Contact Symplectic Fibrations and Fiber Connected Sum

Contact Symplectic Fibrations and Fiber Connected Sum

Abstract.

We consider certain type of fiber bundles with odd dimensional compact contact base, exact symplectic fibers, and the structure group contained in the group of exact symplectomorphisms of the fiber. We call such fibrations “contact symplectic fibrations”. By a result of Hajduk-Walczak, some of these admit contact structures which are “compatible” (in a certain sense) with the corresponding fibration structures. We show that this result can be extended to get a compatible contact structure on any contact symplectic fibration, and also that isotopic contact structures on the base produce isotopic contact structures on the total space. Moreover, we prove that the fiber connected summing of two contact symplectic fibrations along their fibers results in another contact symplectic fibration which admits a compatible contact structure agreeing with the original ones away from the region where we take fiber connected sum, and whose base is the contact connected sum of the original contact bases.

Key words and phrases:
Contact structure, fiber bundle, symplectic fibration, fiber connected sum
2010 Mathematics Subject Classification:
58D27, 58A05, 57R65
The author is partially supported by TUBITAK grant 1109B321200181 and also by NSF FRG grant DMS-1065910

1. Introduction

In what follows, all manifolds, maps and fiber bundle structures will be assumed to be differentiable. Suppose that is any locally trivial fiber bundle such that the fibers are of even dimension but the base space could have dimension in either parity. Let be the fibration map and the structure group. Let us denote such a fiber bundle by the symbol

(1)

If there exist a symplectic structure on the fiber and all transition maps are symplectomorphisms of (i.e., ), then the fiber bundle in (1) is called a symplectic fibration. Note that, for a symplectic fibration, each fiber is equipped with a symplectic structure obtained from pulling back using a local trivialization around , and this construction is independent of the choosen trivialization as transition maps lies in .

In order to motivate the reader, let’s recall some related work in even dimensions. Given a symplectic fibration as above suppose that the total space admits a symplectic form . (In particular, is assumed to be even dimensional.) Then is said to be compatible with the fibration if it restricts to a symplectic form on every fiber . Note that the compatibility condition implies that symplectically embeds into for each . This definition is motivated by the result which states that on a locally trivial fiber bundle as in (1) with a connected base if admits a symplectic form which restricts to a symplectic form on every fiber, then is a symplectic fibration compatible with . Conversely, due to a result of Thurston, for a symplectic fibration with a connected symplectic base if the symplectic form represents a cohomology class in , then one can construct a symplectic form on which is compatible with . For more details about symplectic topology and symplectic fibrations with even dimensional base, we refer the reader [6].

In this note, we consider similar approaches to build “compatible” contact structures on the total spaces of exact symplectic fibrations (see below for definitions). In particular, the base spaces will be odd dimensional and admit contact structures. Here we only consider co-oriented contact structures, that is, those which appear as kernels of certain globally defined -forms: A positive contact form on a smooth oriented -dimensional manifold is a -form such that (i.e., is a positive volume form on ). The hyperplane field (of rank ) of a contact form is called a positive (co-oriented) contact structure on . The pair (or sometimes ) is called a contact manifold. 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 (Gray’s Stability). Note that isotopic contact structures give contactomorphic contact manifolds by Gray’s Theorem. More details about contact topology can be found in [2].

Through out the paper, we will assume the fibers of any fiber bundle to be noncompact or compact with nonempty boundaries, (and, therefore, the symplectic structures on them will be exact). In Section 2, we will show the existence of a compatible contact structure on a fiber bundle with exact symplectic fibers, compact contact base and structure group contained in the group of exact symplectomorphisms of the fiber. One should note that Hajduk-Walczak [4] showed the existence of contact structures on some of such fiber bundles which are covered by the corresponding theorem of the present paper. Therefore, one can say that the related results of this note generalize the corresponding statements in [4]. After constructing contact structures on total spaces of bundles, in Section 3 we also discuss how they are affected if we vary the contact structures on the bases in their isotopy classes.

Fiber sum operation on fiber bundles or more generally fiber connected summing of manifolds along their submanifolds is a well known technique to obtain new fiber bundles and manifolds out of given ones. The results from [3] and [5] show that under suitable conditions, fiber sum process can be done in the symplectic category. Geiges [1] proved that an analog result holds also in contact category under certain conditions. In Section 4, by following the fiber connected sum description given in [1] we will see that under suitable assumptions one can take the fiber connected sum of two fiber bundles (equipped with compatible contact structures) along their fibers which results in another fiber bundle whose total space admiting a compatible contact structure which agrees with the original ones away from the region where we take fiber connected sum, and whose base is the contact connected sum of the original contact bases.

