23 research outputs found
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
Nx{0} in the model space NxR^{2k}.
In this article we make the observation that if (N,\xi_N) is a 3-dimensional
overtwisted submanifold with trivial normal bundle in (M,\xi), and if its model
neighborhood is sufficiently large, then (M,\xi) does not admit an exact
symplectic filling.Comment: 19 pages, 2 figures; added example of manifold that is not fillable
by neighborhood criterium; typo
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.Comment: 6 pages, no figure
Compact Lie group actions on contact manifolds
The main result in this thesis is the classification of SO(3)-actions on contact 5-manifolds. Using properties of the moment map, one can reduce the manifold to a 3-dimensional contact manifold with an S^1-action. This works everywhere outside of the singular orbits. For the singular orbits three models can be given that describe all possible cases. The 5-manifold is then obtained by gluing the singular set onto the 3-dimensional S^1-manifold in a compatible way. As it is well-known, S^1-bundles over a closed surface are classified by an integer called the Euler number. A similar invariant can be recovered in our 3-dimensional setting. We call it the Dehn-Euler number
The Weinstein conjecture in the presence of submanifolds having a Legendrian foliation
Helmut Hofer introduced in '93 a novel technique based on holomorphic curves
to prove the Weinstein conjecture. Among the cases where these methods apply
are all contact 3--manifolds with . We modify Hofer's
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.Comment: 11 pages, 2 figure
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.Comment: This is the version published by Algebraic & Geometric Topology on 15
December 200
Loose Legendrians and the plastikstufe
We show that the presence of a plastikstufe induces a certain degree of flexibility in contact manifolds of dimension 2n + 1 gt; 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
Brieskorn manifolds as contact branched covers of spheres
We show that Brieskorn manifolds with their standard contact structures are
contact branched coverings of spheres. This covering maps a contact open book
decomposition of the Brieskorn manifold onto a Milnor open book of the sphere.Comment: 8 pages, 1 figur
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.Comment: 68 pages, 5 figures. v2: Some attributions clarified, and other minor
edits. v3: exposition improved using referee's comments. Published by Invent.
Mat
5-dimensional contact SO(3)-manifolds and Dehn twists
In this paper the 5-dimensional contact SO(3)-manifolds are classified up to
equivariant contactomorphisms. The construction of such manifolds with singular
orbits requires the use of generalized Dehn twists.
We show as an application that all simply connected 5-manifoldswith singular
orbits are realized by a Brieskorn manifold with exponents (k,2,2,2). The
standard contact structure on such a manifold gives right-handed Dehn twists,
and a second contact structure defined in the article gives left-handed twists.Comment: 16 pages, 1 figure; simplification of arguments by restricting
classification to coorientation preserving contactomorphism