On Legendrian Embbeddings into Open Book Decompositions
Selman Akbulut, M. Firat Arikan

TL;DR
This paper proves that Legendrian submanifolds in certain contact manifolds supported by Weinstein open books can be isotoped to avoid the pages, revealing new flexibility properties of Legendrian embeddings.
Contribution
It establishes a method to Legendrian isotope submanifolds to be disjoint from open book pages in contact manifolds with Weinstein pages, under isotopic contact structures.
Findings
Legendrian submanifolds can be isotoped to avoid open book pages
Existence of a contactomorphism making the submanifold disjoint from a page
Applicable to contact structures supported by Weinstein open books
Abstract
We study Legendrian embeddings of a compact Legendrian submanifold sitting in a closed contact manifold whose contact structure is supported by a (contact) open book on . We prove that if has Weinstein pages, then there exist a contact structure on , isotopic to and supported by , and a contactomorphism such that the image of any such submanifold can be Legendrian isotoped so that it becomes disjoint from the closure of a page of .
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Taxonomy
TopicsGeometric and Algebraic Topology · Geometric Analysis and Curvature Flows · Mathematical Dynamics and Fractals
On Legendrian Embbeddings into Open Book Decompositions
Selman Akbulut
Department of Mathematics, Michigan State University, Lansing MI, USA
and
M. Firat Arikan
Dept. of Mathematics, Middle East Technical University, Ankara, TURKEY
Abstract.
We study Legendrian embeddings of a compact Legendrian submanifold sitting in a closed contact manifold whose contact structure is supported by a (contact) open book on . We prove that if has Weinstein pages, then there exist a contact structure on , isotopic to and supported by , and a contactomorphism such that the image of any such submanifold can be Legendrian isotoped so that it becomes disjoint from the closure of a page of .
Key words and phrases:
contact, convex symplectic, Weinstein, Liouville, Lefschetz fibration, open book
2000 Mathematics Subject Classification:
57R65, 58A05, 58D27
The first author is partially supported by NSF FRG grant DMS- 0905917
The second author is partially supported by NSF FRG grant DMS-1065910, and also by TUBITAK grant 1109B321200181
1. Introduction
A contact manifold is a pair where is a smooth manifold and is a totally non-integrable -plane field distrubution on . The distrubution is called a contact structure on , and is said to be co-oriented if it is the kernel of a globally defined -form with the property . Such a -form is called a contact form on . Here we always assume that is a co-oriented positive contact structure, that is, and with respect to a pre-given orientation on . We say that two contact manifolds and are contactomorphic if there exists a diffeomorphism such that . Two contact structures on a are said to be isotopic if there exists a -parameter family () of contact structures joining them.
A submanifold is said to be isotropic if , and an isotropic submanifold is called Legendrian if . A Legendrian embedding is an embedding of a smooth manifold such that the image is Legendrian. A smooth 1-parameter family of Legendrian submanifolds is called a Legendrian isotopy. Equivalently, a Legendrian isotopy is a smooth 1-parameter family of Legendrian embeddings.
An open book (decomposition) on a closed manifold is determined by a pair where is codimension submanifold with trivial normal bundle, and is a fiber bundle projection. The neighborhood of should have a trivialization , where the angle coordinate on the disk agrees with the fibration map . The manifold is called the binding, and for any a fiber is called a page of the open book.
The following definition is due to Giroux [4]: A contact structure on is said to be supported by (or carried by, or compatible with) an open book on if there exists a contact form for such that
- (i)
is a contact manifold.
- (ii)
For any , the page is a symplectic manifold with symplectic form .
- (iii)
If denotes the closure of a page in , then the orientation of induced by its contact form coincides with its orientation as the boundary of .
We will say that an open book on is called a contact open book if it carries a contact structure on .
In the smooth category, given an -dimensional submanifold of a closed -manifold admitting an open book , it is, in general, not possible to isotope so that it becomes disjoint from a page of . However, if the spine of is -dimensional, the general position argument shows that we can make them disjoint. Moreover, it is known (see below for details) that any Weinstein domain of dimension is homotopy equivalent to its core which is an -dimensional CW-complex. Here we prove:
Theorem 1.1**.**
Let be a contact manifold which admits a contact open book supporting . If the pages of are Weinstein, then there exist a contact structure on , which is isotopic to and supported by , and a contactomorphism such that the image of any compact embedded Legendrian submanifold of can be Legendrian isotoped until it becomes disjoint from the closure of a page of .
