Every contact manifold can be given a nonfillable contact structure
Abstract.
Recently Francisco Presas Mata constructed the first examples of closed contact manifolds of dimension larger than that contain a plastikstufe, and hence are nonfillable. Using contact surgery on his examples we create on every sphere , , an exotic contact structure that also contains a plastikstufe. As a consequence, every closed contact manifold (except ) can be converted into a contact manifold that is not (semipositively) fillable by taking the connected sum .
K. Niederkrüger] O. van Koert]
Most of the natural examples of contact manifolds can be realized as convex boundaries of symplectic manifolds. These manifolds are called symplectically fillable. An important class of contact manifolds that do not fall into this category are socalled overtwisted manifolds ([Eli88], [Gro85]). Unfortunately, the notion of overtwistedness is only defined for –manifolds. A manifold is overtwisted if one finds an embedded disk such that , an overtwisted disk . This topological definition gives an effective way to find many examples of contact –manifolds that are nonfillable.
Until recently no example of a nonfillable contact manifold in higher dimension was known, but Francisco Presas Mata recently discovered a construction that allowed him to build many nonfillable contact manifolds of arbitrary dimension ([Pre06]). He showed that after performing this construction on certain manifolds, they admit the embedding of a plastikstufe. Roughly speaking, a plastikstufe can be thought of as a diskbundle over a closed –dimensional submanifold , where each fiber looks like an overtwisted disk. As shown in [Nie06b], the existence of such an object in a contact manifold excludes the existence of a symplectic filling.
In this paper we extend Presas’ results to a much larger class of contact manifolds. The idea is to start with one of his examples and to use contact surgery to simplify the topology and convert it into a contact sphere^{1}^{1}1In the scope of this article a contact sphere will be a smooth sphere carrying a contact structure. that admits the embedding a plastikstufe. If is any other contact manifold, then carries a contact structure that also has an embedded plastikstufe and is hence nonfillable.
Definition.
Let be a cooriented –dimensional contact manifold, and let be a closed –dimensional manifold. A plastikstufe with singular set in is an embedding of the –dimensional manifold
that carries a (singular) Legendrian foliation given by the –form satisfying:

The boundary of the plastikstufe is the only closed leaf.

There is an elliptic singular set at .

The rest of the plastikstufe is foliated by an –family of stripes, each one diffeomorphic to , which are spanned between the singular set on one end and approach on the other side asymptotically.
The importance of the plastikstufe lies in the following theorem.
Theorem 1.
Let be a contact manifold containing an embedded plastikstufe. Then does not have a semipositive strong symplectic filling. In particular, if , then does not have any strong symplectic filling at all.
Definition.
A contact manifold is called –overtwisted if it admits the embedding of a plastikstufe.
Whether –overtwisted can be taken as the general definition of overtwisted in higher dimensions, has yet to be clarified in the future (see Remark 6).
In dimension , the definition of –overtwisted is identical to the standard definition of overtwistedness. Using the Lutz twist, it is easy to convert a tight contact structure on a –manifold into an overtwisted one. Until recently, no example of a closed –overtwisted contact manifold of dimension larger than was known, but Francisco Presas Mata found a beautiful construction, which allowed him to create such examples in arbitrary dimension [Pre06].
Theorem 2 (F. Presas Mata).
Let be a contact manifold, which contains a –overtwisted contact submanifold of codimension . Assume that has trivial normal bundle. Then we can glue via a fiber sum to , and the resulting manifold
supports a –overtwisted contact structure that coincides on outside a small neighborhood of the gluing area with the original structure. More precisely, if contains a plastikstufe , then contains .
With this construction, he was immediately able to find –overtwisted contact manifolds in every odd dimension greater than .
Corollary 3.

There is a contact form on the –sphere that is obtained by taking an open book with a single lefthanded Dehntwist (see Section 1). It restricts on the standard embedding of to an overtwisted contact structure. Hence, one finds on the manifold
a –overtwisted contact structure.

