Embedding contact 3–manifolds

Embedding all contact –manifolds in a fixed contact 5–manifold

Abstract.

In this note we observe that one can contact embed all contact 3–manifolds into a Stein fillable contact structure on the twisted –bundle over and also into a unique overtwisted contact structure on . These results are proven using “spun embeddings” and Lefschetz fibrations.

1. Introduction

A basic question in geometric topology is the embedding problem of manifolds: given two (smooth) manifolds and , can one find a smooth embedding of into ? In particular, given what is a simple space in which can be embedded? Seeing an abstract manifold as a submanifold of a simple space can provide concrete ways to describe the manifold as well as new avenues to study the manifold. Whitney proved, by using transversality arguments and the famous “Whitney trick” that any –manifold embeds in [33]. This is the smallest possible Euclidean space for which one can prove such a general theorem, though for specific manifolds and specific values of one can do better. For example, Hirsch [17] proved that all oriented –manifolds embed in , and Wall [32] then removed the orientability assumption. There are homological (and other) obstructions to embedding –manifolds in , though Freedman [7] did show that all homology –spheres topologically, locally flatly embed in . (To be clear, in this paper we will always be considering smooth embeddings unless explicitly stated otherwise.) Since one cannot embed all –manifolds in one might ask is there a, let us say, compact 4–manifold into which all –manifolds embed? Shiomi [28] answered this in the negative. So can certainly be said to be the simplest manifold into which all –manifolds embed. Let us also mention that once an embedding is found, one could also ask the question of whether it is unique (up to isotopy). For example, the Schoenflies problem which concerns smooth embeddings of into is among the most famous open problems of topology.

This paper is concerned with the contact analog of the above discussion: What is the simplest contact manifold into which all co-oriented contact –manifolds embed? (Throughout this paper all contact structures will be assumed to be co-oriented.) Recall that a smooth embedding is said to be a contact embedding of the contact manifold into if and . In the case is -dimensional and is -dimensional, a contact embedding is also known as a transverse knot.

The analog of Whitney’s theorem, proven by Gromov [15], is that any contact –manifold can be embedded in the standard contact structure on . In particular, a contact –manifold can be embedded in . As having an embedding in is the same as having one in and we prefer to work with compact manifolds we will switch to consider spheres instead of Euclidean spaces. This result was reproved in [22], based on work in [23], using open book decompositions (which will be a key ingredient in the work presented in this paper too, but they will be used in a fairly different way). Showing that Hirsch’s result does not generalize to the contact category we have the following result of Kasuya.

Theorem 1.1 (Kasuya, 2014 [20]).

If embeds in , in and , then .

Recall that a contact structure has a symplectic structure given by and associated to this symplectic structure there is a unique homotopy class of a compatible complex structure on . When referring to the Chern classes of we are using this complex structure. We will reprove and extend Theorem 1.1 in Section 4.1.

From this theorem we see that there are many contact –manifolds that do not embed in . One might hope that a contact –manifold embeds in if and only if it has trivial Chern class. While this is still an open question the following partial results are obtained in a recent work of the first author and Furukawa.

Theorem 1.2 (Etnyre-Furukawa, 2017 [6]).

If is a –manifold with no –torsion in its second homology then an overtwisted contact structure embeds in if and only if its first Chern class is zero.

If is , , or a lens space with odd and arbitrary, or when is even and or respectively, then a contact structure on embeds in if and only if it has trivial first Chern class.

Kasuya [20] also proved that given a contact –manifold with trivial first Chern class, there is some contact structure on into which it embeds, but the contact structure on could depend on the contact –manifold and it may not necessarily be standard at infinity. Slightly extending Kasuya’s proof by using recent work of Borman, Eliashberg, and Murphy [2] one can show the following.

Theorem 1.3 (Etnyre-Furukawa, 2017 [6]).

A contact –manifold embeds into the unique overtwisted contact structure on if and only if it has trivial first Chern class.

The overtwisted contact structure on is somewhat mysterious (though it has a simple description in terms of open book decompositions), so it would still be of great interest to have a similar theorem for embedding into . It is at least known that any contact 3–manifold embeds in a symplectically fillable contact 5–manifold [4] and also a hyper-tight contact 5–manifold [12], but it should be noted that in these results the contact 5–manifold depends on the contact 3–manifold being embedded.

