Contact Manifolds with Flexible Fillings

Contact Manifolds with Flexible Fillings

Oleg Lazarev
Abstract.

We prove that all flexible Weinstein fillings of a given contact manifold with vanishing first Chern class have isomorphic integral cohomology; in certain cases, we prove that all flexible fillings are symplectomorphic. As an application, we show that in dimension at least 5 any almost contact class that has an almost Weinstein filling has infinitely many different contact structures. Similar methods are used to construct the first known infinite family of almost symplectomorphic Weinstein domains whose contact boundaries are not contactomorphic. We also prove relative analogs of our results, which we apply to Lagrangians in cotangent bundles.

Contents

1. Introduction

This paper is mainly concerned with the two related problems of distinguishing contact structures and classifying symplectic fillings of a given contact structure. We focus on distinguishing contact structures that have the same bundle-theoretic data, or almost contact class, and hence cannot be distinguished via algebraic topology. Non-contactomorphic contact structures in the same almost contact class are called exotic. This problem has a long history. Bennequin [5] constructed the first example of an exotic contact structure in the standard almost contact class on . In higher dimensions, Eliashberg [31] constructed an exotic contact structure in the standard almost contact class on for . This was generalized by Ustilovsky [65] who proved that every almost contact class on has infinitely many different contact structures; also see [37, 43]. van Koert [66] showed that many simply-connected 5-manifolds have infinitely many contact structures in the same almost contact class; see [23, 52, 64] for more examples.

One way to study contact manifolds is through their symplectic fillings, i.e symplectic manifolds whose contact boundary is contactomorphic to the given contact manifold. In this paper, we consider only fillings that are Liouville domains, which are certain exact symplectic manifolds, or Weinstein domains, which are Liouville domains that admit a compatible Morse function; see Section 2.1. One phenomenon is that if a contact manifold has a symplectic filling that satisfies an h-principle, i.e. is governed by algebraic topology, then the contact manifold itself is very rigid and remembers the topology of its fillings; see the discussion at the end of Section 1.1. The first result of this type is the Eliashberg-Floer-McDuff theorem [50]: all Liouville fillings of the standard contact structure are diffeomorphic to . Using this result, Eliashberg proved that the contact structures on in [31] are exotic. Similarly, quite a lot is known about contact manifolds with subcritical Weinstein fillings, which satisfy an h-principle; see Section 2.1. For example, M.-L.Yau [70] showed that linearized contact homology, which a priori depends on the filling, is a contact invariant for such contact manifolds. This can be used to prove that all fillings with vanishing symplectic homology of such contact manifolds have the same rational homology. Later Barth, Geiges, and Zehmisch [4] showed that in fact all Liouville fillings of simply-connected subcritically-filled contact manifolds are diffeomorphic; since has a subcritical filling , this result generalizes the Eliashberg-McDuff-Floer theorem.

However not much was known beyond the subcritical case. There are contact manifolds with many different fillings [56, 57] so the results in the subcritical setting do not hold in general. The main purpose of this paper is to extend those results to flexible fillings, which generalize subcritical fillings and also satisfy an h-principle; see Section 2.2. Flexible Weinstein domains are only defined for and so many of our results below require .

1.1. Contact manifolds with flexible fillings

In this paper, we will denote almost contact structures by and almost symplectic structures by ; see Section 2.1. We also assume the first Chern classes always vanish, even when this is not stated explicitly. Many of our results concern only Weinstein fillings and so it will often suffice to assume ; indeed by Proposition 2.1, if is a Weinstein filling of with , then vanishes if and only if does.

We begin by stating our main geometric result and some of its applications; its proof will be briefly discussed at the end of this section.

Theorem 1.1.

All flexible Weinstein fillings of a contact manifold with have isomorphic integral cohomology; that is, if are flexible fillings of , then as abelian groups.

Remark 1.2.

Theorem 1.1 also holds with any field coefficients and with homology instead of cohomology.

The cohomology long exact sequence of the pair and the fact that for for Weinstein domains show that for . In particular, for because of purely topological reasons. Therefore, Theorem 1.1 is interesting only for and . On the other hand, its stronger variant Corollary 4.2 applies to a larger class of Liouville domains and so is interesting for all . The long exact sequence also shows that for Weinstein , the rank of the intersection form equals

So Theorem 1.1 shows that the contact boundary also remembers the intersection form rank of its flexible fillings.

