Generalizations of planar contact manifolds to higher dimensions
GGENERALIZATIONS OF PLANAR CONTACT MANIFOLDSTO HIGHER DIMENSIONS
BAHAR ACU, JOHN B. ETNYRE, AND BURAK OZBAGCIA
BSTRACT . Iterated planar contact manifolds are a generalization of three dimen-sional planar contact manifolds to higher dimensions. We study some basic topo-logical properties of iterated planar contact manifolds and discuss several exam-ples and constructions showing that many contact manifolds are iterated planar.We also observe that for any odd integer m > , any finitely presented group canbe realized as the fundamental group of some iterated planar contact manifoldof dimension m . Moreover, we introduce another generalization of three dimen-sional planar contact manifolds that we call projective. Finally, building symplecticcobordisms via open books, we show that some projective contact manifolds admitexplicit symplectic caps.
1. I
NTRODUCTION
The groundbreaking work of Giroux [26], that characterized contact manifoldsin terms of open book decompositions with symplectic pages, has had a remark-able impact in the study of the global topology of contact manifolds. By studyingthe properties of the pages of an open book, one can learn a lot about the adaptedcontact structure. For example, given a contact –manifold ( M, ξ ) , one may askwhat the minimal genus is for an open book that supports ξ . This is the supportgenus sg ( ξ ) of ξ , [21]. If sg ( ξ ) = 0 , then the contact structure ξ and the contact –manifold ( M, ξ ) , as well as the supporting open book, are all called planar. Sofar there is a great deal known about planar contact manifolds (in dimension 3, bydefinition), which we itemize below.(1) All overtwisted contact structures are planar and hence a contact structurewith positive support genus is tight, [20].(2) Any (weak) symplectic filling ( X, ω ) of a planar contact manifold must haveconnected boundary, and b +2 ( X ) = b ( X ) = 0 , [20].(3) In particular, any planar contact manifold has a symplectic cap that containsa symplectic sphere of positive self-intersection and any symplectic filling Mathematics Subject Classification.
Primary 57R17; Secondary 53D35.
Key words and phrases. open book decompositions, Lefschetz fibrations, iterated planar contactmanifolds. a r X i v : . [ m a t h . S G ] J un BAHAR ACU, JOHN B. ETNYRE, AND BURAK OZBAGCI of a planar contact manifold embeds as a convex domain in a rational sur-face.(4) Planar contact manifolds do not embed as weak nonseparating contact-typehypersurfaces in closed symplectic –manifolds (cf. [5, 39]).(5) The Weinstein conjecture is true for planar contact manifolds, [1]. Moregenerally, it is now known to be true for all contact –manifolds [44], butthe conjecture was first established for planar contact manifolds.(6) Given a symplectic filling of a contact manifold, a planar open book sup-porting the contact structure can be extended to a Lefschetz fibration overthe symplectic filling, [39, 48].(7) The previous fact allows us to classify fillings of certain planar contact man-ifolds [29, 34, 42].(8) Any weak symplectic filling of a planar contact manifold can be deformedinto a Stein filling [39, 48].(9) Planarity of a contact manifold can be obstructed by the contact element inHeegaard Floer homology, [41].In this article, we discuss two notions that can be thought of as generalizationsof planar open books to higher dimensions: • iterated planar open books and • projective open books.We will mainly focus on basic properties and examples of these open books, andmore importantly, the contact manifolds supported by such open books. A longterm goal is to see how many of the items above can be proven for these specialcontact manifolds in higher dimensions. As there is no notion of Heegaard Floertheory in higher dimensions, we will say nothing more about the last item, andconcentrate on the others.1.1. Iterated planar open books and adapted contact manifolds.
Iterated planarcontact manifolds (which are of dimension greater than three, by definition) wereintroduced by the first author in [3], as a natural generalization of planar contactmanifolds. Briefly, an open book is called iterated planar, if its page has an iteratedplanar Lefschetz fibration, and the notion of iterated planarity for a Lefschetz fi-bration is defined inductively by saying that its fibers are iterated planar, with thebase case being a –dimensional fiber that is a planar surface. See Section 2.2 for aprecise definition. A contact manifold supported by an iterated planar open bookis called iterated planar.To date only a few of the above results are known for iterated planar contactmanifolds. The first substantial result for iterated planar contact manifolds wasthe verification of the Weinstein conjecture by the first author. TERATED PLANAR CONTACT MANIFOLDS 3
Theorem 1.1 (Acu 2017, [3]) . If ( M, ξ ) is an iterated planar contact manifold then anyReeb vector field for ξ has a periodic orbit. Another proof of this result was given by the first author and Moreno, who alsoproved the following results.
Theorem 1.2 (Acu-Moreno 2018, [4]) . Any symplectic filling of an iterated planar con-tact manifold must have connected boundary.
Theorem 1.3 (Acu-Moreno 2018, [4]) . An iterated planar contact manifold cannot be anonseparating weak contact-type hypersurface in any closed symplectic manifold.
Items (4), (5), and the first part of Item (2) are known to be true for iterated planarcontact manifolds.So far there has been little systematic investigation of how common it is for acontact manifold to be iterated planar. In Section 3, we show that there are iteratedplanar contact manifolds in all odd dimensions (greater than three) and we alsoshow that there is no obstruction to iterated planarity coming from the fundamen-tal group.
Theorem 1.4.
Let G be a finitely presented group. Then for each n ≥ , there is an iteratedplanar contact (2 n + 1) –manifold whose fundamental group is G . Remark 1.5.
It is clear that for n ≥ , any finitely presented group G can be realizedas the fundamental group of a contact (2 n + 1) –manifold, since there exists a closedorientable ( n +1) –manifold X with π ( X ) = G and the unit cotangent bundle ST ∗ X equipped with its canonical contact structure is a contact (2 n + 1) –manifold with π ( ST ∗ X ) ∼ = π ( X ) ∼ = G. This argument obviously breaks down for n = 2 . Nevertheless, A’Campo andKotschick [2] observed that any such G can be realized as the fundamental groupof a contact –manifold, via performing surgery on the contact –manifold k ( S × S , ξ std ) , where k is the number of generators in the presentation of G and ξ std is induced by the Stein –manifold S × B . Notice that Theorem 1.4 gives analternate uniform proof of these results as well using a different method.Iterated planarity is preserved under connected sums. Theorem 1.6.
The connected sum of iterated planar contact manifolds is iterated planar.
We now turn our attention to contact –manifolds. We begin by recalling thatin general, a subcritical Stein filling of a contact (2 n − –manifold is a Stein fillingthat does not have handles of index n . BAHAR ACU, JOHN B. ETNYRE, AND BURAK OZBAGCI
Theorem 1.7.
The following subcritically Stein fillable contact –manifolds are iteratedplanar. (1) Consider a –manifold M with a finite cyclic fundamental group (including thetrivial group). If the order of the group is odd, then any subcritically Stein fillablecontact structure on M is iterated planar. If the order of the group is even, thenat least half of the subcritically Stein fillable contact structures on M are iteratedplanar. (2) The unique contact structure admitting a symplectically aspherical subcritical Steinfilling on S × S is iterated planar. (3) If ( M, ξ ) is a subcritically Stein fillable contact –manifold, then M admits a sub-critically Stein fillable iterated planar contact structure ξ (cid:48) in the same homotopyclass of almost contact structures. Moreover, if M has infinite second cohomologythen it has infinitely many such contact structures. Remark 1.8.
We note that a –manifold M can have a subcritically Stein fillablecontact structure ξ and another contact structure ξ (cid:48) that is not subcritically Steinfillable. In general, we can say nothing about the iterated planarity of ξ (cid:48) , but weexpect there to be cases where it is not.For example, let X be the Stein manifold obtained form B by attaching a Stein –handle along the maximal Thurston-Bennequin right-handed Legendrian torusknot in ( S , ξ std ) , and let ( M, ξ ) be the boundary of the subcritical Stein manifold X × D . We observe that the open book with page X and trivial monodromy,that supports ( M, ξ ) , is not iterated planar because otherwise the planar contactmanifold ∂X would have a filling X with b ( X ) = 1 , which contradicts to theresult discussed in Item (2) above. Although, this does not yet prove that the con-tact –manifold ( M, ξ ) is not iterated planar, we present some further supportingevidence. Notice that any symplectically aspherical filling of ( M, ξ ) must be dif-feomorphic to X × D by [8, Theorem 1.5], and it is hard to see how any open bookinduced from a Lefschetz fibration on X × D could be iterated planar. Thus weconjecture that ( M, ξ ) is not iterated planar. One can clearly create infinitely manyother such examples as well by attaching a Stein –handle to B along any Legen-drian knot in ( S , ξ std ) with Thurston-Bennequin invariant larger than zero. Thisdiscussion leads to the following question. Question 1.9.
Are all subcritically Stein fillable contact manifolds iterated planar? Isthere some criteria that determines when they are?
We note that in dimension 3, a subcritically Stein fillable contact structure is theconnect sum of some number of copies of S × S with its unique tight contactstructure. This is well-known to be planar, so the above question asks if this obser-vation generalizes to higher dimensions. TERATED PLANAR CONTACT MANIFOLDS 5
On the other hand, there are many non-subcritically Stein fillable contact mani-folds which are iterated planar.