The proof of the theorem will be given in Section 3. One of the arguments used in the proof is making the given contact open book “-standard” (see the next definition) by altering the compatible contact structure in its isotopy class.
Definition 1.2**.**
Let an exact symplectic manifold, and be a manifold with a open book structure with pages , supporting a contact structure . Then we say that the is -standard if there exists a contact form for , such that restricts to on every page.
We remark that any abstract contact open book is -standard with respect to the contact form constructed by gluing the contact form on the mapping torus with the one on . This construction (which will be given in Remark 3.1) is originally due to Thurston and Winkelnkemper [6] for dimension three, and to Giroux [5] for higher dimensions.
Acknowledgments. The authors would like to thank NSF and TUBITAK for their supports.
2. Preliminaries
A Liouville domain (or a compact convex symplectic manifold) is a pair where is a compact manifold with boundary, together with a Liouville structure (or a convex symplectic structure), which meant there is a -form on such that is symplectic and the -dual vector field of defined by (or, equivalently, ), where denotes the interior product and denotes the Lie derivative, should point strictly outwards along . Since and (resp. and ) together uniquely determine (resp. ), one can replace the notation with the triple (resp. ). Here is called Lioville vector field. The -form is contact (i.e., ), and the contact manifold is called the convex boundary of .
In the non-compact case (i.e., when is an open manifold), if we further assume that is complete (i.e., its flow exists for all times), and also that there exists an exhaustion by compact domains such that each is a Liouville domain with the convex boundary for all , then is called a Liouville manifold (see [1] for details). A Liouville cobordism is a compact cobordism with a Liouville structure such that points outwards along and inwards along . Note that any Liouville domain is a Liouville cobordism with .
If denotes the contracting flow of , then the core (or skeleton) of the Liouville manifold is defined to be the set
[TABLE]
The interior of the core of any Liouville manifold is empty (Lemma 11.1, [1]), and so the core of any Liouville domain is compact. If denotes the convex boundary , then one can see that the negative half of the symplectization symplectically embeds into (as a collar neighborhood of in ) so that its complement in is and the embedding matches the positive -direction of with . The completion of a Liouville domain is obtained from by gluing the positive part of the symplectization of its convex boundary. Two Liouville domains are said to be Liouville isomorphic if there exists a diffeomorphism , called a Liouville isomorphism, between their completions such that where is some compactly supported smooth function on .
To define Weinstein manifolds and domains, we need three preliminary definitions:
Definition 2.1**.**
(i) A vector field on a smooth manifold is said to gradient-like for a smooth function if away from the critical point of .
(ii) A real-valued function is said to be exhausting if it is proper and bounded from below.
(iii) An exhausting function on a symplectic manifold is said to be -convex if there exists a complete Liouville vector field which is gradient-like for .
Definition 2.2**.**
A Weinstein manifold is a symplectic manifold which admits a -convex Morse function whose complete gradient-like Liouville vector field is . The triple is called a Weinstein structure on . A Weinstein cobordism is a Liouville cobordism whose Liouville vector field is gradient-like for a Morse function which is constant on the boundary . A Weinstein cobordism with is called Weinstein domain.
Remark 2.3**.**
Any Weinstein manifold can be exhausted by Weinstein domains where is an increasing sequence of regular values of , and therefore, any Weinstein manifold is a Liouville manifold. In particular, any Weinstein domain is a Liouville domain. Also note that any Weinstein domain has the convex boundary , and the completion of a Weinstein domain is a Weinstein manifold.
The following topologically characterizes Weinstein domains and will be used later.
Theorem 2.4** ([7], see also Lemma 11.13 in [1]).**
Any Weinstein domain of dimension admits a handle decomposition whose handles have indices at most .