There are analogs of Theorem 1.1 in smooth topology, where fillings play a similar role. For example, Kervaire and Milnor [46] showed that the diffeomorphism type of an exotic -dimensional sphere bounding a parallelizable manifold is determined by the signature of this manifold (modulo some other integer). In particular, the smooth structure remembers the signature of its parallelizable fillings (again, modulo some fixed integer). Similarly, Theorem 1.1 shows that the contact structure remembers the cohomology of its flexible fillings. Work in progress with Y. Eliashberg and S. Ganatra [33] shows that the contact structure also remembers the signature of its fillings as an integer (not just as a residue modulo some other integer).

We also note that there are contact manifolds that do not have any flexible fillings, e.g. overtwisted manifolds. In this case, Theorem 1.1 is vacuous. In fact, there are contact structures that have Weinstein fillings but no flexible Weinstein fillings, e.g. if is simply-connected and spin or the Ustilovsky contact structures on ; see Remark 1.13 and Remark 1.15.

As seen in the Eliashberg-Floer-McDuff theorem [50], certain contact manifolds remember the diffeomorphism type of their fillings. This is because in some special cases cohomology is enough to determine diffeomorphism type. Sometimes cohomology can even determine almost symplectomorphism type. Since almost symplectomorphic flexible Weinstein domains are genuinely symplectomorphic (see Section 2.2), we can also use Theorem 1.1 to prove results about symplectomorphism type.

One application is to simply-connected 5-manifolds. In [62], Smale showed that any simply-connected 5-manifold with admits a smooth 2-connected filling that has a handle decomposition with only 0 and 3-handles. Smale also showed that such fillings of are unique up to boundary connected sum with ; we will suppose that every filling of is of the form , the boundary connected sum of with n copies of , for some . Note that is determined by its cohomology. We also note that admits a unique almost complex structure by obstruction theory since vanishes. Hence by the uniqueness h-principle in Section 2.2, admits a unique flexible Weinstein structure. More precisely, all flexible Weinstein structures on have symplectomorphic completions; the completion of is the open symplectic manifold with a conical symplectic form on the cylindrical end , see Section 2.1 for details. So in this case, the cohomology of the flexible Weinstein domain determines the symplectomorphism type of its completion.

The boundary of has a contact structure , which by definition has as a filling; this was first proven by Geiges in [40]. Since , these structures have . Geiges also showed that there is a unique almost contact structure with so these are all in the same almost contact class. Using Theorem 1.1, we can prove that these contact structures have unique flexible fillings.

Corollary 1.3.

Any contact structure on with a 2-connected flexible filling is of the form for some and all 2-connected flexible fillings of have completions that are symplectomorphic to . In particular, are in the same almost contact class but are pairwise non-contactomorphic.

We will provide more examples of exotic contact structures in Section 1.2.

Theorem 1.1 can also be used to show that the contact boundary remembers the symplectomorphism type of its fillings when the original flexible filling is smoothly displaceable in its completion, i.e. there is an embedding smoothly isotopic to the standard inclusion such that . To prove this result, we follow the approach of Barth, Geiges, and Zehmisch [4] of finding an h-cobordism between the standard filling and the new filling.

Corollary 1.4.

Suppose that has a flexible filling such that is smoothly displaceable in its completion and all vanish. Then all flexible Weinstein fillings of have symplectomorphic completions.

The condition that is smoothly displaceable in its completion restricts the topology of ; for example, the intersection form of must vanish.

Example 1.5.

If is a closed manifold with and , then all flexible Weinstein fillings of have symplectomorphic completions. In particular, for n odd, all flexible fillings of have symplectomorphic completions. Similarly, all flexible fillings of have symplectomorphic completions.

We will now sketch the proof of Theorem 1.1. The main technical tool in this paper is a certain Floer-theoretic invariant of Liouville domains called positive symplectic homology ; see Section 2.4. If we fix a contact manifold , then a priori positive symplectic homology of a Liouville filling of depends on . In Section 3, we define a certain collection of asymptotically dynamically convex contact structures, which generalize the dynamically convex contact structures from [3, 20, 44], and show that is independent of the filling for these structures and is therefore a contact invariant; see Definition 3.6 and Proposition 3.8. Our main result is that asymptotically dynamically convex contact structures are preserved under flexible contact surgery, i.e. surgery along a loose Legendrian [54]. This extends a similar result of M.-L.Yau [70] for subcritical surgery; see Theorems 3.15, 3.17, and 3.18. In particular, contact manifolds with flexible Weinstein fillings are asymptotically dynamically convex and so have independent of the filling; see Corollary 4.1. It is a standard fact that of a flexible domain always equals the singular cohomology of the domain (see Proposition 2.9) and so all flexible fillings have the same cohomology. We note that positive symplectic homology is the (non-equivariant) Floer-theoretic version of linearized contact homology and so this generalizes the result of M.-L.Yau mentioned earlier.