Theorem 1.10. If ( M , ξ ) is an iterated planar contact –manifold with c ( ξ ) = 0 , andhas a flexible Stein filling, then it admits infinitely many iterated planar contact structureswhich are filled by Stein manifolds that cannot be subcritical. A Stein manifold is flexible if it admits a handle decomposition with all the crit-ical handles attached along loose Legendrian spheres [38]. In particular, a subcrit-ical Stein manifold is flexible, by definition. Thus combining Theorem 1.7 (3) andTheorem 1.10, we conclude that if a –manifold M with infinite second cohomol-ogy admits a subcritically Stein fillable contact structure with c = 0 , then there areinfinitely many iterated planar contact structures on M which are not subcriticallyStein fillable. There are many –manifolds satisfying these assumptions. For exam-ple, if X is the result of attaching a Stein –handle to B along a Legendrian knotwith rotation number zero, then the boundary of the subcritical Stein -manifold X × D is a contact –manifold with infinite second cohomology such that c = 0 .We now observe that Item (8) above does not hold for iterated planar contactmanifolds. Proposition 1.11.
There are iterated planar contact manifolds that are strongly symplec-tically fillable but not Stein fillable.
Turning to the opposite end of the spectrum we consider overtwisted contactstructures.
Theorem 1.12. If M is a –manifold with finite cyclic fundamental group of odd order(including the trivial group) that is the boundary of a subcritical Stein domain, then everyovertwisted contact structure on M is iterated planar.More generally, if M is any –manifold that admits a subcritically fillable contact struc-ture, then it admits an overtwisted contact structure that is iterated planar. If H ( M ; Z ) is infinite, then there are infinitely many homotopy classes of almost contact structures forwhich every overtwisted contact structure in any of these classes is iterated planar. We note that Theorem 1.12, for example, says that any overtwisted contact struc-ture on • S , • S × S , • S × S , or • S (cid:101) × S (the total space of the nontrivial S –bundle over S )is iterated planar. This gives some evidence that the analog of Item (1) may be true,so we ask: BAHAR ACU, JOHN B. ETNYRE, AND BURAK OZBAGCI
Question 1.13.
Is every overtwisted contact manifold iterated planar?
Another natural class of –manifolds to consider is the simply-connected ones.In [7], Barden gave a classification of closed, oriented, smooth, simply-connected –manifolds, up to diffeomorphism. Note that the vanishing of the third integralStiefel-Whitney class W ( M ) is a necessary condition for the existence of an al-most contact structure on a (not necessarily simply-connected) –manifold M , andMassey [36] showed that it is also sufficient. Moreover, Geiges [22] proved thatif M is simply-connected, then it admits a contact structure in every homotopyclass of almost contact structures. These results imply that any simply-connected –manifold which carries a contact structure can be uniquely decomposed into theconnected sum of prime manifolds M k for ≤ k ≤ ∞ , with possibly one extrasummand X ∞ = S (cid:101) × S . Here M = S , M ∞ = S × S , and M k is characterized bythe property that H ( M k ; Z ) = Z k ⊕ Z k .The results above imply that all overtwisted contact structures on M = S , M ∞ = S × S , and X ∞ = S (cid:101) × S are iterated planar and each of these manifoldshas infinitely many Stein fillable contact structures that are iterated planar. Question 1.14.
Are all the contact structures on S , S × S , and S (cid:101) × S iterated planar? Question 1.15.
For ≤ k < ∞ , does the manifold M k admit any iterated planar contactstructures? We discuss Question 1.15 further in Section 4 and suggest that M k admits a con-tact structure that we conjecture is not iterated planar.Many of the other items listed above for planar contact manifolds follow fromfinding symplectic caps for these manifolds. While we do not have specific resultsto mention here, below we will discuss some methods to build symplectic cobor-disms (that might be a step towards building symplectic caps) and build symplec-tic caps for a very restricted class of iterated planar contact –manifolds. We notethat recently, Conway and the second author [13] and, independently, Lazarev [31]showed that strong symplectic caps exist for arbitrary contact manifolds, but thereis little one can guarantee about the topology of these caps, and hence it does notseem likely they can be used to study iterated planar contact manifolds as the capsin Item (2) were used to restrict planar contact manifolds.We have seen that there are many contact manifolds that are iterated planar, butwe end this section by pointing out that not all higher dimensional contact mani-folds are iterated planar. In [37], Massot, Niederkr ¨uger, and Wendl gave examplesof symplectic manifolds with disconnected weakly convex boundary. It followsthat none of the boundary components of these examples are iterated planar byTheorem 1.2. TERATED PLANAR CONTACT MANIFOLDS 7
Projective open books and adapted contact manifolds.
There is another pos-sible generalization of planar open books to higher dimensions. The idea for thiscomes from noticing that S is also C P and the page of a planar open book natu-rally embeds in C P . Definition 1.16.
We say that an open book for a (2 n + 1) –manifold is projective if itsWeinstein page embeds as a convex domain in C P n k C P n for some k . A contactmanifold supported by a projective open book is called projective.A more natural term for such open books would be “rational”, but as this termis already used for something else when discussing open books [6] we have optedfor “projective”.Our first observation is that in dimension 5, iterated planar open books are alsoprojective, and hence iterated planar contact –manifolds are projective. Theorem 1.17.
Any iterated planar contact –manifold is projective. Combining Theorem 1.17 with the results in Section 1.1, we conclude that thereare many projective contact –manifolds, which leads us to ask: Question 1.18.
Are iterated planar contact manifolds of dimension greater than five alsoprojective?
Our main result about projective open books is that Eliashberg’s “capping pro-cedure” [17] can partially be carried out in this context (see Section 5) and in somecases fully carried out.
Theorem 1.19. If ( M , ξ ) is supported by an open book whose page embeds as a convexdomain in C P n C P for n ≤ , then ( M, ξ ) has a symplectic cap that contains anembedded C P n C P . Example 1.20.
It is well known that there is a Lagrangian sphere in S × S with thesymplectic form ω ⊕ ω and thus there is one in C P C P ∼ = ( S × S ) C P withits blown up symplectic form. As Lagrangians have standard neighborhoods wesee that the unit cotangent bundle DT ∗ S of S embeds in C P C P . Thus anycontact structure on a –manifolds supported by an open book with page DT ∗ S is a rational open book satisfying the hypothesis of this theorem. We see in Ex-ample 4.3 below that S has infinitely many distinct contact structures that aresupported by an open book with page DT ∗ S . So all of these contact structureshave a cap containing C P C P . Example 1.21.
Let W d be the Stein domain which is the complement of a neigh-borhood of a symplectic hypersurface Σ d of degree d > in CP and let τ d denote BAHAR ACU, JOHN B. ETNYRE, AND BURAK OZBAGCI the fibered Dehn twist in W d . The lens space L d give by S / Z d is given by the openbook with page W d and monodromy τ d , see [11, page 423]. Moreover the supportedcontact structure ξ d is simply the one induced from S by taking the quotient bythe group action. So from Theorem 1.19 we see that ( L d , ξ d ) has a symplectic capthat contains a copy of C P .1.3. Building symplectic cobordisms using open books.
Here we discuss twomethods to build symplectic cobordisms based on information about open bookdecompositions.
Theorem 1.22.
Let ( M, ξ ) be a (2 n + 1) –dimensional closed contact manifold supportedby an open book decomposition with page F , binding B = ∂F , and monodromy φ . Supposethat X is an exact symplectic cobordism such that ∂X = − B ∪ B (cid:48) . Then there exists astrong symplectic cobordism W = M × [0 , (cid:91) B × D ×{ } = ∂ − X × D X × D such that ∂W = ∂ − W ∪ ∂ + W with ∂ − W = − M and ∂ + W = M (cid:48) where ( M (cid:48) , ξ (cid:48) ) issupported by an open book with page F ∪ X , binding B (cid:48) , and monodromy φ (cid:48) which isidentity on X and agrees with φ on F . We hope that Theorem 1.22 might be useful in studying symplectic cobordismsbetween contact manifolds. The next theorem does not give a symplectic cobor-dism between contact manifolds but it rather gives a symplectic cobordism from acontact manifold to a symplectic fibration. This will be used in conjunction withthe information about the symplectomorphism group of some symplectic mani-folds to prove Theorem 1.19.
Theorem 1.23 (D ¨orner, Geiges and Zehmisch [16]) . Suppose that ( M, ξ ) is a closedcontact manifold supported by an open book decomposition with page F , binding B = ∂F ,and monodromy φ , and let X be a symplectic cap for B . Then there exists a symplecticcobordism W = M × [0 , (cid:91) B × D ×{ } = ∂ − X × D X × D such that ∂W = ∂ − W ∪ ∂ + W with ∂ − W = − M and ∂ + W = M (cid:48) where M (cid:48) fibers overthe circle with symplectic fibers. The fibers of M (cid:48) are F ∪ X and the monodromy of thebundle is the identity on X and agrees with φ on F . Acknowledgements.
The second author was partially supported by NSF grantDMS-1906414.
TERATED PLANAR CONTACT MANIFOLDS 9
2. I
TERATED P LANAR C ONTACT M ANIFOLDS
In this section, we introduce iterated planar Lefschetz fibrations, iterated planaropen books, and iterated planar contact manifolds, after recalling the necessarybackground on open books and Lefschetz fibrations.2.1.
Open book decompositions and Lefschetz fibrations.
We begin by recallingtwo notions of open book decompositions.
