extend it to M so that it vanishes as one approaches M P along the interface M I .Let φ sE denote the time- s flow of the Liouville vector field X E , choose a small ν ≥
0, and let M + ,ν = { φ νH ( p ) E ( p ) : p ∈ M + } ⊂ E ∞ , ∞ . This is a ν -perturbation of M + inside E ∞ , ∞ , in the Liouville direction, diffeomor-phic to M . Denote by M + ,νS the piece of M + ,ν corresponding to M + S under theperturbation, and similarly for M + ,νI , corresponding to M + I .Define ( Y, ω Y ) to be the compact and connected symplectic submanifold of( E ∞ , ∞ , ω E ) bounded by M − (cid:70) M + ,ν . It is clearly trivial as a smooth cobordism.See Figure 10.By construction, X E is inwards pointing to Y at M − , and so ( M − , ξ − = ker λ E | M − )is a concave contact-type boundary component of Y , whose contact structure is sup-ported by the Bourgeois SOBD and isotopic to ( M, ξ BO ).We now construct the stable Hamiltonian structure H A at M + ,ν , as follows. Picksome small 0 < (cid:15) < δ , and deform the Liouville vector field X E to a vector field X SH of the form X SH = γ ( r, t ) X λ + V σ , where γ : (1 − δ, + ∞ ) × ( − δ, + ∞ ) → R is a smooth function satisfying: • γ ≡ { r ≥ − (cid:15), t ≤ − δ + (cid:15) } . • γ ≡ { ≤ r ≤ (cid:15), t ≤ } (cid:83) { t ≥ − (cid:15) } . • ∂ r γ < ∂ t γ ≡ { (cid:15) < r < − (cid:15), t < − (cid:15) } . • ∂ t γ > ∂ r γ ≡ {− δ + (cid:15) < t < − (cid:15) } .71 � � δ - - γ ( � , � ) M P Figure 13: The bump function γ .See Figure 13 and Figure 10.By construction, X SH coincides with X E away from a neighbourhood of M + P inside Y , and with V σ in a smaller such neighbourhood.We define H A := H νA := ( β ν , Ω ν ) := ( i X SH ω E | M + ,ν , ω E | M + ,ν )One may explicitly check that X SH is a stabilizing vector field near M + ,ν , and so H A is indeed a SHS, which is in fact a confoliation (see below for explicit formulas).This concludes the construction which proves Lemma A.1. (cid:3) Construction of codimension-4 holomorphic foliation
In this section, we carry out the construction of the foliation described in Section 7.1,in all dimensions, by an adaptation of the construction of the model A foliation in[Mor17a] to the Bourgeois supporting SOBD.On M , consider the SHS H A = H νA = ( β ν , Ω ν ) of Lemma A.1, for ν ≥
0, asconstructed in the previous section. Denote its kernel by ξ ABO = ker β ν (which is ν -independent), and its Reeb vector field by R ν . Symplectic connection.
On the symplectic fibration (cid:98) π P : (cid:99) M P → T , there is anatural homotopy class of symplectic connections so that the parallel transport isthe flow associated to the monodromy ψ . This flow is generated by the horizontallift of ∂ θ ∈ T S , which we denote by X = (cid:101) ∂ θ . In other words, we have a splitting T (cid:99) M P = Hor ⊕ Vert , where Vert = ker dπ P is the vertical distribution, and Hor = (cid:104) X (cid:105) ⊕ T T ∼ = T T isthe symplectic connection, the isomorphism being induced by π P .72 xplicit formulas. Let us write explicit expressions for the data associated to H A . Along M + ,νI , we write the time- νH flow of X SH = ∂ t + V σ in coordinates as r (cid:55)→ Φ V σ ( νH ( r ) , r ) t (cid:55)→ t + νH ( r ) , where Φ V σ ( s, · ) is the time s flow of V σ . The r and t coordinates on M + ,νI are then r = Φ V σ ( νH ( F ( ρ ) + 2) , F ( ρ ) + 2) =: F ν ( ρ ) t = G ( ρ ) + 1 + νH ( F ( ρ ) + 2) =: G ν ( ρ )and so ∂ r = ∂ ρ F (cid:48) ν , ∂ t = ∂ s G (cid:48) ν We may then drop the superscripts from the notation for M + ,ν , and work intrinsi-cally on M . We write β ν = e νH ( r ) ( eλ B + λ std ) , on M S ,γ ( F ν ( ρ ) , G ν ( ρ )) e G ν ( ρ ) λ B + σ ( F ν ( ρ )) e F ν ( ρ ) λ T , on