Continuing with the search for a simple contact –manifold into which all contact –manifolds can embed we must consider -manifolds other than the –sphere. The next simplest class of manifolds to consider are products of spheres. Below we will see that a theorem analogous to Theorem 1.1 implies that there is no contact structure on into which one can embed all contact –manifolds. In addition, if there is a contact structure on into which all contact –manifolds embed, it must have first Chern class , see Corollary 4.2. Indeed, we show that there is such a contact structure:

Theorem 1.4.

There is (up to contactomorphism) a unique overtwisted contact structure on into which all contact –manifolds embed with trivial normal bundle.

Remark 1.5.

At the moment the authors do not know if there are embeddings of oriented 3–manifolds into with non-trivial normal bundles.

One would still like an embedding theorem with a “nicer” or more “standard” contact structure. To this end we ask:

Question 1.6.

Is there a symplectically fillable contact structure on into which all contact –manifolds embed?

While we expect the answer to this question is yes, we can prove an analogous result for the unique non-trivial -bundle over . (Note that since , there are exactly two -bundles over ).

Theorem 1.7.

There is a Stein fillable contact structure on the (unique) twisted -bundle over into which all contact –manifolds embed.

Our main approach to this theorem is via open book decompositions and Lefschetz fibrations. In particular, we explore contact “spun embeddings” (see Section 3) where one embeds one manifold into another by embedding the pages of an open book for one into the pages for the other. Such embeddings have been studied under the name of spun knots [8], and they have even been studied in the context of contact geometry. Specifically by Mori (in [23] and unpublished work) and a few observations were made about spun embeddings in [6] and how they relate to braided embeddings. For more recent results also see [25].

Acknowledgments: We are grateful to Patrick Massot and the referee for many helpful suggestions. The first author was partially supported by NSF grant DMS-1608684. The second author was partially supported by the Royal Society (URF) and the NSF grant DMS-1509141.

2. Lefschetz fibrations and open book decompositions

In this section we recall the notion of an open book decomposition and its relation to contact structures on manifolds. We then recall some basic facts about Lefschetz fibrations and symplectic manifolds. We end this section by reviewing overtwisted contact structures in high dimensions.

2.1. Open book decompositions

Given a manifold with boundary and a diffeomorphism whose support is contained in the interior of the manifold one can consider the mapping torus

where is the relation . It is easy to see that and thus there is an obvious way to glue to to obtain a manifold, which we denote by . We say that a manifold has an open book decomposition if is diffeomorphic to . We call the monodromy of the open book.

Notice that the manifold , where is the center of , is a sub-manifold of with neighborhood and that fibers over the circle

so that the fibration on is simply projection to the coordinate of (where is given polar coordinates ). So the image of in also has these properties, we denote this submanifold of by too, and the fibration of its complement by . We call an open book decomposition of too. Notice that, up to diffeomorphism, and each determine the other, so we can describe open books using either definition depending on the situation. We call the binding of the open book and a page of the open book for any .

Example 2.1.

As a simple example consider any manifold and . One may easily see that is diffeomorphic to glued to in the obvious way. (That is, is glued to by the identity map, the two copies of are glued by the identity map, and is glued to by the map .) But of course this is simply the double of .

As a concrete example, consider the -bundle over with Euler number . Then is the -bundle over with Euler number (there are only two such bundles determined by the residue mod ). The double of is clearly and the double of the twisted -bundle over is clearly the twisted -bundle over , .

The fundamental connection between open books and contact structures is given in the following theorem.

Theorem 2.2 (Thurston-Winkelnkemper 1975, [31] for and Giroux 2002, [13] for arbitrary ).

Given a compact –manifold , a diffeomorphism with support contained in the interior of and a -form on such that

  1. is a symplectic form on ,

  2. the Liouville vector field defined by

    points out of , and

  3. ,

then admits a unique, up to isotopy, contact structure whose defining -form satisfies

  1. is a positive contact from on the binding of the open book and

  2. is a symplectic form when restricted to each page of the open book.

The contact structure guaranteed by the theorem is said to be supported by or compatible with the open book. This terminology is due to Giroux [13] and the uniqueness part of the theorem, even in dimension , was established by Giroux.

As we will need the details of the proof for our work we sketch Giroux’s proof of this theorem here.

Sketch of proof.