Definition 2.1. An abstract open book decomposition is a pair ( F, φ ) , where F is a com-pact manifold with boundary, called the page and φ : F → F is a diffeomorphismwhich restricts to the identity on ∂F , called the monodromy . Definition 2.2. An open book decomposition of a closed oriented manifold M is a pair ( B, π ) , where B is a codimension submanifold of M with trivial normal bundle,called the binding of the open book and π : M \ B → S is a fiber bundle such that π agrees with the angular coordinate θ on the normal disk D when restricted to aneighborhood B × D of B .For any θ ∈ S , the closure F θ := π − ( θ ) is a manifold with boundary ∂F θ = B called the page of the open book. The holonomy of the fiber bundle π determinesa conjugacy class in the orientation-preserving diffeomorphism group of a page F θ fixing its boundary, i.e. in Diff + ( F θ , ∂F θ ) which we call the monodromy . Since M is oriented, pages are naturally co-oriented by the canonical orientation of S and hence are naturally oriented. Also, the binding inherits an orientation fromthe open book decomposition of M . We assume that the given orientation on B coincides with the boundary orientation induced by the pages. It is well-knownthat an abstract open book decomposition gives a closed manifold with an openbook decomposition and vice-versa. Definition 2.3.
A contact structure ξ on a compact manifold M is said to be sup-ported by an open book ( B, π ) of M if it is the kernel of a contact form λ satisfying thefollowing:(1) λ is a positive contact form on the binding and(2) dλ is positively symplectic on each fiber of π .If these two conditions hold, then the open book ( B, π ) is called a supportingopen book for the contact manifold ( M, ξ ) and the contact structure ξ , as well as thecontact form λ , is said to be adapted to the open book ( B, π ) . Definition 2.4. A Liouville domain is a compact manifold W with boundary, to-gether with a –form λ such that ω = dλ is symplectic and the Liouville vector field Z λ on W defined by ω ( Z λ , · ) = λ points transversely outward along ∂W . Moreover, if ϕ : W → R is a (generalized) Morse function for which Z λ is gradient-like and ∂W is a regular level set of ϕ , then the quadruple ( W, ω, Z λ , ϕ ) is called a Weinsteindomain .Note that the primitive λ in Definition 2.4 is called a Liouville form and ker λ is acontact structure on ∂W , by definition. We also say that ∂W is convex.Let ( W, β ) be a Liouville domain and let φ : W → W be an exact symplectomor-phism which is identity near ∂W . Exactness of φ means that φ ∗ β − β = dh for somepositive function h : W → R , which is constant on each component of ∂W . Remark 2.5. If dim W = 2 , then any diffeomorphism of W can be isotoped to a sym-plectomorphism of ( W, dβ ) [23, Lemma 7.3.2]. When dim W ≥ , exactness of a symplectomorphism can be achieved by an isotopy [23, Lemma 7.3.4].Note that ker( β | ∂W ) is a contact structure on ∂W . Then α := β + dϕ , where ( r, ϕ ) are polar coordinates on D , descends to a contact form on ( W × [0 , / ∼ , whichcan be extended over ∂W × D as h ( r ) β + h ( r ) dϕ , for some suitable functions h and h (see [23, page 348]). Let λ denote the resulting contact form on the manifold M described by the abstract open book ( W, φ ) . Proposition 2.6 (Giroux 2002, [26]) . The contact structure ξ = ker λ described aboveon M is adapted to the abstract open book whose page is the Liouville domain ( W, β ) andwhose monodromy is the exact symplectomorphism φ . Conversely,
Theorem 2.7 (Giroux 2002, [26]) . Every closed contact manifold admits an adapted openbook with Weinstein pages.
Definition 2.8.
Let E be a compact n –dimensional manifold with corners whoseboundary is the union of two faces, namely the horizontal boundary ∂ h E and thevertical boundary ∂ v E , meeting in a codimension 2 corner. Let ω = dλ be an exactsymplectic form on E such that both faces of the boundary are convex. Let f : E → D be a proper smooth map with finitely many critical points Crit ( f ) ⊂ E and let F denote a regular fiber. We say that f is an exact symplectic Lefschetz fibration on ( E, ω = dλ ) if it satisfies the following properties:(1) Conditions on the boundary : We require that ∂ v E = f − ( ∂D ) and f | ∂ v E : ∂ v E → ∂D is a surjective smooth fiber bundle. Moreover, there is a neighborhood N of ∂ h E such that f | N : N → D is a product fibration D × nhd ( ∂F ) where ω | N and λ | N are pullbacks from the second factor of this product. In particular, ∂ h E = (cid:83) z ∈ D ∂ ( f − ( z )) and f | ∂ h E : ∂ h E → D are surjective smooth fiber bundles.(2) Conditions on the fibers : Let E z denote the fiber f − ( z ) for any z ∈ D . Werequire that the restriction of ω to E z \ Crit ( f ) is symplectic. In particular, the TERATED PLANAR CONTACT MANIFOLDS 11 boundary of each fiber is convex and each regular fiber of f is an exact symplecticmanifold with convex boundary. We also require that there is at most one criticalpoint at each fiber.(3) Conditions on the critical points : For any p ∈ Crit ( f ) , there are orientationpreserving local complex coordinates about p on E and f ( p ) on D such that, withrespect to these coordinates, f is given by the map f ( z , ..., z n ) = z + · · · + z n . It follows by Definition 2.8 that the critical points of f are isolated and belong tothe interior of E . We also allow Crit ( f ) to be the empty set. More importantly, thecorners of E can be rounded off to obtain a Liouville domain ( W, λ ) so that one canspeak about an exact symplectic Lefschetz fibration f : ( W, λ ) → D such that eachregular fiber is again a Liouville domain equipped with the restriction of λ .According to a recent result of Giroux and Pardon [27], any Stein (respectivelyWeinstein) domain admits a Stein (respectively Weinstein) Lefschetz fibration over D , which by definition implies that the regular fiber of the fibration is Stein (re-spectively Weinstein). So, we can consider these type of fibrations as special casesof the exact symplectic fibrations we defined on Liouville domains—which mightbe called Liouville Lefschetz fibrations in the same vein. Construction 2.9.
We make a few observations about constructing open books andLefschetz fibrations that will be used repeatedly below. The first simple observa-tion is that if f : ( W, λ ) → D is an exact Lefschetz fibration, then ∂W is convex andhence has an induced contact structure ξ . Moreover, if we take polar coordinateson D and let B = f − (0) ∩ ∂W , then ∂W \ B is fibered by composing f with pro-jection to the θ coordinate. Once may easily check that this open book supports ξ .In addition, if Λ is an exact Lagrangian sphere in a fiber of f , then there is an asso-ciated Legendrian sphere in ∂W [46]. By attaching a Weinstein handle to W along Λ , we get a new symplectic manifold W (cid:48) with an exact Lefschetz fibration that hasone additional vanishing cycle given by Λ , [27, Section 6.2]. In other words, givenan exact symplectic (2 n − –manifold X , a general exact Lefschetz fibration indimension n can be built by successively attaching Weinstein n –handles to thetrivial Lefschetz fibration X × D along Lagrangian spheres embedded in distinctfibers above ∂D .2.2. Iterated planar open books.
Once an exact symplectic Lefschetz fibration ona Liouville domain is given, one can further consider an exact symplectic Lefschetzfibration on the codimension two Liouville fiber, and iterate this dimension reduc-tion until the fiber is –dimensional. This leads to the following definition. Definition 2.10. An iterated planar Lefschetz fibration on a n –dimensional Liouvilledomain ( W, λ ) is a sequence of exact symplectic Lefschetz fibrations f i : ( W i , λ i ) → D for i = n, n − , . . . , , with the following properties:(1) ( W, λ ) = ( W n , λ n ) and f = f n (2) ( W i − , λ i − ) is a regular fiber of f i , for i = n, n − , . . . , (3) f : ( W , λ ) → D is planar, i.e. the regular fiber of f is a genus zero surfacewith nonempty boundary.Although we formulated the definition of an iterated planar Lefschetz fibrationby reducing the dimension at each step, it is also possible to start with the lowestpossible dimension and build up higher-dimensional fibrations as follows. Sup-pose that W is a smooth –manifold with nonempty boundary which admits a smooth planar Lefschetz fibration over D . Then by standard methods one canequip W with an exact symplectic form ω = dλ such that the smooth planar Lef-schetz fibration turns into an exact symplectic Lefschetz fibration f : ( W , λ ) → D with planar fibers. Similarly, using Construction 2.9 one can construct a Liou-ville domain ( W , λ ) of dimension which admits an exact symplectic Lefschetzfibration f : ( W , λ ) → D , whose fiber is ( W , λ ) . This process, of course, can beiterated as many times as desired. Note that an iterated planar Lefschetz fibrationon a –dimensional Liouville domain is nothing but a planar Lefschetz fibration. Remark 2.11.
It is not true that every –manifold with nonempty boundary admitsa planar Lefschetz fibration over D . For example, suppose that T × D admitssuch a fibration. Then there must be a planar Stein fillable contact structure on ∂ ( T × D ) = T , but the unique Stein fillable contact structure on T [18] is knownto be not planar [20]. Definition 2.12.