As we explain in Definition 3.6, asymptotically dynamically convex contact structures are essentially characterized by the fact that their Reeb orbits have positive degree. Hence to prove our main result that these structures are preserved under flexible surgery, we use Proposition 5.3 (see [8]) which gives a correspondence between the new Reeb orbits after flexible surgery and words of Reeb chords of the loose Legendrian attaching sphere. So it is enough to prove that Reeb chords of loose Legendrians have positive degree (possibly after Legendrian isotopy), which we do in our main geometric result Lemma 5.7. This lemma depends crucially on Murphy’s h-principle for loose Legendrians [54].

In certain special cases, there is an alternative approach to Theorem 1.1 which may provide some more geometric intuition for why it is true. In these cases, Theorem 1.1 for flexible domains can be reduced to the subcritical case. The idea is that flexible domains obey an embedding h-principle [36] so any flexible domain satisfying the appropriate topological conditions can embedded into a subcritical domain. Hence if its contact boundary had many different fillings, then the contact boundary of the subcritical domain would also have many different fillings as well by a cut-and-paste argument, violating known results in the subcritical case [4, 70, 55]. So if there is any contact rigidity (in the sense that some contact manifold remembers the topology of its filling), then the contact boundary of a flexible domain should also inherit this rigidity.

We will give a proof of this result here since it is independent of all other results in this paper. In the following, an almost symplectic embedding is an embedding such that can be deformed through non-degenerate two-forms to ; this is a purely topological notion.

Proposition 1.6.

Suppose has a flexible Weinstein filling that has an almost symplectic embedding into a subcritical Weinstein domain. Then all Liouville fillings of have the same integral homology as .

Remark 1.7.

The condition that has a smooth embedding into a subcritical domain implies that its intersection form vanishes.

Proof.

By the embedding h-principle [36], the topological condition that has an almost symplectic embedding into a subcritical Weinstein domain implies the geometric condition that admits a symplectic (in fact Liouville) embedding . Suppose is another filling of and consider the cut-and-pasted domain , which is another filling of . By Theorem 1.2 of [4], for all . Using the Mayer-Vietoris long exact sequence of the pair and the fact that for , we get for . Similarly, the Mayer-Vietoris sequence of the pair shows that . So for . By the classification of finitely-generated abelian groups, we can cancel from both sides and get for . For the case , we follow the same approach but use the relative Mayer-Vietoris sequences of the pairs and and the facts that and both vanish for ; the latter fact comes from Theorem 1.2 of [4] which shows that is an isomorphism for and . ∎

Example 1.8.

Any closed manifold embeds into and let be its tubular neighborhood. For example, is diffeomorphic to for any embedding . Then admits a Morse function with critical points of index at most and an almost complex structure obtained by restricting the almost complex structure on . By the existence h-principle, has a flexible Weinstein structure; see Section 2.2. This flexible structure automatically has an almost symplectic embedding into the subcritical domain and so Theorem 1.6 shows that all Liouville fillings of (with the induced contact structure) have the same homology. Also, the flexible domain has an almost symplectic embedding into the subcritical domain and so all Liouville fillings of have the same homology.

The proof of Proposition 1.6 works more generally than just for flexible Weinstein domains; any contact manifold with a Liouville filling that symplectically embeds into a subcritical domain remembers the homology of its fillings. But not any Weinstein domain that has an almost symplectic embedding into a subcritical domain has a genuine symplectic embedding there (and not every contact manifold has a unique filling [56]). So the flexible condition cannot be removed. We also note that Proposition 1.6, when it applies, is stronger than Theorem 1.1. For example, it restricts all Liouville fillings of , not just flexible Weinstein fillings. Furthermore, the condition that vanishes is not necessary in Proposition 1.6. Also, it is possible to combine Proposition 1.6 and Corollary 1.4 to get new examples of contact manifolds where all Liouville fillings are diffeomorphic.

Despite the strength of Proposition 1.6, the approach used to prove Theorem 1.1 has several important advantages, particularly for applications. First of all, not all flexible Weinstein domains admit smooth embeddings into subcritical domains; the non-vanishing of the intersection form obstructs this. Hence the condition in Proposition 1.6 is quite special. The constructions of exotic contact structures in the next section (see Theorem 1.9) all require using flexible domains with non-degenerate intersection forms and so these results cannot be proven using Proposition 1.6. Perhaps more importantly, the approach even works for certain non-flexible domains. For example, in Theorem 1.14 we will study non-flexible domains that are constructed by attaching flexible handles to asymptotically dynamically convex contact structures. These domains do not symplectically embed into subcritical domains even if they admit almost symplectic embeddings.