3. Legendrian Embeddings into Contact Open Books
In this section we’ll prove Theorem 1.1. Let us start with by recalling how an open book on a closed manifold determines an abtract open book : By definition is a codimension-two subset of with a trivial normal bundle and is a fiber bundle map agreeing with the angular coordinate on the -factor. To describe an abstract open book, we first pick a page for some . Then the monodromy can be read from the first return map of a vector field on transversal to the fibers of . Now using the resulting abstract open book we can construct a closed manifold
[TABLE]
where is the mapping torus determined by . Observe that the construction defines an open book decomposition on , and also that can be identified with via some diffeomorphism respecting the fibration maps on and .
Next, from a collection of results from [4] and [5] (see also Section 7.3 of [3] for a detailed explanation) we recall the construction of a contact form (under suitable assumptions) on the manifold in the following remark.
Remark 3.1**.**
Given an abstract open book , consider the closed manifold . Suppose that there exists a Liouville form on and . Then one can construct a contact structure on so that becomes -standard. Here we recall the explicit construction of the contact form defining : Since , the form is closed, and it can be made exact by deforming through symplectomorphisms which are identity near . Such a deformation of changes in its diffeomorphism class, and so we may assume that for some smooth function . Adding a large enough constant, one can assume that is strictly positive everywhere on , and so we can use to construct a smooth mapping torus
[TABLE]
Consider the contact form on , and let be the diffeomorphism defining , that is, . Then we compute
[TABLE]
which shows that decends to a contact form, say , on . Then using appropriate cut-off functions, one can construct a contact form on
[TABLE]
by smoothly gluing with the contact form on the normal bundle .
The key observation for the proof of Theorem 1.1 is the following:
Theorem 3.2**.**
Let be a Weinstein domain of dimension whose underlying Liouville structure is given by the Liouville form . For , consider the contact manifold as in Remark 3.1. Let be a Legendrian embedding of a compact Legendrian submanifold . Then can be Legendrian isotoped (through Legendrian embeddings) to another embedded Legendrian submanifold which is disjoint from the closure of a page of on .
In order to prove this theorem we will make use of contact vector fields. A vector field on a contact manifold is said to be contact if its flow preserves the contact distrubution. The following fundamental lemma in contact geometry characterizes contact vector fields on a given contact manifold. More details can be found, for instance, in [3].
Lemma 3.3**.**
Let be any contact manifold and denote the Reeb vector field of . Then there is a one-to-one correspondence between the set of all contact vector fields on and the set of all smooth functions on . The correspondence is given by ( is called the “contact Hamiltonian” of the contact vector field ), and where is the contact vector field uniquely determined by the equations and . ∎
We note that a similar statement also holds between locally defined contact vector fields and locally defined smooth functons.
Proof of Theorem 3.2.
For simplicity we will write and . We may assume that Legendrian submanifold is connected since all the argument used below can be adapted (or generalized) to the case where has more than one connected component. So we have a Legendrian embedding of a compact connected manifold in the open book on corresponding to the abstract open book . So we have
[TABLE]
where is the binding, , and is the mapping torus as above. We may consider as the contact manifold
[TABLE]
where are the coordinates on the unit disk . Consider the following vector fields:
[TABLE]
where is the -dual vector field of , (resp. ) denotes the Reeb vector field of (resp. ), and is the -coordinate in the fibration determined by . Here , and so on…. Note that the first two are defined on , and the third is defined on the contact manifold where is any fixed page of . Here we note that restricts to on every page of (from its construction described in Remark 3.1) and , and so, in particular, we may also write where is the -coordinate on . It is easy to check that:
[TABLE]
So they are all contact vector fields on the regions where they are defined. In fact, if denotes the contact Hamiltonian function corresponding to as in Lemma 3.3, then we have
[TABLE]
We first see that one can make transverse to the binding :
Lemma 3.4**.**
In any pre-given -neighbourhood of , one can Legendrian isotope (in a small neighbourhood of the binding ) so that it becomes everywhere transverse to .
Proof.