An open book decomposition whose (Weinstein) page admits aniterated planar Lefschetz fibration is called an iterated planar open book . An iteratedplanar contact manifold is a closed contact manifold of dimension at least 5, which issupported by an iterated planar open book decomposition.3. G
ENERAL RESULTS ABOUT ITERATED PLANAR CONTACT MANIFOLDS
There have been few examples of iterated planar contact manifolds but therewere a few implicit in the first authors work [3].
Example 3.1.
The simplest examples of iterated planar contact manifolds are(1) the standard contact sphere ( S n +1 , ξ std ) , TERATED PLANAR CONTACT MANIFOLDS 13 (2) the unit cotangent bundle ( ST ∗ S n , η ) of S n equipped with its canonical con-tact structure η , and(3) the convex boundary ( S n × S n +1 , ξ ) of the Weinstein domain DT ∗ S n × D ,where DT ∗ S n denotes the unit disk cotangent bundle of S n .To see this we will first show that DT ∗ S n has the structure of an iterated planarLefschetz fibration, which immediately implies that ( ST ∗ S n , η ) is iterated planar.We will make use of Construction 2.9 repeatedly in the discussion below. Let L be a linear Lagrangian disk in the standard symplectic ball B n so that ∂L = Λ is a Legendrian sphere in the standard contact S n − . If we attach a Weinstein n –handle to B n along Λ it is easy to see that we obtain F = DT ∗ S n that contains aLagrangian S n , that we denote L (cid:48) . Now D × F is a subcritical Weinstein manifoldand in its boundary we can take L (cid:48) to be a Legendrian sphere. Attaching a (2 n +2) –dimensional ( n +1) –handle to D × F along L (cid:48) will result in B n +2 with a symplecticform deformation equivalent to the standard symplectic form. Thus B n +2 hasthe structure of a Lefschetz fibration with fiber F and one vanishing cycle givenby L (cid:48) . But now attaching another (2 n + 2) –dimensional handle to B n +2 along acopy of L (cid:48) will result in DT ∗ S n +1 with a Lefschetz fibration with fiber DT ∗ S n andtwo vanishing cycles. That is for all n ≥ , DT ∗ S n +1 has a Lefschetz fibrationwith fiber DT ∗ S n and in particular for n = 1 , the fiber DT ∗ S = S × [ − , isplanar. Therefore, we conclude that DT ∗ S n (and hence B n +2 ) has the structure ofan iterated planar Lefschetz fibration.Now the iterated planar Lefschetz fibration structure on DT ∗ S n yields an iter-ated planar Lefschetz fibration structure on DT ∗ S n × D and hence induces aniterated planar open book supporting its contact boundary ( S n × S n +1 , ξ ) .The iterated planar Lefschetz fibration structure on B n +2 constructed above in-duces an iterated planar open book supporting ξ std on S n +1 . The ”iterated” pagesare DT ∗ S n and the ”iterated” monodromy is a Dehn twist about L (cid:48) . Example 3.2.
The Brieskorn manifold Σ n − ( k, , . . . , ⊂ C n +1 is defined as theintersection of the sphere S n +1 with the zero set of the polynomial z k + z + · · · + z n .Viewed as a singularity link, Σ n − ( k, , . . . , carries a canonical contact structure η k . The contact manifold (Σ n − ( k, , . . . , , η k ) is supported by an open book withpage DT ∗ S n − , and monodromy the k -fold right-handed Dehn twist along the zerosection [47]. It follows that (Σ n − ( k, , . . . , , η k ) is iterated planar by Example 3.1.Note that the corresponding Milnor fiber admits a Lefschetz fibration with fiber DT ∗ S n − , and hence it admits an iterated planar Lefschetz fibration structure.More generally, we can show that any finitely presented group is the fundamen-tal group of some iterated planar contact manifold. Proof of Theorem 1.4.
First we prove the case n = 2 . For each finitely presentedgroup G , there is a planar Lefschetz fibration W → D with π ( W ) ∼ = G (see [25,Proposition 6.1]). Let M be the contact –manifold supported by the open bookwith page is W and monodromy the identity map. Then M is the boundary ofthe Stein domain W × D and π ( M ) ∼ = π ( W × D ) ∼ = π ( W ) ∼ = G. Therefore, M is an iterated planar contact –manifold whose fundamental groupis G . Now, consider the contact –manifold M supported by the open book withpage W × D and monodromy the identity map. The discussion above shows that π ( M ) ∼ = G and M is iterated planar by definition. It is clear that this process canbe iterated for all n ≥ , to construct an iterated planar contact (2 n + 1) –manifoldwhose fundamental group is G . (cid:3) Next we will show that the connected sum preserves iterated planarity, as acorollary of Lemma 3.3.
Lemma 3.3.
Suppose that f i : ( W i , λ i ) → D is an exact Lefschetz fibration with regularfiber F i , for i = 0 , . Then a –handle can be attached to W ∪ W to obtain a Weinsteinmanifold W which admits an exact Lefschetz fibration with regular fiber F ∪ F with a –handle attached.Proof. The boundary of W i naturally splits into two pieces: the vertical boundary f − i ( ∂D ) and the horizontal boundary ( ∂F i ) × D , where F i denotes a fiber of f i . Note that ( ∂F i ) × D has a natural contact structure with contact form α i + x dy − y dx , where α i is a contact form for ∂F i and ( x, y ) are coordinates on D .Supposing that B i is a Darboux ball in ∂F i , we can attach a n –dimensional –handle to W ∪ W along ( B × D ) ∪ ( B × D ) . More specifically, if we take a (2 n − –dimensional Weinstein –handle D × D n − that is attached along ( ∂D ) × D n − ,then D × D n − × D is a n –dimensional –handle that can be thought of as a D ’sworth of (2 n − –dimensional Weinstein –handles. Therefore, a n –dimensional –handle can be attached to [( ∂F ) ∪ ( ∂F )] × D by attaching ( ∂D ) × D n − × { p } to [( ∂F ) ∪ ( ∂F )] × { p } for each p ∈ D . That is for each p ∈ D , we attach a (2 n − –dimensional –handle to f − ( p ) ∪ f − ( p ) . We conclude that the Weinsteinmanifold W = W ∪ W ∪ (1 –handle ) admits an exact Lefschetz fibration with fiber F ∪ F ∪ (1 –handle ) . (cid:3) Proof of Theorem 1.6.
Suppose that ( M i , ξ i ) is a contact manifold supported by aniterated planar open book ( W i , φ i ) , for i = 0 , . We claim that we can attach a –handle to W ∪ W to obtain a Weinstein manifold W that has the structureof an iterated planar Lefschetz fibration. If φ : W → W denotes the symplecto-morphism that restricts to φ i on W i and is the identity on the –handle, then one TERATED PLANAR CONTACT MANIFOLDS 15 may readily check using [46, Proposition 4.2] that the open book ( W, φ ) supports ( M M , ξ ξ ) — which gives a proof of the desired result assuming the claim.To prove our claim, we first observe that W and W both have the structureof an iterated planar Lefschetz fibration, by definition. Then a –handle can beattached to W ∪ W to obtain a Weinstein manifold W which admits an exact Lef-schetz fibration, whose fibers are obtained from the fibers of the Lefschetz fibra-tions on W and W by Lemma 3.3. Iterating this construction until we get to the –dimensional Lefschetz fibration case, we see that each fiber is simply obtainedfrom the –dimensional fibers for W and W together with a –handle attached toconnect them. But since these –dimensional fibers for W and W are both planar,so is the –dimensional fiber for W . Therefore, the exact Lefschetz fibration on W is indeed iterated planar. (cid:3) Remark 3.4.
We will see an alternate proof of Theorem 1.6 in the –dimensionalcase below.4. I TERATED PLANAR CONTACT MANIFOLDS IN DIMENSION
Lemma 4.1.
A contact –manifold is iterated planar if and only if it admits an adaptedopen book whose binding is planar.Proof. If a contact –manifold is iterated planar, then by definition, it admits a sup-porting open book whose Weinstein page admits an exact symplectic Lefschetzfibration over D with planar fibers, and hence the binding of this open book is aplanar contact –manifold. Conversely, if a contact –manifold admits an adaptedopen book with planar binding, then since the Weinstein page of this open bookis a strong filling of its binding and any strong symplectic filling of a planar con-tact –manifold admits an exact symplectic Lefschetz fibration over D by Wendl’swork [48], we conclude that the contact –manifold at hand is iterated planar. (cid:3) Using this lemma, we can give an alternate proof of the fact that the connectedsums preserve iterated planarity in dimension 5.
Alternate proof of Theorem 1.6 in dimension 5. If ( M i , ξ i ) is iterated planar, then bydefinition it admits an adapted open book with Weinstein page W i so that thebinding ∂W i is planar. Note that the contact connected sum ( M M , ξ ξ ) issupported by the open book whose page is obtained as the boundary connectedsum W (cid:92)W . Hence the binding of this open book is the contact connected sum ∂W ∂W . The connected sum of planar contact manifolds is planar. There-fore, the binding of the open book supporting ( M M , ξ ξ ) is planar and thusLemma 4.1 implies that the open book is iterated planar. (cid:3) We now consider contact –manifolds with subcritical Stein fillings. Recall thatour first theorem along these lines is the following. Theorem 1.7.