1.2. Exotic contact structures

Theorem 1.1 and its variants can also be used to construct exotic contact structures. As we noted in the Introduction, there are infinitely many contact structures in the standard almost contact class and on certain -manifolds [31, 65, 66]. Also, McLean [52] showed that if , , has a certain special Weinstein filling (an algebraic Stein filling with subcritical handles attached), then has at least two Weinstein-fillable contact structures in the same almost contact class. Here we extend these results by constructing infinitely many contact structures in a general setting.

In the following theorem, an almost Weinstein manifold is an almost symplectic manifold with a compatible Morse function; see Section 2.1 for details. Note that this is a purely algebraic topological notion. Also, let denote the boundary connected sum of and .

Theorem 1.9.

Suppose with , has an almost Weinstein filling . Then for any almost Weinstein filling of , there is a contact structure such that the following hold

  • is almost contactomorphic to

  • if , then are non-contactomorphic

  • has a flexible Weinstein filling almost symplectomorphic to .

There are infinitely many such with different integral cohomology and hence there are infinitely many contact structures in with flexible fillings.

Remark 1.10.

Using work of McLean [51], Cieliebak and Eliashberg [19] proved that any almost Weinstein domain admits infinitely many non-symplectomorphic Weinstein structures . The contact boundaries are in the same almost contact class but it is unknown whether they are contactomorphic. We show that at most of the Cieliebak-Eliashberg-McLean contact structures can have flexible fillings and so at most finitely many of them can coincide with the structures from Theorem 1.9.

The second part of Theorem 1.9 fails for (the first part does not make sense for since flexible Weinstein domains are defined only for ). For example, , the 3-torus and the lens space have finitely many Weinstein-fillable contact structures; see Chapter 16 of [19]. Also note that the condition is interesting only in degree since are Weinstein fillings of and hence have zero cohomology except in degree and ; see Remark 4.3 for a generalization where all degrees matter.

We also note that the question of whether an almost contact manifold admits an almost Weinstein filling, and hence a flexible one, is purely topological. This was first explored by Geiges [41] and then by Bowden, Crowley, and Stipsicz [12, 13], who gave a bordism-theoretic characterization of such almost contact manifolds. Combining this with Theorem 1.9, we get a topological criterion for almost contact manifolds to admit infinitely many different contact structures in the same formal class. The following application follows immediately from Theorem 1.9 and [12], [41].

Corollary 1.11.

If with is a simply-connected 5-manifold or a simply-connected 7-manifold with torsion-free , then has infinitely many different contact structures with flexible Weinstein fillings.

Also see Corollary 1.3. The 5-dimensional case of Corollary 1.11 (without the statement about flexible fillings) was proven by van Koert in Corollary 11.14 of [66]. As we noted before, there is at most one almost contact structure with on a simply-connected 5-manifold (see Lemma 7 of [40]) and hence there is at most one structure to which Corollary 1.11 applies. On the other hand, Bowden, Crowley, and Stipsicz [13] showed that many manifolds admit almost contact structures that have no almost Weinstein fillings, in which case Theorem 1.9 does not apply; for example, admits an almost contact structure which has no almost Weinstein filling.

Another application of Theorem 1.9 is to Question 6.12 of [49], where Kwon and van Koert asked whether there are infinitely many different contact structures in for . Ustilovsky [65] proved this for and Uebele [64] proved this for . Since is a Weinstein filling of , Theorem 1.9 provides an affirmative answer to this question, even within the possibly smaller class of Weinstein-fillable contact structures.

Corollary 1.12.

For , has infinitely many contact structures with flexible Weinstein fillings. For odd, this is true for all almost contact classes; furthermore, these contact structures are not contactomorphic to the Ustilovsky structures.

Remark 1.13.

The proof of this corollary also shows that although the Ustilovsky contact structures have Weinstein fillings, they do not have any flexible Weinstein fillings.

For an almost Weinstein filling of , the contact structure in Theorem 1.9 has a flexible Weinstein filling almost symplectomorphic to . Furthermore, with different are non-contactomorphic. Therefore different have fillings that are not even homotopy equivalent. Indeed, if the were almost symplectomorphic for different , then they would be Weinstein homotopy equivalent by the h-principle for flexible Weinstein domains and so their contact boundaries would be contactomorphic; see Section 2.1.

In light of this, one can ask whether there are exotic contact structures that bound almost symplectomorphic Weinstein domains, i.e. exotic Weinstein domains whose contact boundaries are also exotic. One such example is provided by , where are the exotic Weinstein structures on constructed by McLean [51] that are non-symplectomorphic for different ; see Remark 1.10. Although and are almost contactomorphic and admit almost symplectomorphic Weinstein fillings, Oancea and Viterbo [55] show that they are not contactomorphic. However, as noted in Remark 1.10, it is unknown whether are non-contactomorphic for different . More generally, it was unknown whether there exist infinitely many different contact structures bounding almost symplectomorphic Weinstein domains. Here we show that such examples do exist.