Acknowledgments. The author would like to thank TÜBİTAK, The Scientific and Technological Research Council of Turkey, for supporting this research, and also Yıldıray Ozan for helpful conversations.

2. Contact Symplectic Fibrations

A contact manifold is called strongly symplectically filled by a symplectic manifold if there exists a Liouville vector field of defined (at least) locally near such that is transversally pointing out from and on . In such a case, we also say that is the convex boundary of .

An exact symplectic manifold is a noncompact manifold (or a compact manifold with boundary), together with a symplectic form and a -form satisfying . If is a compact manifold with boundary, then we also require that the -dual vector field of defined by should point strictly outwards along , in particular, is a contact form which makes the convex boundary of . Since determines both and , it suffices to write an exact symplectic manifold as a pair . The -form is called an exact symplectic structure on . Since any symplectic form determines an orientation, any exact symplectic manifold is (symplectically) oriented. An exact symplectomorphism between two exact symplectic manifolds is a diffeomorphism such that is exact.

Let us start with defining the analog of the above compatibility notion (given in the introduction) in odd dimensions.

Definition 2.1.

Let be a fiber bundle with odd dimensional base such that the total space admits a contact structure . Suppose that the fiber space is a noncompact manifold or a compact manifold with boundary. Then is said to be compatible with if there exists a contact form for such that restricts to an exact symplectic structure on every fiber of .

Given an exact symplectic manifold , one can define the group

of exact symplectomorphisms which is a subgroup of the symplectic group

Also if , one can consider the subgroup

of exact symplectomorphisms which are identity near .

We can now formally introduce our main objects of interest:

Definition 2.2.

A contact symplectic fibration is a fiber bundle

with exact symplectic fiber and the structure of group of exact symplectomorphisms (i.e., ) such that the base space is compact and admits a co-oriented contact structure.

From the definition one might understand that the phrase “contact symplectic fibration” refers to a fiber bundle with a compact contact base and an exact symplectic fibers, and also that the adjective “contact” in the phrase only emphasizes that the base space is contact. However, the main result of this section (Theorem 2.3) shows that the word “contact” has, indeed, a more global meaning.

Given a fiber bundle , if the base and/or fiber spaces are manifolds with boundaries, then the boundary of consists of two parts: The vertical boundary component , and the horizontal boundary component where is the fiber over .

Theorem 2.3.

The total space of a contact symplectic fibration admits a (co-oriented) contact structure which is compatible with . Moreover, If has a boundary and , then the contact form is equal to a product form on some collar neighborhood of .

Note that this theorem can be regarded as the analog of Thurston’s theorem mentioned above in odd dimensions. Before proving the theorem, let us see some examples of contact symplectic fibrations appeared under a different name in the literature. To this end, we need to recall exact symplectic fibrations from [7, 8]. Here for our purposes we define them in a slightly different way.

Definition 2.4.

An exact symplectic fibration is a fiber bundle

equipped with a -form on such that each fiber with is an exact symplectic manifold. An exact symplectic fibration is said to be trivial near horizontal boundary if and the following triviality condition near is satisfied: Choose a point and consider the trivial fibration with the form which is the pullback of , respectively. Then there should be a fiber-preserving diffeomorphism between neighborhoods of in and of in which maps to , equals the identity on , and

Lemma 1.1 of [8] implies that for an exact symplectic fibration

the structure group falls into where is the restriction of the -form on to any fixed fiber . The same result also implies that if we have an exact symplectic fibration which is trivial near horizontal boundary, then its structure group is contained in . Also as discussed after Lemma 1.1, using the fact that is nondegenerate on for any , one can define a preferred (or compatible) connection on any exact symplectic fibration. Moreover, one can proceed in the other direction as well, that is, any fiber bundle with structure group contained in (resp. ) and equipped with a compatible connection admits a structure of exact symplectic fibration (resp. a structure of exact symplectic fibration which is trivial near horizontal boundary). Putting these observations together for exact symplectic fibrations we obtain

Lemma 2.5.

Any exact symplectic fibration is equipped with a fiber bundle structure with transition maps contained in , that is, , where is the restriction of to any fixed fiber . If it is trivial near horizontal boundary, then . ∎

As an immediate consequence of this lemma, we have

Corollary 2.6.

Any exact symplectic fibration with a compact base which admits a co-oriented contact structure can be equipped with a structure of a contact symplectic fibration. ∎

For completeness let us provide the following elementary fact which will be used in the proof of Theorem 2.3:

Lemma 2.7.

The product of a contact manifold with an exact symplectic manifold admits a contact structure.

Proof.