Let be any -neighbourhood of . We will use the fact that Legendrian (more generally, isotropic) submanifolds stays Legendrian (isotropic) under the flows of contact vector fields. Note that for any constants the vector field is contact with the contact Hamiltonian . If intersects transversally, then there is nothing to prove. If not, let be the region where they don’t intersect transversally. Consider on . For any , the tangent space
[TABLE]
does not lie in (otherwise and would intersect transversally at ). Therefore, there exists a vector (for some constants ) which is everywhere transverse to . We consider as the smooth extension of to the whole . Note that will stay transverse to in a small neighbourhood for some . Let be a small neighbourhood of in . Now choose a regular value of the composition
[TABLE]
where is the projection and , such that lies on the line segment joining and , and the above composition has no critical value other than on the line segment, say , joining and . Note that, by construction, intersects the identical copy of transversally, and is everywhere transverse to . Let be a cut-off function such that on a neigbourhood of in , and on outside of a slightly larger neigbourhood (see Figure 1).
Next consider the contact vector field corresponding to the contact Hamiltonian . By the choice of , agrees with on and it is identically zero outside . Using the backward flow of we define the following -parameter smooth family:
[TABLE]
where is the time elapsed during the points of are moved to their final images in under the backward flow of . (Note that all the points of reach at the same time because the “horizontal” components of is defined by the constants .) Observe that , for each we have outside and is Legendrian, and is everywhere transverse to the binding as depicted in Figure 1. Finally, by choosing small enough, one can guarantee that the isotopy stays in the pre-given -neighbourhood of . ∎
By the above lemma we may assume that the Legendrian submanifold is transverse to the binding . Next, by picking a regular value for the projection we can assume that for the page of , the intersection
[TABLE]
is a properly imbedded dimensional submanifold meeting the binding along an dimensional submanifold (for simplicity we will write for ).
Lemma 3.5**.**
In any pre-given -neighbourhood of , one can Legendrian isotope (in a small neighbourhood of the page ) so that becomes disjoint from .
Proof.
Since is Weinstein (by assumption), by Theorem 2.4. Also we have . Hence, by the general position in , we can (topologically) isotope to a nearby copy which is disjoint from . This means that there exists a vector field on which is transverse to both and along their intersection . In what follows, using contact vector fields which are compactly supported near (and which are generated from ), we will construct an isotopy which transforms to some nearby copy (disjoint from ), and recognize this isotopy (in ) as the restriction of a local Legendrian isotopy (in ) moving to another Legendrian submanifold, say .
Recall that there is a canonical contact model (Legendrian Neighbourhood Theorem) for the tubular neighbourhood of in . That is, there exists a contactomorphism
[TABLE]
from the -jet bundle where are the standard coordinates on and is the real coordinate. (Here maps the zero section onto .) Observe that, on , there are linearly independent contact vector fields:
[TABLE]
The corresponding contact Hamiltonian functions (as in Lemma 3.3), respectively, are
[TABLE]
We will use these contact vector fields for local Legendrian isotopies in that we need for our purpose.
Let where ’s are (disjoint) connected components. Note that is compact as both and are compact. Moreover, the core is compact, and so is also compact from which we conclude that each is a compact CW-complex of finite type. Denote by the -skeleton of for . In particular, we have (Figure 2-a). Using the isotopies mentioned above, we will first make the closure of every -cell in each disjoint from , and then do the same for -cells, and so on…
Let be the set of all -cells in . For any , consider the following open cover for the closure (recall the vector field in the proof of Lemma 3.5):
[TABLE]
Clearly, covers which is compact. So, has a finite subcover
[TABLE]
for some finite number of points in . We label these points in such a way that the neighbourhood of any point has nonempty intersection with the union of the neighbourhoods of the preceding points in the list as depicted in Figure 2-b (for the case ).
We will first make disjoint from . From the definition of , we have a constant vector field which is everywhere transverse to . (Here at every point of , we have the same vector .) Observe that, since is a linearly independent set, the pull-back vector field can be written as the unique linear combination
[TABLE]
for some unique constants . Since these constants depend on the point , we set the notation
[TABLE]
Let us write for the pre-image . Observe that is a contact vector field on with the contact Hamiltonian
[TABLE]
and also that it is everywhere transverse to .