Let denote the closed Riemann surface of genus , and let be a closed –overtwisted contact manifold. Then the manifold also supports a –overtwisted contact structure.
In this note, we apply contact surgery to examples that are similar to the manifold , and we obtain the following corollary.
Corollary 4.
Every sphere with supports a –overtwisted contact structure. More precisely, on , with , exists a contact structure which admits the embedding of a plastikstufe (with and ).
Corollary 5.
It is possible to modify the contact structure of a contact manifold with in an arbitrary small open set in such a way that the new contact structure is –overtwisted.
Proof.
Attach one of the contact spheres obtained in Corollary 4 via connected contact sum to the manifold . ∎
Remark 6.
The results above make it tempting to claim that –overtwisted is the proper definition of overtwistedness in higher dimensions. The second author proved together with Frédéric Bourgeois that contact manifolds having an open book decomposition of a certain type have vanishing contact homology, which as folklore tells also implies that they are nonfillable. In dimensions, overtwisted can equivalently be defined via open book decompositions, and thus in higher dimensions it would also be interesting to find the precise relation between the definition using a plastikstufe or an open book decomposition.
In particular all of the spheres defined in Section 1 have vanishing contact homology, and hence are nonfillable even before applying the Presas gluing.
Acknowledgments
This article was written at the Université Libre de Bruxelles, where we are both being funded by the Fonds National de la Recherche Scientifique (FNRS). We thank Hansjörg Geiges for fruitful discussions.
1. The contact spheres
In this section, we will describe an exotic contact structure on each sphere with , with the interesting property that they can all be stacked into each other in a natural way.
Let be the unit sphere in with coordinates , and let be the polynomial
The –form
defines a contact structure on , which is obtained by using an open book decomposition with lefthanded Dehntwist (see Remark 2). The first term of is just the standard contact form on the sphere, and it is the second term that is responsible for changing the properties of .
Proposition 7.
The sphere is a contact manifold.
Proof.
Consider the –form
Its restriction to the unit sphere is equal to . We will show that its exterior differential, the –form
is symplectic on . To compute the –fold product , note that the last term can appear at most once in each term of the total product. Furthermore since the first terms always couple a with , this eliminates most mixed forms of the last term. Using this, one easily computes
This form does not vanish, because
Now, it follows that (and hence also ) is a contact form by using that the Liouville field
for satisfies . ∎
Remark 8.

We obtain sequences of contact embeddings
where each map is just given by
The normal bundle of every –sphere in the following –sphere is of course trivial.