We begin by assuming that for some function . Of course, by adding a constant we can assume that is bounded above by a large negative constant. We notice that is a contact form on where has coordinate . There is an action of on generated by the diffeomorphism . Clearly is invariant under this action and descends to a contact form on which is canonically diffeomorphic to . One may easily find a non-decreasing function that is equal to near , and a positive but non-increasing function such that the contact form

on , where is given polar coordinates, can be glued to the contact from on to get a contact form on , cf. [10].

Now if is not exact then let be the vector field on satisfying

and its flow. We note for future reference that as is equal to the identity near we know is zero there and is also supported on the interior of . A computation shows that satisfies and of course is diffeomorphic to since is isotopic to .

One may easily see that the constructed contact structure is compatible with the open book. As the details are not needed here, we leave the uniqueness of the compatible contact structure to the reader. For a somewhat different proof of this theorem see [14]. ∎

Remark 2.3.

We make a few observations for use later. First notice that one may use the from , for any , to construct the contact structure on the mapping torus part of . In addition, notice that when the monodromy is the identity, then construction in the proof is particularly simple.

Finally we observe that if is -dimensional then we can find a 1-parameter family of 1-forms on interpolating between and . Then one easily checks that for large enough is a contact form on the mapping torus part of that can be extended over the binding as was done in the proof above.

Example 2.4.

Continuing Example 2.1 let be a Stein domain in the Stein manifold . Then the Stein structure on gives a canonical 1–form on as in Theorem 2.2. Let be the contact structure on supported by the open book . From Example 2.1 we know that is the boundary of . We note that can be given the structure of a Stein domain (indeed on consider , where is a strictly pluri-subharmonic function. It is clear that is also strictly pluri-subharmonic and will define a domain diffeomorphic to ). It is also easy to check that the Stein domain is a filling of (cf. [5, Prop. 3.1]).

Giroux also proved the following “converse” to the Theorem 2.2.

Theorem 2.5 (Giroux 2002, [13]).

Every contact structure on a closed –manifold is supported by some open book decomposition.

2.2. Lefschetz fibrations

A Lefschetz fibration of an oriented 4–manifold is a map where is an oriented surface and is surjective at all but finitely many points , called singular points, each of which has the following local model: each point has a neighborhood that is orientation preserving diffeomorphic to an open set in , has a neighborhood that is orientation preserving diffeomorphic to in and in these local coordinates is expressed as the map . We say is an achiral Lefschetz fibration if there is a map as above except the local charts expressing as do not have to be orientation preserving.

We state a few well known facts about Lefschetz fibrations, see for example [24]. Let be a Lefschetz fibration with the set of singular points.

  1. Setting and the map is a fibration with fiber some surface .

  2. Fix a point and for each , let be a path in from to whose interior is in . Then there is an embedded curve on that is homologically non-trivial in but is trivial in the homology of . The curve is called the vanishing cycle of (though it also depends on the arc ). We will assume that for all .

  3. For each let be a disk in containing in its interior, disjoint from the for , and intersecting in a single arc that is transverse to . The boundary is a -bundle over . Identifying a fiber of with using , the monodromy of is given by a right-handed Dehn twist about .

  4. If the surface is the disk then let be a disk containing and intersecting each in an arc transverse to . The manifold can be built from by adding a –handle for each along sitting in where with framing one less than the framing of given by in . Conversely, any 4-manifold constructed from by attaching –handles in this way will correspond to a Lefschetz fibration.

  5. If is an achiral Lefschetz fibration then the above statements are still true but for an “achiral” singular point the monodromy Dehn twist is left-handed and the –handle is added with framing one greater than the surface framing.

The following theorem is well-known and provides a way to study symplectic/contact topology in low-dimensions via Lefschetz fibrations, in turn, via the theory of mapping class groups of surfaces.

Theorem 2.6.

Suppose that is a 4–manifold that admits a Lefschetz fibration . If the fiber of is not null-homologous then admits a symplectic structure in which each fiber of is symplectic. If moreover, the fibers of are surfaces with boundary, and the vanishing cycles are non-separating, can be taken to be exact and a strong filling of a contact structure on .

There is an open book for supporting that can be described as follows:

  1. We may assume is the unit disk in and is a regular value of . The binding is .

  2. The projection is simply the composition of restricted to and projection to the coordinate of (where is given polar coordinates ).

  3. There is a subdisk of the base containing and all the critical points of such that is a neighborhood of the binding and if is the Liouville field for then the contact form is given by on each component of .

Sketch of proof.

