Klaus Niederkrüger

The plastikstufe – a generalization of the overtwisted disk to higher dimensions

In this article, we give a first prototype-definition of overtwistedness in higher dimensions. According to this definition, a contact manifold is called overtwisted if it contains a plastikstufe, a submanifold foliated by the contact structure in a certain way. In three dimensions the definition of the plastikstufe is identical to the one of the overtwisted disk. The main justification for this definition lies in the fact that the existence of a plastikstufe implies that the contact manifold does not have a (semipositive) symplectic filling.

Weak and strong fillability of higher dimensional contact manifolds

For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five), while also being obstructed by all known manifestations of “overtwistedness”. We also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense. These depend on a higher dimensional analogue of Giroux torsion, which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary.

Some remarks on the size of tubular neighborhoods in contact topology and fillability

The well-known tubular neighborhood theorem for contact submanifolds states that a small enough neighborhood of such a submanifold N is uniquely determined by the contact structure on N , and the conformal symplectic structure of the normal bundle. In particular, if the submanifold N has trivial normal bundle then its tubular neighborhood will be contactomorphic to a neighborhood of Nf 0g in the model space N R 2k . In this article we make the observation that if .N; N/ is a 3‐dimensional overtwisted submanifold with trivial normal bundle in .M;/ , and if its model neighborhood is sufficiently large, then .M;/ does not admit a symplectically aspherical filling. 57R17; 53D35 In symplectic geometry, many invariants are known that measure in some way the “size” of a symplectic manifold. The most obvious one is the total volume, but this is usually discarded, because one can change the volume (in case it is finite) by rescaling the symplectic form without changing any other fundamental property of the manifold. The first non-trivial example of an invariant based on size is the symplectic capacity (see Gromov [15]). It relies on the fact that the size of a symplectic ball that can be embedded into a symplectic manifold does not only depend on its total volume but also on the volume of its intersection with the symplectic 2‐planes. Contact geometry does not give a direct generalization of these invariants. The main difficulties stem from the fact that one is only interested in the contact structure, and not in the contact form, so that the total volume is not defined, and to make matters worse the whole Euclidean space R 2nC1 with the standard structure can be compressed by a contactomorphism into an arbitrarily small open ball in R 2nC1 . A more successful approach consists in studying the size of the neighborhood of submanifolds. This can be considered to be a generalization of the initial idea since contact balls are just neighborhoods of points. In the literature this idea has been pursued by looking at the tubular neighborhoods of transverse circles. Let .N; N/ be a closed contact manifold. The product N R 2k carries a contact structure given as

Resolution of symplectic cyclic orbifold singularities

In this paper we present a method to obtain resolutions of symplectic orbifolds arising from symplectic reduction of a Hamiltonian S^1-manifold at a regular value. As an application, we show that all isolated cyclic singularities of a symplectic orbifold admit a resolution and that pre-quantisations of symplectic orbifolds are symplectically fillable by a smooth manifold.

Open book decompositions for contact structures on Brieskorn manifolds

In this paper, we give an open book decomposition for the contact. structures on some Brieskorn manifolds, in particular for the contact structures of Ustilovsky. The decomposition uses right-handed Dehn twists as conjectured by Giroux.

Every Contact Manifolds can be given a Nonfillable Contact Structure

Recently Francisco Presas Mata constructed the first examples of closed contact manifolds of dimension larger than 3 that contain a plastikstufe, and hence are non-fillable. Using contact surgery on his examples we create on every sphere S^{2n-1}, n>1, an exotic contact structure \xi_- that also contains a plastikstufe. As a consequence, every closed contact manifold M (except S^1) can be converted into a contact manifold that is not (semi-positively) fillable by taking the connected sum of M with (S^{2n-1},\xi_-).

THE WEINSTEIN CONJECTURE IN THE PRESENCE OF SUBMANIFOLDS HAVING A LEGENDRIAN FOLIATION

Helmut Hofer introduced in 1993 a novel technique based on holomorphic curves to prove the Weinstein conjecture. Among the cases where these methods apply are all contact 3-manifolds (M, ξ) with π2(M) ≠ 0. We modify Hofers argument to prove the Weinstein conjecture for some examples of higher-dimensional contact manifolds. In particular, we are able to show that the connected sum with a real projective space always has a closed contractible Reeb orbit.

Towards a good definition of algebraically overtwisted

Abstract Symplectic field theory (SFT) is a collection of homology theories that provide invariants for contact manifolds. We show that vanishing of any one of either contact homology, rational SFT or (full) SFT are equivalent. We call a manifold for which these theories vanish algebraically overtwisted.

Loose Legendrians and the plastikstufe

We show that the presence of a plastikstufe induces a certain degree of flexibility in contact manifolds of dimension 2nC1> 3. More precisely, we prove that every Legendrian knot whose complement contains a “nice” plastikstufe can be destabilized (and, as a consequence, is loose). As an application, it follows in certain situations that two nonisomorphic contact structures become isomorphic after connect-summing with a manifold containing a plastikstufe. 57R17

Subcritical contact surgeries and the topology of symplectic fillings

By a result of Eliashberg, every symplectic filling of a three-dimensional contact connected sum is obtained by performing a boundary connected sum on another symplectic filling. We prove a partial generalization of this result for subcritical contact surgeries in higher dimensions: given any contact manifold that arises from another contact manifold by subcritical surgery, its belt sphere is zero in the oriented bordism group Ω∗^{SO}(W ) of any symplectically aspherical filling W , and in dimension five, it will even be nullhomotopic. More generally, if the filling is not aspherical but is semipositive, then the belt sphere will be trivial in H∗(W ). Using the same methods, we show that the contact connected sum decomposition for tight contact structures in dimension three does not extend to higher dimensions: in particular, we exhibit connected sums of manifolds of dimension at least five with Stein fillable contact structures that do not arise as contact connected sums. The proofs are based on holomorphic disk-filling techniques, with families of Legendrian open books (so-called “Lobs”) as boundary conditions.