For the following theorem, let denote the set of smooth manifolds such that

  • is closed, simply-connected, and stably parallelizable

  • if is even, and if is odd,

Here denotes the semi-characteristic . Note these conditions are purely algebraic topological. We will also use to denote the free loop space of .

Theorem 1.14.

Suppose with , has an almost Weinstein filling . Then for any , there is a contact structure such that

  • is in for odd and in some fixed for even (depending only on and not )

  • if for some , then are non-contactomorphic

  • has a Weinstein filling almost symplectomorphic to , where is a certain plumbing of depending only the dimension (and not ).

In particular, for there are infinitely many different contact structures in or that admit almost symplectomorphic Weinstein fillings.

Remark 1.15.
  1. The fillings are not flexible since otherwise they would be Weinstein homotopic by the h-principle for flexible domains and so would be contactomorphic. In fact, does not have any flexible fillings. As a result, the contact structures in Theorem 1.14 are different from the structures in Theorem 1.9.

  2. In the last part of Theorem 1.14 involving the infinite collection of contact structures, we do not claim that all fillings of a given contact manifold in this collection are almost symplectomorphic but rather than every contact manifold in this collection admits some Weinstein domain filling and these particular Weinstein domains are all almost symplectomorphic.

1.3. Legendrians with flexible Lagrangian fillings

We also prove relative analogs of our results for Legendrians. In particular, we define the class of asymptotically dynamically convex Legendrians, show that positive wrapped Floer homology (the relative analog of ) is an invariant for these Legendrians, and prove that Legendrians with flexible Lagrangian fillings are asymptotically dynamically convex; see Definition 7.1, Proposition 7.8, and Corollary 7.22.

We now give some geometric applications of these results. In this paper, we will assume all Legendrians and Lagrangians are connected, oriented, spin, and and vanish.

As for contact manifolds, it is known that certain Legendrians remember the topology of their exact Lagrangian fillings. For example, a relative analog of the Eliashberg-McDuff-Floer theorem states that any exact Lagrangian filling of the standard Legendrian unknot in is diffeomorphic to ; see [16], [1]. Recently, Eliashberg, Ganatra, and the author introduced the class of flexible Lagrangians [34]. These Lagrangians are the relative analog of flexible Weinstein domains. In fact, they are always contained in flexible Weinstein domains and so are defined only for . Problem 4.13 of [34] asked whether the Legendrian boundary of a flexible Lagrangian remembers the topology of the filling. The following relative analog of Theorem 1.1 gives an affirmative answer to this question assuming some minor topological conditions.

Theorem 1.16.

If , has a flexible Lagrangian filling and , then all exact Lagrangian fillings of in all flexible Weinstein fillings of have the isomorphic cohomology, i.e. if is a flexible filling of and is an exact Lagrangian filling of , then .

In fact, Theorem 1.16 holds for if admits a proper Morse function whose critical points have index less than ; for , this is equivalent to by Smale’s handle-trading trick. The result also holds with any field coefficients. Note that here does not have be flexible. So this result is slightly stronger than its contact analog Theorem 1.1 which is concerned only with flexible Weinstein fillings. We also note that work of Ekholm and Lekili [28] implies that certain Legendrians (with special Reeb chord conditions that should be satisfied for Legendrians with flexible fillings) also remember the homology of the based loop space of their Lagrangian fillings.

As in the contact case, we can use Theorem 1.16 to construct infinitely many non-isotopic Legendrians in the same formal Legendrian class, i.e. with the same algebraic topological data; see Section 2.2. The first exotic examples were 1-dimensional Legendrians in constructed by Chekanov [17]. Higher dimensional examples were found in [25] by Ekholm, Etnyre, and Sullivan who produced infinitely many non-isotopic Legendrians spheres and tori in in the same formal class. However these Legendrians are all nullhomologous in . Björklund [6] showed that if is a closed surface, there are arbitrarily many 1-dimensional Legendrians in that are formally isotopic representing any class in . The following analog of Theorem 1.9 generalizes these results. Here is as in Theorem 1.14.

Theorem 1.17.

Suppose is a formal Legendrian that has a formal Lagrangian filling in a flexible filling of and . Then for any , there is a Legendrian such that

  • is formally isotopic to

  • if , then are not Legendrian isotopic

  • has a flexible Lagrangian filling diffeomorphic to .

In particular, there are infinitely many non-isotopic Legendrians in that are formally isotopic to and have flexible Lagrangian fillings in .

Remark 1.18.

A similar result was proven by Eliashberg, Ganatra, and the author in [34]. In their situation, and if , then are not Legendrian isotopic.