The following subcritically Stein fillable contact –manifolds are iteratedplanar. (1) Consider a –manifold M with a finite cyclic fundamental group (including thetrivial group). If the order of the group is odd, then any subcritically Stein fillablecontact structure on M is iterated planar. If the order of the group is even, thenat least half of the subcritically Stein fillable contact structures on M are iteratedplanar. (2) The unique contact structure admitting a symplectically aspherical subcritical Steinfilling on S × S is iterated planar. (3) If ( M, ξ ) is a subcritically Stein fillable contact –manifold, then M admits a sub-critically Stein fillable iterated planar contact structure ξ (cid:48) in the same homotopyclass of almost contact structures. Moreover, if M has infinite second cohomologythen it has infinitely many such contact structures.Proof. We begin with Item (2). The work of Barth, Geiges, and Zehmisch [8] showsthat there is a unique symplectically aspherical subcritically Stein fillable contactstructure on S × S . It is filled by the Stein manifold S × D obtained by attach-ing a Stein –handle to the standard symplectic B . This induces an open booksupporting the contact structure with pages S × D , which admits a Lefschetzfibration with fiber S × D . Thus the open book, and hence the aforementionedcontact structure on S × S is iterated planar.For Item (1) we recall that subcritical Stein fillings of such manifolds have beenclassified by Ding, Geiges, and Zhang [15] as follows. We first define L n to be thelens space L ( n, with an open –ball removed. Suppose that ( M , ξ ) is subcriti-cally Stein fillable contact –manifold whose fundamental group is Z /n Z . Let r bethe rank of H ( M ; Z ) . If n is odd then M is diffeomorphic to ∂ ( L n × D ) r ( S × S ) or ∂ ( L n × D ) S (cid:101) × S ) r − ( S × S ) , depending whether M is spin or not. If n is even and M is spin then it is diffeo-morphic to ∂ ( L n × D ) r ( S × S ) , and if M is not spin then it is diffeomorphic to ∂ ( L n (cid:101) × D ) r ( S × S ) or ∂ ( L n × D ) S (cid:101) × S ) r − ( S × S ) , Moreover, in each homotopy class of almost contact structures, there is a uniquesubcritically Stein fillable contact structure.
TERATED PLANAR CONTACT MANIFOLDS 17
Given Theorem 1.6 concerning connected sums of iterated planar contact struc-tures (and the uniqueness results above), it will suffice to prove that each homo-topy class of almost contact structures on the summands above, is realized by asubcritically Stein fillable contact structure that is supported by an iterated planaropen book.We begin with S × S and first recall that the possible framings on an S in a –manifold are given by π ( SO (4)) = Z / Z . Thus if we consider a –dimensionalhandlebody H , then H × D is a 6-dimensional handlebody and the framings onthe –handles are reduced modulo 2. Let W denote the Stein –manifold obtainedfrom B by attaching a Stein –handle along a Legendrian unknot L ⊂ ( S , ξ std ) with odd Thurston-Bennequin invariant. Then the Stein –manifold W × D isdiffeomorphic to S × D and thus its boundary is S × S . As we can realize anypossible Chern class on W by the appropriate choice of L , we see that we can alsorealize any possible choice of Chern class on W × D and hence on S × S . Asthere is no –torsion in the homology of S × S this means that all almost contactstructures are realized by this construction [15]. Of course the binding of the openbook coming form the product Lefschetz fibration W × D is simply ∂W . Since thisis a lens space with a tight contact structure it is known to be planar [43]. Thus byLemma 4.1, all the subcritical Stein fillable contact structures on S × S are iteratedplanar.The same argument works for S (cid:101) × S except that one uses a Legendrian unknot L ⊂ ( S , ξ std ) with even Thurston-Bennequin invariant.We now turn to L n × D . As L n is a –manifold built by a single 0-, –, and –handle, where the attaching region for the –handle is a circle that is a ( n, curveon the solid torus -handle ∪ -handle, we see that W in Figure 1 is a handlebodypicture for L n × [0 , . − − k − n − nn − − n − k F IGURE
1. The –manifold W k shown on the left. The –handle runs n times over the –handle. On the right, is the –manifold boundaryof W k . The thin red curve is a regular fiber in the fibration and hascontact framing with respect to the fibration framing. As noted above if we change the framing on the –handle by some even num-ber then we get a –manifold W k , that when crossed with D is diffeomorphic to L n × D . Notice that Figure 1 can be thought of as a Stein S × D with a Legen-drian knot L in its boundary, and L has Thurston-Bennequin invariant − n . Thusstabilizing once and attaching a Stein handle to L gives the manifold W . Similarlyif we stabilize L , k times before attaching a handle we obtain a Stein realizationof W k +1 . We notice that ∂W k +1 is the Seifert fibered space shown on the right ofFigure 1. In [35], Lisca and Stipsicz showed that all contact structures with zerotwisting are given by the surgery diagram in Figure 2. Zero twisting means that (+1)(+1)( − k + 1)( − n + 1) (1 − nn − ) F IGURE
2. A surgery diagram for ∂W k the regular fiber in the Seifert fibration can be realized by a Legendrian knot withcontact framing agreeing with the framing coming form the fibration. The thin redcurve in Figure 1 is a regular fiber and we see that it has zero twisting. As noted in[35] such a contact structure can easily be seen to be supported by a planar openbook. Therefore, the boundaries of W k +1 × D have contact structures that aresupported by iterated planar open books.So we get Stein manifolds W k × D diffeomorphic to L n × D , and we can realizeall possible Chern classes of L n × D . Thus as above, when n is odd, so that thereis no –torsion in the homology of L n × D , we see that all subcritically fillablecontact structures on L n × D can be realized by iterated planar open books.Turning to the case when n is even we recall that the Chern class does notuniquely specify the almost contact class of a contact structure. However, the ob-struction to two almost contact classes being homotopic is half the difference intheir Chern classes. Thus in our situation there are at most two contact structuressharing the same Chern class and since we can realize all possible Chern classeswith our construction, we have shown at least half of the subcritically fillable con-tact structures are iterated planar.We now turn to Item (3) in the statement of the theorem. In the proof of this wewill need the following result. TERATED PLANAR CONTACT MANIFOLDS 19
Lemma 4.2 (Onaran 2020, [40]) . If L is a topological link in k S × S then there isa planar open book for k S × S supporting the unique tight contact structure on thismanifold such that L can be realized as a Legendrian link on a page of the open book. Now if ( W, ω ) is a subcritical Stein filling of a given contact –manifold ( M, ξ ) ,then we know from Cieliebak [12] that ( W, ω ) is obtained from a Stein –manifold ( X, ω ) by taking the product with D . The –manifold ( X, ω ) is obtained from ( (cid:92) k S × D , ω std ) by attaching Weinstein –handles along a Legendrian link L in ( k S × S , ξ std ) . By Lemma 4.2, there is some other Legendrian link L (cid:48) that issmoothly isotopic to L and sits on a page of a planar open book for ( k S × S , ξ std ) .After stabilizing L (cid:48) more (and still calling it L (cid:48) ) we can assume the parity of thecontact framing of L agrees with the parity of the contact framing of L (cid:48) and that thecomponents of the links have the same rotation number. Let ( X (cid:48) , ω (cid:48) ) be the Steindomain obtained from (cid:92) k S × D by attaching Stein –handles along the Legendrianlink L (cid:48) . The boundary of X (cid:48) is a planar contact manifold, since the surgery link L (cid:48) sits on a page of a planar open book for ( k S × S , ξ std ) . So ( X (cid:48) , ω (cid:48) ) admits aplanar Lefschetz fibration. As discussed above, X (cid:48) × D is diffeomorphic to X × D (because of our condition on the parity of contact framings) and the correspondingsymplectic manifolds have homotopic almost complex structures (because of ourcondition on the rotation numbers). Thus ∂ ( X (cid:48) × D ) = ∂ ( X × D ) = ∂W = M hasan iterated planar contact structure in the same homotopy class of almost contactstructures as ξ . To prove the last part of Item (3) we notice that by changing therotation numbers on L (cid:48) we can achieve different Chern classes for X (cid:48) × D . Thussince H ( X (cid:48) × D ; Z ) ∼ = H ( ∂ ( X (cid:48) × D ); Z ) we see that we can realize infinitelymany different homotopy classes of almost contact structures by iterated planarcontact structures. (cid:3) In order to prove Theorem 1.10 concerning iterated planar contact manifoldswith flexible Stein fillings, we first need to consider open book decompositions forsome contact structures on S . Example 4.3.