In the case when the fibers are closed, and there exists a class in that integrates to a positive number along the fiber, the existence of a symplectic form is due to Gompf [11], which in turn relies on a classical argument of Thurston [30]. The case when the fibers are surfaces with boundary (or more generally are exact symplectic manifolds) and the vanishing cycles are non-separating (or more generally exact Lagrangian spheres) is studied extensively by Seidel in [27, Chapter 3]. Note that on a punctured surface equipped with an exact symplectic form, we can isotope any non-separating curve to an exact Lagrangian (unique up to Hamiltonian isotopy), then by [27, Lemma 16.8], we can construct an exact symplectic structure on and an exact Lefschetz fibration on . The properties (1)-(3) listed above are consequences of the definition of an exact Lefschetz fibration given in [27, Section 15(a)]. In particular, for the property (3) see [27, Remark 15.2], where the triviality of the symplectic connection along the horizontal boundary can be arranged in our case since the base of our exact Lefschetz fibration, being , is contractible. ∎

Proposition 2.7.

Let be a surface of genus and the curves shown in Figure 1. Then the disk bundle over with Euler number has a Lefschetz fibration with vanishing cycles , and the disk bundle over with Euler number has a Lefschetz fibration with vanishing cycles . The Lefschetz fibration gives these manifolds an exact symplectic structure that is independent of the genus of . Similarly has a Lefschetz fibration with vanishing cycles .

{overpic}

surface

Figure 1. A surface of genus with one boundary component and curves marked.
Proof.

One may easily describe a handle presentation for the 4–manifold described by the Lefschetz fibration in the proposition, see [24]. The handlebody picture in Figure 2 is one such description when the genus is . There is an obvious extension of this picture to the higher genus case. The 1 and –handles for higher genus surfaces all cancel and do not interact with and .

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stein

Figure 2. Handle presentation describing manifolds in Proposition 2.7 when the genus is . All –handles have framing . The 1–handles are paired as indicated by the numbers.

Notice that all the 1 and –handles cancel if we do not have the vanishing cycles and . Thus the manifold is . After this cancellation the curves and are both unknots with the former having framing and the later having framing . Thus the first Lefschetz fibration in the theorem gives disk bundle over the sphere with Euler number and the Lefschetz fibration with replacing results in the disk bundle over the sphere with Euler number .

It is well known how to turn these handlebody diagrams into Stein handlebody diagrams, see again [24], and thus our manifolds have an exact symplectic structure. As the and –handles for higher genus surfaces symplectically cancel we see this structure is independent of the genus. ∎

2.3. Overtwisted contact structures

Recall an almost contact structure on a –dimensional manifold is a reduction of its structure group to . This is easily seen to be a necessary condition for a manifold to admit a contact structure. We will be mainly considering contact structures in dimension so a reduction of the structure group corresponds to a section of the -bundle associated to the tangent bundle of . It is known that is diffeomorphic to , for this and other facts below see for example [10], and thus the only (and primary) obstruction to the existence of an almost contact structure lives in and turns out to be the integral second Stiefel-Whitney class (which vanishes if and only if the ordinary second Stiefel-Whitney has an integral lift). Moreover, two almost contact structures are homotopic if and only if they are homotopic on the –skeleton of . If has no -torsion then an almost contact structure is determined up to homotopy by its first Chern class .

Borman, Eliashberg, and Murphy [2] defined a notion of overtwisted contact structure in all dimensions and proved a strong version of the -principle for such structures, in particular they showed that any almost contact structure can be homotoped to a unique overtwisted contact structure. The precise definition of an overtwisted contact structure will not be needed here, but see [2] for the definition and [3] for alternate (possibly simpler) definitions. Here, we content ourselves with stating the main theorem that we need.

Theorem 2.8 (Borman, Eliashberg, and Murphy 2014, [2]).

Let be an almost contact structure on a manifold that defines a contact structure on some neighborhood of a closed (possible empty) subset . Then there is a homotopy of , fixed on , through almost contact structures to an overtwisted contact structure on . Moreover, any other contact structure that agrees with on and is overtwisted in the complement of is isotopic, relative to , to .

3. Topological embeddings

In this section we discuss a general procedure for constructing embeddings using open book decompositions.

3.1. Smooth embeddings via open books

The simplest way to embed one manifold into another using open book decompositions is the following.

Lemma 3.1.