Denote by a small neighbourhood of in . Let be a smooth cut-off function such that near , and on the complement . Now consider the contact vector field whose corresponding contact Hamiltonian is equal to . By the choice of the cut-off function , and so is also transverse to . Using the flow we isotope to its nearby copy for some fixed time . Note that pushing along a transverse contact vector field implies that is disjoint from , and is still isotropic in . Indeed, since the transversality is an open condition, we know that the isotropic image is disjoint from where is a neighbourhood of in such that
[TABLE]
Similarly, we can make the closure of all the other open sets in the above finite subcover of disjoint from (this will isotope the whole closed -cell to some isotropic copy which is disjoint from ). However, for each such closure, the choice of how much we push it (using the flow of the corresponding contact vector field) needs a litle bit of more care: Let us discuss this in an inductive way: Suppose that we have already isotoped the union
[TABLE]
along the contact vector fields (where the smooth cut-off functions are constructed in the same way as above) so that the image of the union
[TABLE]
is isotropic in and is disjoint from where is a small neighbourhood of in . Now we would like to push (i.e., isotope) using . Observe that the region
[TABLE]
has been already made disjoint from , and also that might be tangent to the image of this region at some points, or even its flow might transform some points in the region back to (if we let them flow too much). On the other hand, since transversality is an open condition there exists a codim-[math] subset with a codim-[math] nonempty intersection
[TABLE]
to which is everywhere transverse. Therefore, we can make the union
[TABLE]
disjoint from by pushing (in an appropriate amount) along as shown in Figure 3.
Repeating the above process we can make the union of the closures of all -cells disjoint from by pushing to a nearby isotropic copy in . Note that for any particular closure in the union if some part of it has been already pushed (this might happen if it has a common boundary part with another -cell which has been pushed earlier), then we isotope it in an appropriate amount (as in the above discussion) so that previously pushed regions in the cell would not be moved back to .
Similarly, we can deal with the union of all closed -cells in (for ) in the same way (under the assumption that all closed -cells have been already made disjoint from . Note that we do not need to push any -cell which appears as a part of the boundary of some -cell(s) because such -cells have been already made disjoint from in the previous step. This process deforms the connected component to its image which is isotropic and disjoint from . Repeating this for each connected component, we conclude that there exists an isotopy of the embeddings in from to its nearby isotropic copy which is disjoint from . This isotopy is generated by contact vector fields and compactly supported in the neighbourhood
[TABLE]
Next, we want to extend this local isotopy of in to a local Legendrian isotopy of compactly supported in . (Here since is codimension submanifold of , we can consider the tubular neighbourhood of in as the product such that corresponds to .) Consider the smooth cut-off function
[TABLE]
where is a smooth cut-off function which is equal to near , and [math] near . Also let be the extension of given by
[TABLE]
Denote by the contact vector field on whose coresponding contact Hamiltonian (as in Lemma 3.3) is equal to . Now we isotope to its nearby Legendrian copy by applying the flow maps of in the same order and amount that we apply the flow maps of to isotope to . By construction, this isotopy is compactly supported in , and its restriction to is the isotopy taking to (constructed above) as . In particular, for the new Legendrian submanifold , its intersection is disjoint from the core of .
Finally, we note that by working on a small enough neighbourhood one can guarantee that lies in any pre-given -neighbourhood of in . ∎
By the last lemma we may assume that the transverse intersection of the Legendrian submanifold is disjoint from the core of . To finish the proof of Theorem 3.2, we will construct an isotopy of Legendrian embeddings of which will be compactly supported in a neighbourhood of in and will push completely outside . To this end, we will first isotope along a contact vector field generated from given in the list (1) above so that the part of (recall is the mapping torus of the open book ) is completely pushed into the interior of (tubular neighbourhood of the binding ). Then using of the list (1) we will isotope until completely crosses the binding .
Remark 3.6**.**
So far, when we write we meant the whole page (in particular, was the binding ). However, for what follows it is better to use the abstract open book desription as in the previous paragraph. Therefore, from now on will denote the complement of the collar neighbourhood of the binding in the corresponding page.