The Brieskorn manifold (Σ ( k, , , , η k ) is supported by the openbook with page DT ∗ S , and monodromy the k –fold right-handed Dehn twist alongthe zero-section (see, Example 3.2). It follows that the iterated planar contact –manifold (Σ ( k, , , , η k ) is Stein fillable and hence tight.For k odd, Σ ( k, , , is diffeomorphic to S , [45]. With this identification,we get an infinite set { ( S , η k ) | k odd } of iterated contact manifolds. Moreover,Ustilovsky [45] showed that these contact manifolds are non-contactomorphic. Wealso note that ( S , η k ) has a Weinstein filling X k which is obtained from DT ∗ S × D by attaching k Weinstein –handles along copies of the zero section in DT ∗ S , andby Construction 2.9 we see that X k has the structure of an iterated planar Lefschetz fibration. Since the rank of H ( X k , Z ) is equal to k − , none of these fillings can besubcritical for k > .On the other hand, Σ ( k, , , is diffeomorphic to S × S for k even. It isknown that (Σ (2 , , , , η ) is contactomorphic to ST ∗ S ∼ = S × S equippedwith its canonical contact structure ξ can . We observe that, since the canonical con-tact structure on the unit cotangent bundle of a closed manifold does not admit asubcritical Stein filling by [8, Proposition 3.9], ( S × S , ξ can ) is not contactomor-phic to any of the subcritically Stein fillable contact manifolds that appeared in theproof of Theorem 1.7.With the symplectic fillings X k +1 of S in hand, we are ready to prove Theo-rem 1.10 which we recall here. Theorem 1.10. If ( M , ξ ) is an iterated planar contact –manifold with c ( ξ ) = 0 , andhas a flexible Stein filling, then it admits infinitely many iterated planar contact structureswhich are filled by Stein manifolds that cannot be subcritical.Proof. We begin by recalling a result of Lazarev [30, Theorem 1.9], the proof ofwhich says that if ξ is a contact structure on M with c ( ξ ) = 0 that has a flexible fill-ing W , then for any two flexible fillings W and W of S , the contact structures on M induced by the flexible fillings W (cid:92)W and W (cid:92)W are distinct provided that W and W have distinct homologies. (Here (cid:92) means boundary sum and we can thinkof W (cid:92)W i as being constructed from W ∪ W i by attaching a Weinstein –handle.)We claim that there are flexible Weinstein structures on the manifolds X k +1 andthat the contact structure on their boundaries are all iterated planar. When dis-cussing the flexible Weinstein structure on X k +1 we will denote it X (cid:48) k +1 . Giventhis claim and our hypothesized ( M, ξ ) with c ( ξ ) = 0 that is iterated planar andhas a flexible filling W , we can consider W (cid:92)X (cid:48) k +1 . All the contact structures on M = ∂ ( W (cid:92)X (cid:48) k +1 ) are distinct since, as noted above, all the X (cid:48) k +1 have distincthomologies. Moreover, since the contact structures only differ on a –ball (since ∂X (cid:48) k +1 is diffeomorphic to S ) they are all in the same homotopy class of almostcontact structures. Finally, all the contact structures are iterated planar by Theo-rem 1.6 concerning connected sums, since ξ and ∂X (cid:48) k +1 are.To prove the claim we revisit the description of the Weinstein fillings X k +1 . Re-call that they are built from ( DT ∗ S ) × D by attaching Weinstein –handles along k + 1 copies of the zero section of DT ∗ S . Also recall that ( ∂ ( DT ∗ S × D ) , ξ ) is a contact manifold (see Example 3.1) supported by the open book with page DT ∗ S and monodromy the identity. If λ is the Liouville form on T ∗ S , then dλ isthe symplectic form on T ∗ S . The zero section Z of DT ∗ S is Lagrangian and infact λ | Z = 0 , so it corresponds to a Legendrian sphere on each page of the open TERATED PLANAR CONTACT MANIFOLDS 21 book supporting ( ∂ ( DT ∗ S × D ) , ξ ) . We can get X k +1 by attaching Weinstein –handles to DT ∗ S × D along k + 1 copies of this Legendrian sphere. It is wellknown, see [46], that the resulting open book for ∂X k +1 still have page DT ∗ S butthe monodromy is a composition of k + 1 Dehn twists about Z .Now let Λ be a fiber in the bundle DT ∗ S → S , and U the boundary of Λ . So Λ isa Lagrangian disk in ( DT ∗ S , dλ ) , and U is a Legendrian circle in ( ∂ ( DT ∗ S ) , ker λ ) .Moreover, Λ intersects Z exactly once and λ = 0 on Λ . Let Y be the result ofattaching a Weinstein –handle to DT ∗ S along U . Notice that there is now anexact Lagrangian sphere S in Y that is the union of Λ and the core of the –handle.Let τ S be the Dehn twist about S and τ Z be the Dehn twist about Z . The openbook ( Y, τ Z ◦ τ S ) is a stabilization of the open book ( DT ∗ S , τ Z ) , [46], and thussupports the same contact structure. Notice that ( DT ∗ S , τ Z ) supports the standardcontact S and in that contact manifold Z is the standard Legendrian unknot. Thus ( Y, τ Z ◦ τ S ) also supports ( S , ξ std ) and in this manifold both Z and S are standardLegendrian unknots. This should be clear as Y is the plumbing of two copies of DT ∗ S and S and Z are the zero sections of the two copies. Moreover, this openbook is the boundary of the Lefschetz fibration obtained from Y × D by attachingtwo Weinstein –handles along Z and S . The total space of this Lefschetz fibrationis a Weinstein –manifold, which we denote ( E, ω ) . Notice that DT ∗ S is obtainedby attaching a Weinstein –handle to the standard symplectic B and Y is the resultof attaching a Weinstein –handle to DT ∗ S . It follows that ( E, ω ) is simply thestandard symplectic B , since Y × D is obtained from the standard symplectic B by attaching two Weinstein –handles, and the Weinstein –handles, which areattached to Y × D along Z and S to obtain E , cancel them. We finally notice that X k +1 has an open book decomposition ( Y, τ Z ◦ τ S ◦ τ k +1 Z ) .In [28, Section 4.1] Honda and Huang defined an operation S (cid:93) Z on S and Z ,which produces a new Legendrian sphere that is a stabilization of Z . We note thatthere is a technical hypothesis that S and Z must intersect ξ -transversely, that is,their tangent spaces must span the contact hyperplane at an intersection point. Butthis is clear as the contact hyperplanes are tangent to the fiber at S ∩ Z and the inter-section is transverse in the fiber. Now let X (cid:48) k +1 be the result of attaching Weinstein –handles to W along k + 1 copies of S (cid:93) Z . The sphere S (cid:93) Z is a stabilizationof Z which is the standard unknotted Legendrian sphere, thus it is a loose sphere(a front diagram for S (cid:93) Z consists of “front spinning” a Legendrian arc with threezig-zags, so it has a loose chart [38]). Hence X (cid:48) k +1 is a flexible Weinstein manifoldthat is diffeomorphic to X k +1 because the sphere S (cid:93) Z is smoothly isotopic to Z .Next we claim that the contact structure on the boundary of the Weinstein –manifold X (cid:48) k +1 is supported by the open book ( Y, τ Z ◦ τ S ◦ τ k +1 S (cid:93) Z ) . To establish this we need to see that S (cid:93) Z is realized by a Lagrangian on a page of the open book ( Y, τ Z ◦ τ S ) . We show this in Lemma 4.4 below. Thus we are done with the proofof Theorem 1.10 by noticing that X (cid:48) k +1 is iterated planar by Lemma 4.1, since thebinding ∂Y is a lens space. (cid:3) Lemma 4.4.
Suppose that S and Z are Legendrian -spheres in a contact -manifold ( M , ξ ) which intersect ξ -transversely at a point. Suppose also that S and Z are bothLagrangian on a page of an open book supporting ( M , ξ ) . Then the stabilized Legendriansphere S (cid:93) Z in ( M , ξ ) can be realized as a Lagrangian on a page of this open book.Proof. We begin by recalling the construction in [28, Section 4.1]. Consider R withthe contact form α = dz − y · d x − x · d y (where x is ( x , x ) and y is ( y , y ) ). There isa neighborhood N of the transverse intersection point between the spheres Z and S that is contactomorphic to a neighborhood of the origin in ( R , ker α ) so that N ∩ S goes to the x -plane and N ∩ Z to the y -plane. One then embeds A = S × R intothe hyperplane z = 0 so that it is Legendrian in ( R , ker α ) (and also Lagrangian inthe symplectic hyperplane z = 0 ) and is asymptotic to Z on one end and S on theother end. Looking at the S end, one can see that A is the -jet of some function f S over x . Then using a cutoff function φ , the -jet of φf S deforms A so that it agreeswith S away from the intersection point. This can be done for the Z end too. Thusremoving disks from S and Z and inserting the portion of A we have constructed S (cid:93) Z .Now in our situation we notice that in this local model, the page of the openbook on which S and Z sit can be taken to be the z = 0 hypersurface. Thus A alsosits on the page. Notice that the image of d ( φf ) in z = 0 , is a Lagrangian annulusin the page of the open book and S (cid:93) Z is a graph over this (in the local model,one just needs to change the z coordinate to f ( x ) above the point ( x , df x ) ). Thuswe have constructed a Lagrangian L S (cid:93) Z in the page of our open book and S (cid:93) Z isa graph over this Lagrangian.Now we can isotope our monodromy map so that this construction takes place ina neighborhood of the intersection point on which the monodromy is the identity.If λ were the Liouville form on our original page, then λ t = λ − td ( φf ) is a family ofLiouville forms on the page and induce contactomorphic contact structures on themanifolds. Moreover, L S (cid:93) Z is Legendrian with respect to the contact form inducedby λ and the graph of (1 − t ) φf is Legendrian with respect to the contact forminduced by λ t . Thus we have an isotopy from S (cid:93) Z in our original contact manifoldto the Legendrian L S (cid:93) Z in the contact structure induced by λ , and there L S (cid:93) Z isalso a Lagrangian on the page of the open book. (cid:3) TERATED PLANAR CONTACT MANIFOLDS 23
We now turn to Proposition 1.11 that shows, contrary to what happens in di-mension 3, that in higher dimensions there are iterated planar contact manifoldsthat are strongly symplectically fillable but not Stein fillable.