Given two open book decompositions and and a family of embeddings , , such that

  1. each is proper,

  2. is independent of near , and

  3. ,

then there is a smooth embedding of into .

Proof.

The last condition on guarantees that the embedding

descends to an embedding of mapping torus into . The first two conditions on guarantee that the embedding

extends the embedding of the mapping tori to an embedding of into . ∎

This operation has been extensively studied in the context of knot theory. In particular, given a properly embedded arc in one obtains an embedding of (that is the manifold with open book having page an arc and monodromy the identity) into obtained from the above lemma (where is independent of ) when thinking of as . If is obtained from a knot in by removing a small ball about a point on then knotted is called the spun knot in or is said to be obtained by spinning . If one has a nontrivial family of embeddings of an arc into then the result is frequently called a twist spun knot, [8]. Given this history we call the embedding constructed in the lemma above a spun embedding.

We also notice the converse to the above lemma holds. If has an open book and an embedding is transverse to and the pages of the open book, then there is an induced open book on such that the embedding can be described in terms of a spun embedding.

One can embed open books in a more interesting way by noting that non-trivial diffeomorphisms of a submanifold can be induced by isotopy.

Lemma 3.2.

Suppose that is a (possibly achiral) Lefschetz fibration with fiber surface and vanishing cycles . If is a –manifold described by an open book where can be written as a composition of right and left handed Dehn twists about the , then there is a spun embedding of into , where is any diffeomorphism of (equal to the identity in a neighborhood of ).

Proof.

Choose a regular value of in so that is in the region where is the identity. Also choose paths from to the critical points of which are disjoint apart from at that realize the vanishing cycles. For each we can choose a disk that is a small neighborhood of (and disjoint from the other ). Let isotoped slightly so that it goes through and so that all the are tangent at . The (regular) fibers of are oriented surfaces so an orientation on makes an oriented -manifold and of course if is oriented as the boundary of (which in turn is oriented as a subset of ) then is the -bundle over with monodromy a right-handed Dehn twist about if is an ordinary Lefschetz critical points (and the inverse of this if it is an achiral critical point). Considering the opposite orientation on we see that is the -bundle over with monodromy a left-handed Dehn twist about if is an ordinary Lefschetz point (and the inverse of this if it is an achiral critical point). Now let be isotoped near so that it is positively tangent to at (in particular will be an immersed curve), see Figure 3. Clearly there is an immersion of the previous surface bundle with image .

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curves

Figure 3. The curves and (slightly offset for clarity).

We will construct our embedding in three steps.

Step 1(preliminary embedding of mapping tori): Note that given which can be written as a composition of right and left handed Dehn twists about , there is a path that is a composition of the paths and so that the mapping torus immerses in with image . In other words, there is a map

that when composed with is and induces an immersion on . (More precisely, the pull back of by is a -bundle over with monodromy , now cutting this bundle along a fiber gives the desired immersion.) Notice that the map

is an embedding and induces an embedding (here, it is important that is in the region where is the identity).

Step 2 (fix the embedding near the boundary): We would now like to extend our embedding over the binding. To do this we need to make restricted to each boundary component be independent of . To this end notice that maps into which is a union of solid tori (one for each boundary component of ). We will focus on one of these solid tori which we call , the arguments for the others being the same. If is a neighborhood of a boundary component of corresponding to then since is a trivial fibration restricted to the neighborhood of we see that with being projection to the last factor. Thus restricted to this neighborhood is simply

We now slightly enlarge and by adding collar neighborhoods and use these extensions to make independent of near the boundary components. Specifically, we add to and to . Our monodromy maps can be extended by the identity and then this does not change the diffeomorphism type of and . For each , we let be a straight line homotopy from to . We now extend by the map

This extended now is independent of near and thus descends to an embedding that for each boundary component of , sends to .

Step 3 (extend embedding over binding): To form from we glue in a to each boundary component of and similarly for . Notice that maps into the part of written as in the above coordinates. Thus we can extend over each by the map

to get an embedding of to get an embedding . ∎

3.2. Embedding oriented –manifolds in

Recall from Proposition 2.7 that there is a Lefschetz fibration of with vanishing cycles generating the hyperelliptic mapping class group. Thus our next result immediately follows from Lemma 3.2.

Proposition 3.3.

If is a –manifold supported by an open book with hyperelliptic monodromy (that is the monodromy is a composition of positive and negative Dehn twists about ), then embeds in . In particular if is obtained as the –fold cover of branched over a link, then it has a spun embedding into

We can slightly strengthen the above result as follows.