As for contact manifolds, this theorem fails for . Because of the Bennequin inequality, there are formal Legendrians with no genuine Legendrian representatives and hence the first part of Theorem 1.17 fails; see [19] for example. Furthermore, there are formal Legendrians that have a unique Legendrian representation up to isotopy, in which case the second part fails; for example, all Legendrians in that are topologically unknotted and formally isotopic are Legendrian isotopic [32]. However, like Theorem 1.16, the first part of Theorem 1.17 does hold for if admits a Morse function whose critical points have index less than ; the second part about infinitely many Legendrians also holds for .

We can also use Theorem 1.16 to give a new proof of the homotopy equivalence version of the nearby Lagrangian conjecture for simply-connected Lagrangians intersecting a cotangent fiber once. In the following, let be the projection to the zero-section.

Corollary 1.19.

Suppose is simply-connected and is a closed exact Lagrangian with zero Maslov class intersecting transversely in a single point for some . Then is an isomorphism; hence if is simply-connected, is a homotopy equivalence.

The fact that is homotopy equivalence was first proven by Fukaya, Seidel, and Smith [39], whose result did not require to intersect in a single point. Although our intersection condition is quite restrictive and immediately implies that is surjective, our approach seems to be fairly elementary and does not use Fukaya categories or spectral sequences.. In addition, our approach seems to be adaptable and can be generalized to certain other Weinstein domains like plumbings; see Section 7.3.

This paper is organized as follows. In Section 2, we present some background material. In Section 3, we introduce asymptotically dynamically convex contact structures and prove their main properties. In Section 4, we prove the results stated in Section 1.1 and 1.2, assuming Theorem 3.18 that flexible surgery preserves asymptotically dynamically convex contact structures. In Section 5, we prove our main geometric result about Reeb chords of loose Legendrians. In Section 6, we prove Theorem 3.18 modulo a technical lemma that we prove in the Appendix. In Section 7, we prove the relative versions of our results for Legendrians stated in Section 1.3. In Section 8, we present some open problems.

Acknowledgements

I would like to thank my Ph.D. advisor Yasha Eliashberg for suggesting this problem and for many inspiring discussions. I am also grateful to Kyler Siegel and Laura Starkston for providing many helpful comments on earlier drafts of this paper. I also thank Matthew Strom Borman, Tobias Ekholm, Sheel Ganatra, Jean Gutt, Alexandru Oancea, Joshua Sabloff, and Otto van Koert for valuable discussions. This work was partially supported by a National Science Foundation Graduate Research Fellowship under grant number DGE-114747.

2. Background

2.1. Liouville and Weinstein domains

We first review the relevant symplectic manifolds and their relationship to contact manifolds. All our symplectic manifolds will be exact and either compact with boundary or open. A Liouville domain is a pair such that

  • is a compact manifold with boundary

  • is a symplectic form on

  • the Liouville field , defined by , is outward transverse along

A Weinstein domain is a triple such that

  • is a Liouville domain

  • is a Morse function with maximal level set

  • is a gradient-like vector field for .

Since is compact and is a Morse function with maximal level set , has finitely many critical points. Liouville and Weinstein cobordisms are defined similarly.

The fact that is outward transverse to implies that is a contact manifold. If a contact manifold is contactomorphic to , then we say that is a Liouville or Weinstein filling of . Because is defined on all of and points outward along , the flow of is defined for all negative time. In particular, the negative flow of identifies a subset of with the negative symplectization of ; here is the second coordinate on . We can also glue the positive symplectization of to along . The result is the completion of , an open exact symplectic manifold; here in and in . Note that is a complete vector field in . In order to avoid trivial invariants like volume, one usually speaks of symplectomorphisms of completed Weinstein domains rather than symplectomorphisms of domains themselves. In this paper, the negative symplectization of in will play a more important role than the completion of ; see Remark 3.13.

The natural notion of equivalence between Weinstein domains is a Weinstein homotopy, i.e. a 1-parameter family of Weinstein structures , connecting them, where is allowed to have birth-death critical points. Weinstein domains that are Weinstein homotopic have exact symplectomorphic completions and contactomorphic contact boundaries; see [19]. We note that the notion of Weinstein homotopy between Weinstein manifolds is more general and does not necessarily imply that the contact manifolds at infinity are contactomorphic (or even diffeomorphic); see [21].