Proof of Proposition 1.11.
Let W d be the Stein domain which is the complement of aneighborhood of a symplectic hypersurface Σ d of degree d > in CP and let τ d denote the fibered Dehn twist in W d . Then, according to [11, page 423], the contactmanifold supported by the open book with page W d and monodromy τ d is con-tactomorphic to ( L d , ξ ) , where L d denotes the lens space S / Z d and ξ is obtainedby taking the quotient of ξ st . We conclude by [11, Theorem 6.3] that ( L d , ξ ) is ofBoothby-Wang type and therefore it is symplectically fillable by the correspondingdisk bundle (see [24, Lemma 3]), but this symplectic filling can not be Stein, since,as was observed in [9, Example 6.5], for any d > , the lens space L d does not carryany Stein fillable contact structures at all.The binding of the above open book is given by the convex boundary of theStein page W d . Note that ∂W d can be described as a circle bundle over Σ d of Eulernumber − [Σ d ] = − d , and the induced contact structure on ∂W d is of Boothby-Wang type.For the case d = 2 , we have W ∼ = DT ∗ RP ( ∼ = CP \ a quadratic), and thus ∂W ∼ = ST ∗ RP ∼ = L (4 , . Therefore, ( L , ξ ) is an iterated planar contact –manifold, which is symplectically fillable but not Stein fillable. (cid:3) Turning to overtwisted contact structures we consider Theorem 1.12 which werecall says:
Theorem 1.12. If M is a –manifold with finite cyclic fundamental group of odd order(including the trivial group) that is the boundary of a subcritical Stein domain, then everyovertwisted contact structure on M is iterated planar.More generally, if M is any –manifold that admits a subcritically fillable contact struc-ture, then it admits an overtwisted contact structure that is iterated planar. If H ( M ; Z ) is infinite, then there are infinitely many homotopy classes of almost contact structures forwhich every overtwisted contact structure in any of these classes is iterated planar.Proof. By the proof of Theorem 1.7, we know that each of the homotopy classes ofalmost contact structures mentioned in the theorem is realized by a contact struc-ture ξ on M that is supported by an iterated planar open book. Moreover, a neg-ative stabilization of such an open book supports ( M S , ξ ξ OT ) , and ξ and theovertwisted contact structure ξ ξ OT are in the same homotopy class of almost con-tact structures on M [10, Theorem 1.1]. Notice that since both ( M , ξ ) and ( S , ξ OT ) are iterated planar, so is ( M S , ξ ξ OT ) , by Theorem 1.6. (cid:3) Example 4.5. ( Simply-connected contact –manifolds ) Recall from the introductionthat the overtwisted contact structures on the simply connected –manifolds M = S , M ∞ = S × S , and X ∞ = S (cid:101) × S , are all iterated planar and each of these –manifolds also carry infinitely many iterated planar Stein fillable contact struc-tures. However, we were unable to say anything about M k for < k < ∞ .The –manifold M k carries a Stein fillable contact structure η k , which is describedas a contact open book with Stein pages in [14, Section 7.2]. Specifically let X k bethe Stein domain given by the Legendrian surgery diagram depicted in Figure 3.Notice that both knots in the diagram bound Lagrangian disks in B and hence k full twistsF IGURE
3. The Stein domain X k .there are Lagrangian spheres S and S in X k that come from capping these disksoff with the core disks of the –handles. Now consider the Stein manifold X k × D and attach two Stein –handles to these spheres. This gives a Stein domain W k such that ∂W k = M k . Moreover, we see that the contact structure η k induced on M k is supported by the open book ( X k , τ S ◦ τ S ) .Notice that the intersection form on X k cannot embed in a diagonal negativedefinite form for k > . This says the contact structure on ∂X k is not planar asdiscussed in Item (3) in the introduction. Thus X k does not have a planar Lefschetzfibration and we cannot use the above open book to see that η k is iterated planar.In fact our guess is that η k for < k < ∞ is not iterated planar. Notice that η isobviously iterated planar. For k = 2 the intersection form on X k does embed ina diagonal negative definite form, but we still cannot determine if it has a planarLefschetz fibration.Furthermore, for a simply connected –manifold M , homotopy classes of almostcontact structures are in one-to-one correspondence with integral lifts of w ( M ) and the correspondence is given by associating to an almost contact structure itsfirst Chern class [23, Proposition 8.1.1] (note that this is true for simply connected –manifolds, as pointed out in an erratum, there are more subtleties in the general TERATED PLANAR CONTACT MANIFOLDS 25 case). It follows that M k admits a unique homotopy class of almost contact struc-tures since H ( M k ; Z ) = 0 . Thus M k has a unique overtwisted contact structure.For < k < ∞ it is unclear whether these are iterated planar.5. S YMPLECTIC COBORDISMS
Eliashberg [17] proved that any weak symplectic filling of a closed contact –manifold can be symplectically embedded into a closed symplectic –manifold.(Note that an alternate independent proof of this result was obtained by the secondauthor [19].)As discussed in the introduction, Conway and the second author [13] and, inde-pendently, Lazarev [31] have shown that caps for contact manifolds can always beconstructed (though it is still not clear if weak symplectic fillings of a contact man-ifold can be embedded in closed symplectic manifolds). But in dimension 3, thecaps constructed in [17, 19] are more explicit, and so one might hope for new con-structions of caps in higher dimensions that more closely follow the constructionin dimension 3.After reviewing Eliashberg’s proof briefly, we discuss some partial generaliza-tions to higher dimensions below. The first step in Eliashberg’s proof is the con-struction of a cobordism W equipped with a symplectic form ω such that ∂W = − M ∪ N with the following properties: M is a concave boundary component of W , contactomorphic to the given weakly fillable contact –manifold and N admitsa fibration over S such that the restriction of ω | N to each fiber is symplectic. Thiscobordism is obtained by a symplectic –handle attachment along the binding ofan open book adapted to the weakly fillable contact –manifold at hand. Then hefills in this symplectic fibration over S by a symplectic Lefschetz fibration over D to obtain a symplectic cap.The first step in Eliashberg’s program has already been carried out in higher di-mensions by D ¨orner, Geiges, and Zehmisch [16], which we stated as Theorem 1.23in the introduction. In order to construct a symplectic cap in higher dimensions,one also needs to obtain an analogue of the second step of Eliashberg’s proof,which we formulate as Question 5.1. Question 5.1.
For n > , assume that N is a (2 n + 1) –dimensional closed manifoldwhich is a boundary component of a symplectic manifold ( W, ω ) so that ( N, ω | N ) admits asymplectic fibration over S . Does there exist a symplectic manifold Z so that we can use Z to cap off N ? Theorem 1.19 proved below gives a positive answer to this question in somenontrivial cases. The simplest case when the answer is yes, is when the symplec-tomorphism group of the fiber X in the symplectic fibration N → S is connected. This is because we can take the monodromy of the symplectic fibration to be theidentity and cap off N by ( X × D , ω + ω st ) , relying on Lemma 5.2. Lemma 5.2.
Let X be a closed symplectic manifold and φ i : X → X , for i = 0 , , besymplectomorphisms that are isotopic through symplectomorphisms. Let X φ i denote themapping torus of ( X, φ i ) . Then there is a symplectic manifold ( W, ω ) such that ∂W = − X φ ∪ X φ , and ω induces the symplectic structures on the fibers of the mapping cylinders X φ i .Proof. Let φ t : X → X be the isotopy of symplectomorphisms. Consider X × R × [0 , , and coordinates t ∈ R and s ∈ [0 , on the last two factors. Notice that Ω = ω + dt ∧ ds is a symplectic form on X × R × [0 , . Consider the map Ψ : X × R × [0 , → X × R × [0 , , ( p, t, s ) (cid:55)→ ( φ s ( p ) , t + 1 , s ) . Notice that Ψ ∗ Ω = ω + dt ∧ ds . So the map Ψ generates an action of Z on X × R × [0 , by symplectomorphisms. So the quotient space W of X × R × [0 , bythis Z –action inherits a symplectic structure ω W from Ω . Clearly this is the desiredcobordism carrying a symplectic structure. (cid:3) We now study when we can glue two symplectic cobordisms along a boundarycomponent that is a symplectic bundle over a circle. To that end we recall that a stable Hamiltonian structure on a (2 n +1) –dimensional oriented manifold M is a pair ( ω, λ ) where ω is a closed -form and λ is a -form such that λ ∧ ω n > and thekernel of ω is contained in the kernel of λ . The Reeb vector field of such a structure isthe unique vector field R such that R is in the kernel of ω and λ ( R ) = 1 . Moreover,given a stable Hamiltonian structure ( ω, λ ) on M we have its symplectization , whichis the symplectic manifold ( R × M, ω + d ( tλ )) . If ( ω, λ ) and ( ω (cid:48) , λ (cid:48) ) are stable Hamil-tonian structures on M and M (cid:48) , respectively, then a diffeomorphism f : M (cid:48) → M satisfying f ∗ ω = ω (cid:48) and f ∗ λ = λ (cid:48) induces a symplectomorphism between theirsymplectizations.If p : M → S is a bundle over S and ω is a closed –form on M that is symplecticon each fiber, then ( ω, λ = p ∗ dθ ) is a stable Hamiltonian structure. We note thatsuch bundles have a standard form. Choose a fiber F = p − ( θ ) and let ω denotethe restriction of ω to F . Let φ : F → F be the first return map of the flow of theReeb vector field R . Since it is clear that the flow of X preserves ω , we see that φ isa symplectomorphism of ( F , ω ) . We can now form the mapping cylinder C φ bytaking the quotient of F × R by the action of Z generated by ( p, t ) (cid:55)→ ( φ ( p ) , t − π ) .The pull-back of ω to F × R (which we still denote by ω ) is invariant under thisaction, so induces a form (still denoted ω ) on C φ . Moreover, the from dt descends TERATED PLANAR CONTACT MANIFOLDS 27 to a –form, which we call dθ , on C φ and hence ( ω , dθ ) is a stable Hamiltonianstructure on C φ . One may use the identification of F with a fiber of p : M → S and the flow of R to construct a diffeomorphism f : C φ → M that pulls back ( ω, dθ ) to ( ω , dθ ) . The upshot is that all symplectic bundles over S are equivalent,as stable Hamiltonian structures, to a mapping cylinder.We now observe that symplectic bundles over circles in the boundary of a sym-plectic manifold have standard neighborhoods. This is essentially Lemma 10 from[16]. Lemma 5.3.