From its construction the contact manifold is obtained as the quotient space of using the equivalence relation . Recall that we have
[TABLE]
By translating (i.e., isotoping) along the Reeb direction (which corresponds to on ), we may assume that corresponds to under this identification. Therefore, we can identify a neighbourhood of in with for some real number where is a constant satisfying
[TABLE]
(Note that such exists since is compact and is a strictly positive continuous function). The contact form on is equal to on where takes the form as mentioned earlier. By choosing small enough, we may guarantee that the intersection is equal to ( and intersect transversally), and also that is disjoint from (this is because all cut-off functions which we used to isotope to in the proof of Lemma 3.5 are all equal to near .)
One can think of slightly larger by considering slightly smaller. More precisely, let be a smaller neighbourhood of where is a smaller disk in around the origin. By expanding each in to a larger domain , we get another decomposition where . Note that extending the above identification, a neighbourhood of in can be identified with ( on which the contact form is given as and we have the extension
[TABLE]
of where is the -dual of . We remark that is contact with the contact Hamiltonian . Let be a smooth cut-off function such that near and in the complement for some which will be determined later. Denote by the contact vector field on which corresponds to the contact Hamiltonian .
Now we first push using the flow until the image of lies completely in the interior of as follows: First note that on (as there). Since is disjoint from the core of (which is the same as that of ), for every point there exists a unique flow line of (i.e., of ) passing through . All such flow lines reach the region . Consider the set
[TABLE]
Since is a compact, the set is non-empty. Choose a finite number from . We isotope the Legendrian sphere using the flow maps . Indeed, by the construction of , we only push the region during the isotopy. Observe that by choosing small enough, we can guarantee that the image completely lies in . Therefore, is an embedded Legendrian which is Legendrian isotopic to (see Figure 4).
By the choice of the flow parameter above, we know that is completely lie in the region . For consider the disk
[TABLE]
and the neighborhood of the binding. Then after the above isotopy we know that for some . Let be a point which corresponds to the angular coordinate (recall that . Recall the contact vector fields from the list (1) and their contact Hamiltonians . Then the vector field is also contact whose contact Hamiltonian is given by . Let be a smooth cut-off function such that near and in the complement . Denote by the contact vector field on which corresponds to the contact Hamiltonian . Now we isotope using the flow maps of the contact vector field until completely crosses the binding. Say for the image of under the flow map completely crosses the binding. As a result, the closure of the page of the open book is disjoint from the final image which is Legendrian isotopic to as claimed. This finishes the proof of Theorem 3.2. ∎
Proof of Theorem 1.1.
Let be an open book decomposition carrying a contact structure on a (closed) manifold of dimension . In particular, there exists a contact form for such that restricts to a Liouville form on every page of . By assumption, for any page of , the restriction is, indeed, the underlying Liouville form of a Weinstein structure on .
Pick a page equipped with the Liouville form which is, by assumption, the underlying Liouville form of a Weinstein structure on . Denote by the monodromy of and consider the manifold equipped with the contact structure where is the contact form on constructed as in Remark 3.1. Let
[TABLE]
be the contact structure on obtained by pushing forward using the inverse of the identification map . As explained at the beginning of this section, respects the fibration maps on and associated to the open books and , respectively. Therefore, we have a contactomorphism
[TABLE]
mapping pages of to those of .
Next, observe that and are supported by the same open book (by construction of ), so we know, by Giroux’s work, that there exists an isotopy () of contact structures on connecting and . Then Gray’s Stability (see, for instance, Theorem 2.2.2 of [3]) implies that there is a diffeotopy
[TABLE]
such that for each . In particular, , and hence we obtain two contactomorphisms
[TABLE]
Suppose now we are given a Legendrian embedding of a compact Legendrian submanifold . By pushing forward using the above contactomorphisms, we obtain two Legendrian embeddings
[TABLE]
We set . By Theorem 3.2 we have a smooth -parameter family
[TABLE]
of Legendrian embeddings such that and the Legendrian submanifold is disjoint from the closure of a page of the open book on associated to . By composing with the contactomorphism , we obtain a smooth -parameter family
[TABLE]
of Legendrian embeddings such that . Finally, using this isotopy and the fact that is mapping pages of to the corresponding pages of , we conclude that the Legendrian submanifold
[TABLE]
is Legendrian isotopic to and disjoint from a page of the open book . Thus, setting and finishes the proof of Theorem 1.1. ∎
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