Let be any contact manifold, and an exact symplectic manifold. Consider the product and the standard projections given by

Consider the pull-back forms , and set

Then by the binomial expansion formula and using the facts and , we compute

Therefore, is a volume form on the product manifold because and are volume forms on and , respectively. Hence, is a contact form on . ∎

Remark 2.8.

After the above observations, one can prove Theorem 2.3 in the following way: Given a contact symplectic fibration as in Definition 2.2, as the transition maps are contained in we can glue the forms on each locally trivial piece of the fibration to get a smooth -form, say , on which restricts to an exact symplectic structure on each fiber of . Then by following the lines in the proof of Lemma 2.7, it is easy to check that the pull-back form on defines contact structure which is compatible with . For the purpose of the subsequent parts of the paper we now prove Theorem 2.3 in a different perspective.

Proof of Theorem 2.3.

Let be any given contact symplectic fibration with and . By the fiber bundle structure, we know that up to diffeomorphism the total space is obtained by patching the trivial -bundles, and the transition maps are contained in the structure group where is any fixed fiber of . More precisely, there is an open cover of and a collection of diffeomorphisms such that the diagram

commutes. Moreover, by restricting to any fiber and then projecting onto -factor, we get a smooth map , and the transition maps given by

Using paracompactness, one can find another open cover of such that for any index the closure is contained in for some . Since is compact (by assumption), there is a finite subcover of . Here one can assume that ’s are listed in a specific order so that whenever and belong to the same connected component of . Indeed, the argument below can be done independently for each connected component. Therefore, without loss of generality will be assumed to be connected. Now consider the collection of compact sets defined by

Clearly, forms a partition for the base space , that is, we have

(2)

for any , , and each union intersects with along a smooth compact hypersurface

Observe that for each we have the map obtained by restricting the corresponding . More precisely, from the construction of , we know for each and with that and for some . So one can define the map by patching the restrictions of to the subsets together. As a result, up to diffeomorphism we obtain the description given by

(3)

where each is the smooth gluing map defined by

For each the fact that implies that there exists a smooth function such that . In accordance with the smooth gluing map (or rule) , one can patch all these functions together which yields a smooth function .

Next, in order to construct a compatible contact structure on , we will use the above description as follows: By assumption admits a contact structure, say . For any real number , the -form is contact (indeed, the manifolds and are contactomorphic, and their contact hyperplane distributions coincide everywhere on ). Also from Lemma 2.7 we know that for any the -form is a contact form on , and so, in particular, on each . (For simplicity, we will denote the pull-back of (resp. ) on any product space still by (resp. by ).

For a sufficiently small consider the tubular neighborhood of in , and let be a smooth cut-off function defined by where is the smooth cut-off function such that on for some and for all near as in Figure 1.

Figure 1. Smooth cut-off funtion

For a real number , consider the smooth -form on . By construction we can smoothly glue this form with the contact form on . Also observe that is in accordance with the gluing map because for any fiber over the gluing region (where ) we have . Therefore, is, indeed, a smooth -form on . Moreover, on the region we compute


where are constants detemined by the binomial expansion formula for , and we use the facts and . Observe that if is chosen large enough, the first term on the right hand side is dominant over all other terms since it has the highest degree of . Therefore, for large enough is a volume form on , and hence is a contact form on .

Now we want to extend to a contact form on : For sufficiently small, consider the tubular neighborhood of in such that and . Take a smooth cut-off function defined by where is the smooth cut-off function such that on for some and for all near as in Figure 2.

For a real number , we define the smooth -form on where . On the region , similar to the computation given in Equation (2) we have

Figure 2. Smooth cut-off funtion

which implies that if is chosen large enough, then is contact on . Moreover, the functions and are identical on their overlaping domains because and are constructed using the same transition functions on their common domains. As a result, if we choose , then by gluing with we obtain a smooth contact form on .

Next, we can repeatedly follow exactly the same lines of the last extension argument for the remaining parts of the description of given in Equation (3), and eventually obtain a contact form on .

More precisely, for each and small enough, consider the smooth cut-off function defined by . Here, similar to above, we assume that and . Suppose that the contact form

is already constructed on the union , and it extends . Note that one needs to choose the constants () to be to perform this extension. Then one can similarly check as in Equation (2) that the form

is contact on if is large enough. Moreover, one can see as the extension of to the union once we do the reassignments

Finally, we set . Then by the construction restricts to an exact symplectic structure on each fiber of . So the contact form defines a contact structure on which is compatible with the fibration map .

In order to prove the second statement, observe that if , then for any transition map used in the above construction and for any we have near (in particular, ). Therefore, the smooth maps in the construction are all constant functions near , and so the corresponding functions take constant values near (i.e., near ). As a result, the description of the contact form on given above takes the form near . ∎

3. Bundle Contact Structures

From the previous section any contact symplectic fibration admits a (co-oriented) contact structure. Here we study the flexibility of such contact structures. More specifically, we are interested in how they react if we change the contact structure on the base in its isotopy class. Before we proceed let us first fix some terminonology and notation.