Let ( W, ω ) be a symplectic manifold with a boundary component M suchthat p : M → S is a bundle and ω is a symplectic form on each fiber of p . Let ω (cid:48) denotethe restriction of ω to M . Then there is a neighborhood ( − (cid:15), × M of M in W such that ω takes the form ω (cid:48) + dt ∧ p ∗ dθ. In particular, a neighborhood of M in W is symplectomorphicto a piece of the symplectization of the stable Hamiltonian structure ( ω (cid:48) , λ = p ∗ dθ ) on M . Remark 5.4.
If in the statement of the Lemma 5.3 we had − M being a boundarycomponent of W , then the conclusion would be a neighborhood [0 , (cid:15) ) × M with thesame properties. Proof of Lemma 5.3.
Choose any identification of a neighborhood of M in W with ( − δ, × M , so that we have a projection from this neighborhood to S . Denotethe pull-back of dθ by dθ and let X be the unique vector field on the neighborhoodsuch that ι X ω = dθ . We claim that X transversely points out of M . To see thislet R be the Reeb field of ( ω (cid:48) , λ ) and compute ω ( X, R ) = λ ( R ) = 1 . Therefore, X cannot be tangent to M since R is in the kernel of ω restricted to M . Since theflow of X preserves ω we can use the backwards flow of X to create the desiredneighborhood. (cid:3) The above results will be used in the next section to study caps for some contactmanifolds supported by projective open books. For now we turn to the construc-tion of symplectic cobordisms between contact manifolds. In particular, we proveTheorem 1.22, which says: let ( M, ξ ) be a (2 n + 1) –dimensional closed contact man-ifold supported by an open book decomposition with page F , binding B = ∂F and monodromy φ . Suppose that X is an exact symplectic cobordism such that ∂X = − B ∪ B (cid:48) . Then there exists a strong symplectic cobordism W = M × [0 , (cid:91) B × D ×{ } = ∂ − X × D X × D such that ∂W = ∂ − W ∪ ∂ + W with ∂ − W = − M and ∂ + W = M (cid:48) where ( M (cid:48) , ξ (cid:48) ) is supported by the open book with page F ∪ X , binding B (cid:48) and monodromy φ (cid:48) which is identity on X and agrees with φ on F . Proof of Theorem 1.22. If dλ is the exact symplectic form on the given cobordism X ,then dλ (cid:48) is an exact symplectic form on X × D , where λ (cid:48) = λ + x dy − y dx . Let V denote the Liouville vector field of λ on X . It follows that Z = V + 12 (cid:18) x ∂∂x + y ∂∂y (cid:19) is the Liouville vector field for λ (cid:48) on X × D . Notice that ∂ ( X × D ) = ( X × ∂D ) ∪ ( ∂ − X × D ) ∪ ( ∂ + X × D ) . We consider M × [0 , as a part of the symplectization of ( M, ξ ) and using Z weglue the symplectic manifold ( X × D , dλ (cid:48) ) to M × [0 , by identifiying B × D × { } with ∂ − X × D as follows. The gluing of X × D can be viewed as a generalizedhandle attachment, which we depicted in Figure 4 below. In particular, the darkgreen on the left is obtained by the flow of Z applied to the lower boundary, ∂ − X × D (actually we apply it to ∂ − X × D (cid:48) where D (cid:48) is a slightly smaller disk in D asindicated in Figure 4). Therefore, it is part of the symplectization of ∂ − X × D ,and thus it can be identified with part of the symplectization of B × D = ∂ − X × D in the symplectization of M . As a result (after rounding corners) we get thesymplectic cobordism W indicated in the theorem.F IGURE
4. The ”handle” attachment. The green on the left is X × D .The red lines represent the flow lines of the Liouville field Z . Theblue is the smoothed boundary of the handle we want to attach. Theactual handle is depicted on the right.To finish the proof we need to check that the upper boundary ( M (cid:48) , ξ (cid:48) ) of thecobordism W is supported by the open book described in the theorem. First weobserve that the fibration of the complement of B × D in M extends to a fibration ofthe complement of B (cid:48) in M (cid:48) . This is because while attaching the “handle” X × D , TERATED PLANAR CONTACT MANIFOLDS 29 we perform a surgery on M removing B × D and gluing in ( X × ∂D ) ∪ ( ∂ + X × D ) so that B × { p } is identified with ∂ − X × { p } for each p ∈ ∂D . This shows that M (cid:48) admits an open book with page F ∪ X , binding B (cid:48) and monodromy φ (cid:48) which isidentity on X and agrees with φ on F . The contact structure ξ (cid:48) on M (cid:48) can be givenby the kernel of the -forms λ + x dt − y dx on ∂ + X × D , λ on X × ∂D , andthe original contact form on M minus B × D , respectively. These contact formsare clearly supported by the pages of the open book. One may easily check witha local model that when we “rounded corners” to construct the cobordism thecompatibility is preserved. (cid:3)
6. P
ROJECTIVE CONTACT MANIFOLDS
We begin this section with a quick proof of Theorem 1.17 which says that aniterated planar contact –manifold is projective. See Section 1.2 for the relevantdefinitions. Proof of Theorem 1.17.
The Weinstein page of an iterated planar open book support-ing a contact –manifold ( M , ξ ) is a symplectic filling of the planar binding of theopen book. In the proof of Theorem 4.1 in [19], the second author has shown thatany symplectic filling of a planar contact manifold embeds in a blowup of S × S ,and by possibly blowing up again we can assume that there is at least one blowup.But of course such a manifold is symplectomorphic to C P blowup some numberof times, which shows that the iterated planar open book above is projective, andhence ( M , ξ ) is projective. (cid:3) We now turn to building caps for some projective contact –manifolds. Specif-ically, we will prove Theorem 1.19 which says that if ( M , ξ ) is supported by anopen book whose page embeds as a convex domain in C P n C P for n ≤ , then ( M, ξ ) has a symplectic cap that contains an embedded C P n C P . Proof of Theorem 1.19.
Suppose ( M , ξ ) is supported by the open book ( X, φ ) where X embeds as a convex domain in F = C P n C P for n ≤ . Then we can ap-ply Theorem 1.23, using the symplectic cap Y = F \ X , to build a symplecticcobordism W from ( M, ξ ) to M (cid:48) , where M (cid:48) is a symplectic F –bundle over S . Let Φ : F → F be the monodromy of this bundle.In [33, Theorem 1.8], Li and Wu proved that for any symplectomorphism f : F → F , there are Lagrangian spheres in F such that f acts the same on homologyas a composition of Dehn twists about these spheres. In [32, Theorem 1.1], Li,Li, and Wu showed that for C P n C P with n ≤ , if a symplectomorphism actstrivially on the homology of F then it is symplectically isotopic to the identity map. This of course implies that f is symplectically isotopic to a composition of Dehntwists about the Lagrangian spheres. Let S i denote the Lagrangian spheres onegets when we apply this construction to Φ − .Now one can take the trivial Lefschetz fibration F × D and attach symplectic –handles along the spheres S i in different fibers of F × S to get a symplecticLefschetz fibration W (cid:48) over D so that the monodromy Ψ for ∂W (cid:48) is a compositionof Dehn twists about the S i (see Construction 2.9).Next we apply Lemma 5.2 and 5.3 to the isotopy from Φ to Ψ − to extend theabove symplectic cobordism W to go from ( M, ξ ) to a symplectic F –bundle over S with monodromy Ψ − . Now the diffeomorphism from − M = ∂W (cid:48) to M (cid:48) ⊂ ∂W given by F × [0 , / ∼ Ψ → F × [0 , / ∼ Φ ( p, t ) (cid:55)→ ( p, − t ) is a symplectomorphism on the fibers. Thus we can use Lemma 5.3 to glue W and W (cid:48) together along M (cid:48) to get the symplectic cap for ( M, ξ ) . (cid:3) R EFERENCES [1] Casim Abbas, Kai Cieliebak, and Helmut Hofer. The Weinstein conjecture for planar contactstructures in dimension three.
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EPARTMENT OF M ATHEMATICS , N
ORTHWESTERN U NIVERSITY , E
VANSTON , I
LLINOIS , USA
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EORGIA I NSTITUTE OF T ECHNOLOGY , A
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