2.1.1. Weinstein handle attachment and contact surgery

A Weinstein structure yields a special handle-body decomposition for . First, recall that vanishes on the -stable disc of a critical point ; see [19]. In particular, is isotropic with respect to and so all critical points of have index less than or equal to . If all critical points of have index strictly less than , then the Weinstein domain is subcritical. Also, is transverse to any regular level of and so is a contact manifold; similarly is a Weinstein subdomain of . Since vanishes on , then is an isotropic sphere, where for sufficiently small . Furthermore, comes with a parametrization and framing, i.e. a trivialization of its normal bundle. Note that a framing of is equivalent to the framing of the conformal symplectic normal bundle of ; see [42]. Hence parametrized Legendrians come with a canonical framing.

Suppose that are regular values of and contains a unique critical point of . Then is an elementary Weinstein cobordism between and and the symplectomorphism type of is determined by the symplectomorphism type of along with the framed isotopy class of the isotropic sphere . If the critical values of are distinct, then can be viewed as the concatenation of such elementary Weinstein cobordisms.

On the other hand, one can explicitly construct such cobordisms and use them to modify Liouville domains or contact manifolds. Given a Liouville domain and a framed isotropic sphere in its contact boundary , we can attach an elementary Weinstein cobordism with critical point and to and obtain a new Liouville domain . This operation is called Weinstein handle attachment and is called the attaching sphere of the Weinstein handle. If was Weinstein, then so is . The contact boundary of is the result of contact surgery along and the Weinstein handle gives an elementary Weinstein cobordism between and ; see Proposition 6.3 for details. If the dimension of is less than , the handle attachment, surgery, and itself are all called subcritical. Therefore, any (subcritical) Weinstein domain can be obtained by attaching (subcritical) Weinstein handles to the standard Weinstein structure on ; similarly, the contact boundary of any (subcritical) Weinstein domain can be obtained by doing (subcritical) contact surgery to .

2.1.2. Formal structures

There are also formal versions of symplectic, Weinstein, and contact structures that depend on just the underlying algebraic topological data. For example, an almost symplectic structure on is an almost complex structure on ; this is equivalent to having a non-degenerate (but not necessarily closed) 2-form on . An almost symplectomorphism between two almost symplectic manifolds is a diffeomorphism such that can be deformed to through almost complex structures on . An almost Weinstein domain is a triple , where is a compact almost symplectic manifold with boundary and is a Morse function on with no critical points of index greater than and maximal level set . An almost contact structure on is an almost complex structure on the stabilized tangent bundle of . Therefore an almost symplectic domain has almost contact boundary ; it is an almost symplectic filling of this almost contact manifold. Note that any symplectic, Weinstein, or contact structure can also be viewed as an almost symplectic, Weinstein, or contact structure by considering just the underlying algebraic topological data.

Note that the first Chern class is an invariant of almost symplectic, almost Weinstein, or almost contact structures. In this paper, we will often need to assume that vanishes. The following proposition, which will be used several times in this paper, shows that the vanishing of is often preserved under contact surgery and furthermore implies the vanishing of .

Proposition 2.1.

Let be an almost Weinstein cobordism between . If , the vanishing of and and are equivalent. If , the vanishing of and are equivalent.

Proof.

Let be inclusions. Then so the vanishing of implies the vanishing of and . To prove the converse, consider the cohomology long exact sequences of the pairs and :

By assumption, vanishes and hence is injective. By Poincaré-Lefschetz duality, . Since for and is a Weinstein cobordism, vanishes and hence is also injective. Then if either or vanish, so does .

If , we just need the vanishing of , which holds for . ∎

2.2. Loose Legendrians and flexible Weinstein domains

Murphy [54] discovered that a certain class of loose Legendrians satisfy a h-principle. That is, the symplectic topology of these Legendrians is governed by algebraic topology. There are several equivalent criteria for a Legendrian to be loose, all of which depend the existence of a certain local model inside this Legendrian. We will use the following local model from Section 2.1 of [15]. Let be a unit ball and let be the 1-dimensional Legendrian whose front projection is shown in Figure 1. Let be a closed manifold and a neighborhood of the zero-section . Then is a Legendrian submanifold. This Legendrian is the stabilization over of the Legendrian .

Definition 2.2.

A Legendrian is loose if there is a neighborhood of such that is contactomorphic to .

Remark 2.3.

If is an equidimensional contact embedding and is loose, then is also loose.

Figure 1. Front projection of

A formal Legendrian embedding is an embedding together with a homotopy of bundle monomorphisms covering for all such that and is a Lagrangian subspace of . A formal Legendrian isotopy is an isotopy through formal Legendrian embeddings. Murphy’s h-principle [54] has an existence and uniqueness part:

  • any formal Legendrian is formally Legendrian isotopic to a loose Legendrian

  • any two loose Legendrians that are formally Legendrian isotopic are genuinely Legendrian isotopic.

We now define a class of Weinstein domains introduced in [19] that are constructed by iteratively attaching Weinstein handles along loose Legendrians.