Definition 3.1.

Let be a contact symplectic fibration with the contact base and the structure group . The contact structure on which is defined by the contact form

constructed as in the proof of Theorem 2.3, is called a bundle contact structure for associated to , and will be denoted by .

Note by definition bundle contact structures are compatible with the fibration maps. Next we show that isotopic contact structures on the base space produce isotopic bundle contact structures on the total space.

Theorem 3.2.

Let be a contact symplectic fibration. Suppose co-oriented contact structures , , on are isotopic to each other. Then the corresponding bundle contact structures on are isotopic through bundle contact structures on . In particular, and are contactomorphic.

Proof.

Since are isotopic, there exists a smooth family of contact structures on joining them. Consider a smooth family of contact forms on such that and for all . Since , we have for some smooth function . Then we obtain the (modified) smooth family

of contact forms on joining and and with the property for all .

Consider the bundle contact structure on associated to given by the contact form constructed as in the proof of Theorem 2.3. In order to simplify the notation, we will write for . Let us consider

.

For each the choice of can be done in such a way that ’s varies continuously with respect to . This is possible since is a smooth -parameter family. Therefore, the set can be realized as the range set of a continuous function on the compact set , and hence there exists an upper bound, say , for . We set , and consider three smooth -parameter families:

As (resp. and ) for each , the construction in Theorem 2.3 implies that (resp. and ) is a contact form on for any . Now we define a concatenation by the rule:

Observe that and , and so is a smooth -parameter family of contact forms on joining and . As a result, we obtain a smooth -parameter family

of (bundle) contact structures on joining and . Hence, and are isotopic by Gray’s Stability Theorem (see, for instance, Theorem 2.2.2 of [2]). ∎

4. Fiber Connected Sum

In this section we prove that the fiber connected summing of two contact symplectic fibrations along their fibers results in another contact symplectic fibration, and, in particular, the sum admits a (compatible) bundle contact structure and it fibers over the contact connected sum of the original contact bases. We first recall the fiber connected sum operation in a slightly more general situation than the ones described in [1, 2]. Namely, we also allow the ambient manifolds to be noncompact or compact with boundary.

Definition 4.1.

For an oriented manifolds , let , be two codimension embeddings of an oriented manifold , and for each denote by the normal bundle of in (which is an oriented bundle over ). Let be a fiber-orientation-reversing bundle isomorphism covering the composition . After picking a Riemannian metric and normalization, one can assume that is norm-preserving. For an interval , where , define the subset . Furthermore, for a given consider the orientation-preserving diffeomorphism (or identification)

given explicitly by the rule

(6)

where denotes the fiber component of . Then the fiber connected sum of , along their submanifolds is the oriented manifold given as the quotient space

.

We remark that in the special case when , in order to get a smooth manifold one should also assume that the embeddings are disjoint. Now we are ready to prove

Theorem 4.2.

Let be two contact symplectic fibrations with . For an exact symplectic manifold , assume that there exist embeddings , of the fibers and (which are distinct if and ) of and , respectively, and for each the total space is constructed by regarding . Then if is odd (resp. even), then one can form the fiber connected sum of and (resp. and ) along ’s using some canonical fiber-orientation-reversing bundle isomorhism between normal bundles of ’s such that (resp. ) admits a contact structure which restricts to a bundle contact structure on each piece (resp. on and ). Moreover, there exists a contact symplectic fibration

where if is odd and if is even, such that (resp. ) is the contact connected sum of the original contact bases (resp. ), , on their overlaping domains, and is a bundle contact structure for .

Proof.

First note that each admits a contact form from which we can construct a contact form on using the pull-back form by taking large enough. (See the proof of Theorem 2.3). Also in the case when is even, since the orientation on is opposite of the one on , one should consider the contact form in order to define a positive contact structure on . By assumption both and can be equipped with a fiber bundle structure such that structure groups . Consider the decomposition of each (as in Equation (3)) given by

where we consider the decomposition of each (as in Equation (2)) given by

From differential topology point of view we are free to choose the fibers and along which the fiber connected sum is performed because different choices of fibers produces diffeomorphic total spaces which fibers over diffeomorphic bases. Therefore, by changing the fiber (if necessary), we may assume that the embedding maps onto a fiber which lies in for some . Assume that each is the fiber over the point , that is, .

By Darboux theorem (see, for instance, Theorem 2.5.1 of [2]), for each there exist a neighborhood of