Using similar computations as the ones in [KN05] or [Nie06a], it can be seen that is compatible with the open book , which is equivalent to the abstract open book with page and monodromy map consisting of a single lefthanded Dehntwist. In particular, the –dimensional case is overtwisted, and it is not difficult to localize an overtwisted disk: The intersection between the –sphere and the hyperplane is diffeomorphic to a –sphere, which is foliated by . Using stereographic projection
we obtain for the pullback of the contact form onto this sphere
with . This form does not vanish with exception of the points corresponding to the origin and the circles of radius and . Hence we find an overtwisted disk in each of the hemispheres of (the set and ).
2. supports a –overtwisted contact structure
In this section, we will prove the result stated in Corollary 4 for dimension . To achieve our goal, we will simply start with the manifold (see Corollary 3), and then use contact –surgery to kill the fundamental group. The general proof in Section 4 includes the –dimensional case, but the induction used there is relatively complicated, so that we preferred to work out this case explicitly. The following proposition is well known to topologists, but since it is key to our construction we include a proof.
Proposition 9.
Let be an orientable manifold of dimension . Assume the fundamental group to be generated by the closed embedded paths . Then by using surgery on these circles we obtain a simply connected manifold .
Proof.
We have to show that vanishes. First note that the statement in the proposition needs a little clarification. We want all generators to be disjoint, but this is strictly speaking not possible, because all elements in have at least the base point in common. Instead move each of the circle with a small isotopy to make them disjoint from each other. Choose now new representatives such that consists of a short segment connecting with a point on the boundary of the tubular neighborhood of , the path , and a copy of the first segment but with opposite orientation connecting back with . After the surgery, each of the can be contracted to the point .
Let be a closed path that represents an element in , where we assume that the base point lies outside the surgery area. With a homotopy, we can make it also everywhere disjoint from the surgery area which is essentially an –sphere. This way we obtain a loop that lives not only in but also in , and represents an element in the fundamental group . In , this circle is homotopic to a product of the , and this homotopy can be made disjoint from the surgery regions , because a homotopy of curves is a map , but in an –dimensional manifold with , it is always possible to make a the image of a –manifold by a perturbation disjoint from a –dimensional submanifold. This homotopy can thus be also realized in , and so is also in homotopic to a product of the , which are all contractible in . If follows that represents the trivial element in , and hence . ∎
We start by using the manifold given in Corollary 3, part (1). The homology^{2}^{2}2From now on, we will always assume integer coefficients for the homology groups. of this space is (as already stated by Presas) , and . The fundamental group can be easily computed using the Seifertvan Kampen theorem. We obtain
where are the generators of , and generates the fundamental group of . The relation follows from identifying elements in the intersection .
Represent the two generators by smooth embedded paths of the form
with fixed such that both curves are isotropic and do not intersect the plastikstufe lying in . This can be achieved by choosing both points in the –dimensional binding of (because then both functions used for the definition of on vanish, cf. Section 3). One can find an overtwisted disk in that intersects the binding at only one point. For the construction of , this disk is transported through along two paths which are parallel to or . Hence there is sufficient space to choose in such a way that and do not intersect .
Now apply contact surgery on these generators of , i.e. cut out a tubular neighborhood of the two isotropic curves representing and . This neighborhood is of the form and it has boundary . Glue in two copies of , which have the same boundary as the cavities. As explained in [Wei91], the manifold we obtain this way, carries a contact structure, which coincides outside the surgery loci with the original structure, so in particular still contains a plastikstufe. By Proposition 9, it follows that is simply connected.
2.1. The homology of
We want to show that , because this together with , using the Poincaré conjecture proved by Smale and the nonexistence of exotic –spheres shows that is diffeomorphic to .
All of the computations in this section are standard applications of the MayerVietoris sequence. Use the following notation , , , . Then, because , both MayerVietoris sequences for and for split at that homology group. Using that , the MayerVietoris sequences for the pairs and reduce to
Because , it follows that also vanishes. The top sequence simplifies to
As cannot have torsion in such a short exact sequence, we obtain . With this, the second sequence simplifies to
so that vanishes.
By Poincaré duality and the universal coefficient theorem, we have . This implies that is homeomorphic to and in fact it is even diffeomorphic to , because there are no exotic –spheres. This proves Corollary 4 for dimension .
3. Submanifolds and Presas gluing
In [Pre06], Presas starts out with a manifold which contains a codimension contact submanifold with trivial normal bundle. A neighborhood of in is contactomorphic to . The product manifold also supports a contact structure, namely are the coordinates of the –torus and are certain functions, which are obtained from an open book decomposition of (see [Bou02]). A fiber has a neighborhood that is contactomorphic to , and one can perform the fiber connected sum of onto along and . Presas has shown that if is –overtwisted then so is . This construction can be carried out simultaneously on several embedded contact manifolds. To this end we need a neighborhood theorem that is adapted to such a situation. , where
Proposition 10.
Let be codimension contact submanifolds of with trivial normal bundle. Assume that all of these contact submanifolds intersect and each other transversely, i.e. at every and at every .
Then we find a neighborhood of in that can be represented as such that is in this neighborhood of the form .
Proof.
Start by choosing a metric on , and extend this metric first over all , then to all , and so on until the metric is defined on all . Now define finally the metric on the rest of . By considering the exponential map along , we obtain a tubular neighborhood diffeomorphic to such that every is given by .
The standard neighborhood theorem (see for example [Gei06]) guarantees that we have a contactomorphism from to itself, which deforms the contact form to . This map is generated by a vector field which vanishes on , and hence we can easily extend the contactomorphism to a diffeomorphism from any to itself such that leaves fixed. By suitably extending the vector field successively over all of the submanifolds and then to the rest of the manifold, we finally obtain a diffeomorphism that converts the contact form on into the desired type, that leaves pointwise fixed and all other submanifolds invariant as subsets.
Now use again the neighborhood theorem, this times for in . Note that the Moser vector field vanishes on , because there the contact form was already brought into the desired shape in the previous step. Extend the vector successively over all submanifolds of like before, but take care to choose it to vanish, if possible. The flow of this vector field maps to itself, keeps pointwise fixed, and all other submanifolds invariant as subsets. This map converts the contact form into the desired one on .
Now repeat the process for . Here care has to be taken for the Moser field not to destroy the form on , which was already arranged correctly in the previous step, but if the vector field is extended to vanish wherever possible, this submanifold is not moved at all by the flow. By continuing with this process, one finally proves the statement in the proposition. ∎
Now consider a contact submanifold that has transverse intersection with . The neighborhood of in can be represented as in such a way that is of the form in this neighborhood. Then it follows that the Presas gluing contains the gluing as a contact submanifold. This construction works for several submanifolds that satisfy the assumptions in Proposition 10.
4. The proof of Corollary 4
The general construction to prove Corollary 4 is considerably more complicated than the –dimensional one. The proof works by induction. We start with a contact sphere that contains a –overtwisted contact submanifold of codimension . In each induction step, we raise the dimension of the submanifold by two until finally the sphere itself is –overtwisted.
Let be a contact sphere that contains codimension contact submanifolds (with ) with the following properties (see also Figure 2):

Every is a sphere that is unknotted in .