M I ,Kλ T , on M P . where λ T is the standard contact form on T induced by λ std = e r λ T at r = 1, andΩ ν = d (cid:0) e νH ( r ) ( λ B + λ std ) (cid:1) , on M S ,d (cid:0) e G ν ( ρ ) λ B + σ ( F ν ( ρ )) e F ν ( ρ ) λ T (cid:1) , on M I ,dλ ψ + Kdλ T , on M P . It follows that ξ ABO = ker β ν = ξ B ⊕ (cid:98) T D ∗ T , on M S ,ξ B ⊕ (cid:101) ξ T ⊕ (cid:104) v , v (cid:105) , on M I , Vert ⊕ (cid:101) ξ T , on M P , (8)where ξ B = ker λ B , (cid:98) T D ∗ T := { (cid:98) v := v − λ std ( v ) e − R B : v ∈ T D ∗ T } ∼ = T D ∗ T (theisomorphism being induced by π S ), (cid:101) ξ T = { (cid:101) w : w ∈ ξ T } is the horizontal lift of ξ T = ker λ T with respect to the symplectic connection Hor, v = ∂ ρ , and v = v (cid:48) Ω ν ( v , v (cid:48) ) = A ( ρ ) R T + B ( ρ ) R B , where R T and R B are respectively the Reeb vector fields of λ T and λ B and v (cid:48) = γ ( F ν ( ρ ) , G ν ( ρ )) e G ν ( ρ ) R T − σ ( F ν ( ρ )) e F ν ( ρ ) R B . ν ( v , v (cid:48) ) = e F ν + G ν ( γ ( F ν , G ν ) F (cid:48) ν ( σ (cid:48) ( F ν ) + σ ( F ν )) − σ ( F ν ) G (cid:48) ν ) := Φ( ρ )is strictly positive, and A ( ρ ) = γ ( F ν , G ν ) e G ν Φ ( ρ ) , B ( ρ ) = − σ ( F ν ) e F ν Φ ( ρ )Also, the following expressions hold: R ν = e − νH ( r ) ( R B − ν (cid:100) X H ) , on M S .U ( ρ ) R B + V ( ρ ) R T , on M I .R T /K, on M P . Here, X H is the Hamiltonian vector field of H (defined by i X H dλ std = − dH ), (cid:100) X H isits lift to ξ ABO as defined above, and
U, V : ( − δ, δ ) → R are suitable smooth functionsdepending on F, G, γ, σ and their derivatives.
Remark A.2. R ν coincides with R B along the binding { r = 0 } , and so Reeb orbitsin B are Reeb orbits in M of action 2 πe νH (0) a B , where a B is their action as λ B -orbits. Remark A.3.
Given any fixed action threshold T (cid:29)
0, by choosing K sufficientlylarge, and by choosing σ suitably along M I , we can arrange that all Reeb orbitswhose action is below T is a multiple cover of a binding orbit (as follows by inspectingthe above expression for R ν ). Almost complex structure.
We now construct a H A -compatible almost complexstructure J ν on the symplectization ( R a × M, ω ϕ = d ( ϕ ( a ) β ν )+Ω ν ) of ( M, H A ). Here, ϕ : R → ( − (cid:15), (cid:15) ) satisfies ϕ (cid:48) >
0, and (cid:15) > ω ϕ is symplectic.The almost complex structure J ν has to map ∂ a to R ν , and be R -invariant, so weneed only specify it along ξ ABO .Take an almost complex structure J on (cid:98) Σ which is dλ -compatible and cylindricalalong the cylindrical end N ( B ) ⊂ (cid:98) Σ, and denote its restriction to ξ B along N ( B )by J B . Let J = ψ ∗ J , which is also dλ -compatible and coincides with J on N ( B ),and so is cylindrical. Since the space of dλ -compatible almost complex structureswhich coincide with J along N ( B ) is contractible, we can find a path { J t Σ } t ∈ [0 , of dλ -compatible almost complex structures, which joins J to J , and is t -independenton N ( B ). We may then define a fiber-wise compatible almost complex structure J Σ on (cid:98) Σ ψ by J Σ | [ x,ϕ ] = J ϕ/h ( x )Σ . It is well-defined by construction.Choose a λ std -compatible almost complex structure J on T ∗ T , which agreeswith the standard integrable one along { r ≤ − δ } , and which is cylindrical and74 T -compatible along the cylindrical end { r ≥ − δ } . Denote by J T its restrictionto ξ T along the latter. Observe that J is automatically compatible with σλ std .Along ξ ABO | M I , we make the ansatz J ν ( v ) := g ( ρ ) v , for some smooth positive function g : ( − δ, δ ) → R + to be specified, and we denote J I = J ν | (cid:104) v ,v (cid:105) .Using the splitting (8), we define J ν | ξ ABO := J B ⊕ (cid:98) J , on M S .J B ⊕ (cid:101) J T ⊕ J I , on M I .J Σ ⊕ (cid:101) J T , on M P , where (cid:98) J (cid:98) v := (cid:100) J v defines (cid:98) J as the lift of J via π S , and (cid:101) J T (cid:101) w := (cid:103) J T w defines thehorizontal lift of J T to (cid:101) ξ T .It remains to determine the function g , which we do by interpolating betweenthe two definitions in M S and M P . Along M S ∩ M I , we have J ν ( v ) = (cid:98) J ( F (cid:48) ν ∂ r ) = F (cid:48) ν (cid:98) R T = F (cid:48) ν A ( AR T + BR B ) = F (cid:48) ν A v . Similarly, along M P ∩ M I , we have γ = 0 and so v = − KR B , and hence J ( v ) = J Σ ( G (cid:48) ν ∂ t ) = G (cid:48) ν R B = − G (cid:48) ν K v . We observe that, in both cases, J ( v ) = g ( ρ ) v for some strictly positive function g .We then extend those functions to any choice of positive function g : ( − δ, δ ) → R + interpolating between g = F (cid:48) ν A near ρ = − δ and g = − G (cid:48) ν K near ρ = δ . This finishesthe construction of J ν .We leave it for the reader to check that J ν is compatible with H A (namely thatΩ ν ( · , J ν · ) is a J ν -invariant metric on ξ ABO , cf. [Mor17a, p. 45-48]).
The holomorphic foliation.
Since Vert is an integrable distribution, we have a J ν -holomorphic foliation of R × M P of the form F P = {F ( a, c ) P } ( a, c ) ∈ R × S × T where c = ( θ, q ) ∈ S × T and F ( a, c ) P := { a } × Σ θ × { q } , for Σ θ the θ -page of theopen book. Moreover, one can extend the foliation F P to a holomorphic foliation of R × M by attaching cylindrical ends to its leaves, as follows.75or c = ( θ, q ) ∈ S × T , define γ ν c : [0 , + ∞ ) → D ∗ T , γ ν c ( t ) = ( q, e − νt , θ ) . Then γ ν c (0) = ( q, , θ ) and γ ν c is a positive flow-line of νH , i.e. a solution of ˙ γ ( t ) = ν ∇ H ( γ ( t )) (for the standard flat metric). Moreover, lim t → + ∞ γ ν c ( t ) = ( q, ∈ T ×{ } lies on the zero section. Let also f νa : [0 , + ∞ ) → R be a solution to the equation ˙ f νa ( t ) = e νH ( γ ( t )) , with f νa (0) = a . Define F S := {F ( a, c ) S } ( a, c ) ∈ R × S × T (cid:91) {F q } q ∈ T where F ( a, c ) S := B × { ( f νa ( t ) , γ ν c ( t )) : t ∈ [0 , + ∞ ) }F q = R × B × { q } . It is easy to check that this is a J ν -holomorphic foliation of R × M S . Indeed, letting g ( t ) := ( f ( t ) , γ ( t )) := ( f νa ( t ) , γ ν c ( t )), we have ∂ t g ( t ) = f (cid:48) ( t ) ∂ a + γ (cid:48) ( t ) = e νH ( γ ( t )) ∂ a + ν (cid:100) ∇ H ( γ ( t )) , where we have used that (cid:100) ∇ H ( γ ( t )) = ∇ H ( γ ( t ))(= − r∂ r ) coincides with its lift to ξ ABO . Then, the claim follows from the fact that J ν ∂ t g ( t ) = e νH ( γ ( t )) R ν + ν (cid:100) X H ( γ ( t )) = R B − ν (cid:100) X H ( γ ( t )) + ν (cid:100) X H ( γ ( t )) = R B . Observe that a leaf F ( a, c ) S has as many connected components as B = ∂W ,and glues smoothly to the leaf F ( a, c ) P of F P , defined over M P . In fact, F ( a, c ) S is acylindrical end of F ( a, c ) P , and one checks that it is possible to glue in a holomorphicmanner. Indeed, along M I , the vector fields R ν and J ν v = gv are both linearcombinations of R T and R B for coefficients only depending on ρ , and they are notcolinear. So, we have R B = C ( ρ ) R ν + D ( ρ ) J ν v , for some smooth functions C, D : ( − δ, δ ) → R . Then, J ν R B = − C∂ a − Dv = − C∂ a − D∂ ρ We conclude that
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School of Mathematical Sciences, Monash University, Melbourne, Australia.
Email : [email protected]´ed R´enyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest,Hungary.
Email : [email protected], [email protected] f¨ur Mathematik, Universit¨at Augsburg, Augsburg, Germany.