Proposition 3.4.

Suppose is a 3–manifold supported by an open book with monodromy a composition of positive and negative Dehn twists about and . Then embeds in .

Proof.

Take the Lefschetz fibration over with fiber a surface of genus and vanishing cycles . All will be ordinary vanishing cycles except for which will be achiral. To this we add two achiral vanishing cycles along and two more copies of , both achiral. See Figure 4 for the genus case, for higher genus the extra 1 and 2–handles will all simply cancel.

{overpic}

stein2

Figure 4. The top figure is a handle presentation describing manifolds in Proposition 3.4 when the genus is . All –handles have framing depending on if they are chiral or achiral. The 1–handles are paired as indicated by the numbers. The bottom figure is the result of canceling all the 1–handles with the vanishing cycles to and the chiral .

One may easily check that this 4–manifold is with a disk removed. Saeki [26] has shown that has an open book decomposition with as its page. The result follows from Lemma 3.2. ∎

Question 3.5.

Is there an open book for from which one can give a spun embedding of all –manifolds?

Remark 3.6.

In unpublished work Atsuhide Mori has sketched an idea to construct such spun embeddings into using the open book on with pages. Moreover, while completing a draft of this paper the authors were informed by Dishant Pancholi, Suhas Pandit, and Kuldeep Saha that they could construct similar such embeddings. It would still be interesting to know if one could construct a Lefschetz fibration structure on a page of an open book for to use the techniques in this paper to find embeddings of all 3–manifolds into , cf. Remark 4.5.

4. Contact embeddings

In the first subsection, we discuss an obstruction to embedding contact manifolds that generalizes Kasuya’s theorem discussed in the introduction. In the following subsection, we prove Theorem 1.7 that there is a Stein fillable contact structure on the twisted -bundle over into which all closed, oriented contact 3–manifolds embed. In the final subsection, we make an observation about embedding contact manifolds into the standard contact structure on .

4.1. Obstructions to contact embeddings

We begin with a simple lemma about co-dimension contact embeddings.

Lemma 4.1.

Let be a co-dimension contact embedding with trivial normal bundle. Then .

Proof.

The standard contact neighborhood theorem says that has a neighborhood contactomorphic to a neighborhood of in with the contact form , where is a contact form for and are polar coordinates on . So we see that along has a splitting as a complex bundle into and thus . ∎

We note that Kasuya’s result, Theorem 1.1, follows from this.

Proof of Theorem 1.1.

We first note that the result in [20] actually says that if embeds in , and , then , but the proof gives the stronger version stated in the introduction. To prove the result form the above lemma notice that if is trivial in homology then it bounds a hypersurface that in turn trivializes the normal bundle of , [21]. The result now clearly follows. ∎

We notice Lemma 4.1 has other consequences too. Specifically, when searching for a contact manifold into which all contact –manifolds embed one might consider the simplest ones first, that is the product of two spheres.

Corollary 4.2.

If all contact –manifolds embeds with trivial normal bundle in a manifold with a co-oriented contact structure where is a product of two or fewer spheres, then and the where is a generator.

Proof.

The only products of spheres are , and . In the first two examples notice that an embedding of a –manifold always realizes the trivial homology class and thus Lemma 4.1 implies the Chern class of is trivial. Now consider a co-oriented contact structure on choosing a generator of the second cohomology we see that for some integer since the mod reduction of is . Now given a contact embedding with trivial normal bundle we see . In particular if the Chern class is trivial, if then is divisible by . Since we have contact structures on –manifolds with divisible by only we see that . (Notice that the Chern class of any co-oriented contact structure on a –manifold is even for the same reason as for such structures on .) ∎

4.2. Contact embeddings in the twisted -bundle over

We are now ready to prove our main embedding theorem, Theorem 1.7, that says there is a Stein fillable contact structure on the twisted -bundle over into which all contact –manifolds embed.

Proof of Theorem 1.7.

Let be the disk bundle over with Euler number . From Example 2.1 we see that the manifold is the twisted -bundle over , . We denote the binding of this open book by and the fibration by . As discussed in Proposition 2.7 there is a Lefschetz fibration with fiber genus greater than and by Theorem 2.6 there is an exact symplectic structure (in fact Stein structure) with the properties listed in the theorem (in particular the fibers of are symplectic). Thus, by Example 2.4 the contact structure on is Stein fillable by . Recall as is supported by the open book we know that there is a contact 1–form for so that is a symplectic form on each page of the open book and restricted to the binding is a (positive) contact form on the binding.