Every two spheres and () intersect transversely, and the intersection
of any combination of these spheres is a contact –sphere, which is unknotted in any of the spheres .

Finally the lowest dimensional sphere is –overtwisted.
We are then able to construct a new contact sphere which satisfies the conditions in the above list for instead of . More explicitly we mean that contains unknotted codimension spheres , which are contact submanifolds, all possible intersections are contact spheres, unknotted in any of the other higher dimensional spheres, and is –overtwisted. By induction, we can continue these steps until we find a contact structure on , such that contains a –overtwisted unknotted –sphere. In the next step, we finally obtain then the –overtwisted contact structure on the –sphere itself (the proof in Section 2 amounts to this last step).
4.1. Start of induction
To start the process in arbitrary dimension, consider the manifold defined in Section 1. Every sphere for is contact, and all of the possible intersections also are. The –sphere is overtwisted, see Remark 8. Hence it is possible to start the construction in any odd dimension , and so assume that the induction step is true for some . Then we have to show that we find a –dimensional contact sphere such that the statements are also true for .
4.2. Construction of and
Apply the Presas gluing on along , i.e. construct the manifold
By Section 3, it contains all of the gluings on the other spheres with , along the intersection
and of course also the gluings of any other subspace along the intersection .
In particular, it follows from Theorem 2 that is –overtwisted, because we obtained it by gluing along , which is by our assumptions –overtwisted. By using surgery on all of these submanifolds, we will be able to convert every into a sphere, which will prove property of the list at the beginning of Section 4.
Using that each sphere is unknotted in the next higher dimensional one, we obtain with arguments that are completely analogous to those of Section 2 that , and that the homology is , , and all other homology groups are trivial.
Represent the generators of the fundamental group of the lowest dimensional manifold by the two loops of the form
(4.1) 
with fixed. We cannot only assume that do not intersect the fiber along which we perform the Presas gluing (Figure 3), but by choosing the two points suitably, it can also be achieved that the plastikstufe created in by Theorem 2 is not touched by any of the two paths. The conformal symplectic normal bundles of these loops are trivial, and hence the –principle guarantees that a perturbation of in turns them into isotropic curves, so that they can be used to apply contact surgery. Note that the loops chosen do not only generate but also the fundamental group of any other of the manifolds including the one of the maximal space .
From Proposition 10, we can easily deduce the following corollary.
Corollary 11.
The neighborhood of is contactomorphic to
where is represented by .
Proof.
Consider in . They satisfy the conditions of Proposition 10, and so there is a tubular contact neighborhood of
such that is given by . Repeat the step for in . We obtain a neighborhood of the form
This step can be iterated until one arrives at the neighborhood described in the corollary we want to prove. ∎
With this neighborhood theorem, we will be able to apply contact –surgery on along the curves and .
4.2.1. Surgery compatible with submanifolds
To describe the contact surgery, we briefly recall Weinstein’s picture for surgery [Wei91]. Consider with coordinates and symplectic form
The vector field
is Liouville, and it is transverse to the nonzero level sets of the function
In particular, is a contact hypersurface, and the circle
is isotropic. By a neighborhood theorem, the loops and considered above have a neighborhood that is contactomorphic to a neighborhood of in .
The next step consists of gluing in the –handle lying between and . A piece of the set can be identified via the Liouville flow with a contact manifold that is close to (see Figure 4). The precise construction that is necessary can be found in [Gei07]. Hence we can replace a neighborhood of the curve by the surgery . As the Liouville field is transverse to the set , we see that the surgered manifold is contact.
We still need to show that we can choose the contact surgery on in such a way that it induces contact surgery on all the submanifolds , where is an index set. To see that the surgery can be made compatible, we need to specify the framing more precisely, which can be done by using an induction. Corollary 11 is of relevance here.
Let us denote the curve where we perform surgery by . We start with the contact submanifold . This manifold also contains the curve , which is still isotropic. Therefore we can identify a tubular neighborhood of with as in the above model. We have added the subscript to indicate the different dimensions, i.e. the number of coordinates depends on the size of index set . At this stage there is still some freedom in choosing the framing. For clarity, we will also give a subindex to indicate in what submanifold we consider the curve, i.e. denotes the restriction of the curve to the submanifold .
Next, suppose we have fixed the framing on such that contact surgery on induces the desired contact surgery on submanifolds in indexed by . Let us now look at . A neighborhood of in looks like