Definition 2.4.

A Weinstein domain is flexible if there exist regular values of such that and for all , is a Weinstein cobordism with a single critical point whose the attaching sphere is either subcritical or a loose Legendrian in .

Flexible Weinstein cobordisms are defined similarly. Also, Weinstein handle attachment or contact surgery is called flexible if the attaching Legendrian is loose. So any flexible Weinstein domain can be constructed by iteratively attaching subcritical or flexible handles to . A Weinstein domain that is Weinstein homotopic to a Weinstein domain satisfying Definition 2.4 will also be called flexible. Loose Legendrians have dimension at least so if is the result of flexible contact surgery on , then by Proposition 2.1 vanishes if and only if does. Finally, we note that subcritical domains are automatically flexible.

Our definition of flexible Weinstein domains is a bit different from the original definition in [19], where several critical points are allowed in . There are no gradient trajectories between these critical points and their attaching spheres form a loose link in , i.e each Legendrian is loose in the complement of the others. In this paper, we prefer to work with connected Legendrians, which is why we allow only one critical point in each cobordism ; hence all critical points have distinct critical values. These two definitions are the same up to Weinstein homotopy.

Since they are built using loose Legendrians, which satisfy an h-principle, flexible Weinstein domains also satisfy an h-principle [19]. Again, the h-principle has an existence and uniqueness part:

  • any almost Weinstein domain admits a flexible Weinstein structure in the same almost symplectic class

  • any two flexible Weinstein domains that are almost symplectomorphic are Weinstein homotopic (and hence have exact symplectomorphic completions and contactomorphic boundaries).

2.3. Symplectic homology

In this section, we review the symplectic homology of a Liouville domain . We will follow the conventions for signs and grading in symplectic homology used in [20]. Let be the contact boundary of . As before, let be the completion of obtained by attaching the positive symplectization of to ; here is the cylindrical coordinate on . The contact manifold has a canonical Reeb vector field defined by and ; periodic orbits of are called Reeb orbits. The action of a Reeb orbit is

(2.1)

Note that is always positive and equals the period of . Let denote the set of actions of all Reeb orbits of .

We say that a Reeb orbit of is non-degenerate if the linearized Reeb flow from to for some does not have as an eigenvalue; similarly, we say that the contact form is non-degenerate if all Reeb orbits of are non-degenerate. A generic contact form is non-degenerate and we can assume that any contact form is non-degenerate after a -small modification. If is non-degenerate, then is a discrete subspace of . All Reeb orbits that we discuss in this paper will be non-degenerate. However we only work with orbits below a fixed action and so contact forms do not have to be non-degenerate, i.e there may be high-action orbits that are degenerate.

2.3.1. Admissible Hamiltonians and almost complex structures

To define symplectic homology, we need to equip with a certain family of functions and almost complex structures. Let denote the class of admissible Hamiltonians, which are functions on defined up to smooth approximation as follows:

  • in

  • is linear in with slope in .

More precisely, is a -small Morse function in and in for some function that is increasing convex in a small collar of and linear with slope outside this collar; for example, see [43] for details. Often, we will just say that is increasing convex near , by which we mean in such a collar.

For , the Hamiltonian vector field is defined by the condition . The time-1 orbits of are called the Hamiltonian orbits of and fall into two classes depending on their location in :

  • In , the only Hamiltonian orbits are constants corresponding to Morse critical points of

  • In , we have , where is the Reeb vector field of . So all Hamiltonian orbits lie on level sets of and come in -families corresponding to reparametrizations of some Reeb orbit of with period .

Since the slope of at infinity is not in , all Hamiltonian orbits lie in a small neighborhood of in . After a -small time-dependent perturbation of , the orbits become non-degenerate, i.e. the linearized Hamiltonian flow from to , for some in the Hamiltonian orbit, does not have as an eigenvalue. These non-degenerate orbits also lie in a neighborhood of and so there are only finitely many of them. In fact, under this perturbation, each -family of Hamiltonian orbits degenerates into two Hamiltonian orbits; see for example [10].

We say that an almost complex structure is cylindrical on the symplectization if it preserves is independent of and compatible with , and . Let denote the class of admissible almost complex structures on which satisfy

  • is compatible with on

  • is cylindrical on .

2.3.2. Floer complex

For , the Floer complex is generated as a free abelian group by Hamiltonian orbits of that are contractible in . Note that we can work with integer coefficients rather than Novikov ring coefficients since all the symplectic manifolds in this paper are exact. We will often write this complex as when we do not need to specify .

The differential is given by counts of Floer trajectories. In particular, for two Hamiltonian orbits of , let be the moduli space of smooth maps such that and