Now given a contact -manifold we know that can be supported by an open book decomposition where has connected boundary. We denote the binding of this open book by and the fibration of its complement by . Proposition 2.7 gives us a Lefschetz fibration of with generic fiber and vanishing cycles (from Figure 1). As it is well-known that Dehn twists about these curves generate the mapping class group of a surface, there is a smooth embedding of by Lemma 3.2. We will show by exerting more care in the proof of Lemma 3.2 one may show that sends each page of the open book for to a symplectic surface in a page of the open book (where the symplectic structure comes from ) and the binding of maps to a transverse knot in the binding of . If we can do this then clearly will define a contact structure supported by and since is also supported by we see that is (isotopic to) a contact embedding.

Recall in Step 1 of the proof of Lemma 3.2 we constructed an embedding

that would descent to an embedding of the mapping tori to . Now the contact form on is given by where is the exact symplectic form on . We also know that each fiber of is symplectic and that send to fibers of . Thus, we see that pulls back to a contact form on by Remark 2.3.

Now in Step 2 of the proof of Lemma 3.2 we extended the embedding over a collar neighborhood of the boundary to make (and hence ) independent of . Recall in the proof we write a neighborhood of one of the components of ) as and in these coordinates we can use Theorem 2.6 to say that the exact symplectic structure on is given by ) (here we are use as the angular coordinate on , as the coordinate on , and as polar coordinates on ). Now in Step 2 of the proof of Lemma 3.2 we considered the extension of by

and thus pulling ) back by this map yields

where expressed in polar coordinates is given by . This is clearly a contact form if either is chosen sufficiently large or if is sufficiently large that the derivatives of and are sufficiently small. Moreover, if we choose the to be constant near then it is simply near each boundary component of .

Since we have a standard model for our embedding near , it is now a simple matter to see that the extension of over the neighborhoods of the binding has the desired properties. ∎

4.3. Contact embeddings in

We notice that the same proof as the one given for Theorem 1.7 also yields the following results.

Theorem 4.3.

If is a contact –manifold supported by an open book with hyperelliptic monodromy then embeds in .

We note that this theorem is known, see for example [6], and can be used to show among other things that all tight contact structures with on with odd or even and or can be embedded in .

We can similarly give a criterion guaranteeing a contact structure embeds in a Stein fillable contact structure on .

Theorem 4.4.

If is a contact –manifold supported by an open book with a genus surface with one boundary component and a composition of Dehn twists about the curves and , then embeds in a Stein fillable contact structure on .

Remark 4.5.

We note that Dehn twists around and do not generate the mapping class group of , where is a genus surface with one boundary component. One way to see this is as follows: Consider the spin structure on given by the quadratic form enhancing the intersection product, that assigns for . It automatically follows that since and bound a genus 0 surface (see [18] for the relation between spin structures and quadratic forms). Now, Dehn twists around preserve this spin structure. On the other hand, it is known that the action of the mapping class group on the set of spin structures on a surface has exactly 2 orbits distinguished by the Arf invariant of , [1, 18].

A similar argument also implies that one cannot hope to find a set of curves such that Dehn twists around them generate the mapping class group and also such that the curves are the vanishing cycles for a Lefschetz fibration on some Stein surface with fiber , whose total space is a page of an open book on . Indeed, since is spin, such a construction would induce a spin structure on with (as the vanishing cycles bound thimbles, see [29]), thus the image of the Dehn twists around in can only generate the theta group, a certain subgroup of the generated by anisotropic transvections, see [19].

5. Contact embeddings in overtwisted contact structures

In this section we prove Theorem 1.4 which shows that any contact –manifold contact embeds in a unique overtwisted contact structure on . To this end we first show how to “explicitly”, in terms of handle decompositions, smoothly embed a –manifold in and .

5.1. Topologically embedding –manifolds in and

We begin by embedding a 3–manifold in . Given an oriented –manifold we can find a handlebody decomposition of the form

where is an –handle and the handles are attached in the order in which they appear above. We will build a handlebody structure on in which we see the above handle decomposition for .

Step I: Handle decomposition of . Consider . This is a 5-manifold with an analogous handle decomposition