Examples Of Contact Manifolds Research Articles

A monopole invariant for families of contact structures

We use monopole Floer homology to study the topology of the space of contact structures on a 3-manifold. Our main tool is a generalisation of the Kronheimer–Mrowka–Ozsváth–Szabó contact invariant to an invariant for families of contact structures, and we establish foundational results that describe the interaction between this invariant and the module structure in monopole Floer homology. We apply these results in several examples of contact manifolds, such as the links of non-rational surface singularities, to deduce several applications. Namely, we are able to obstruct the existence of sections of a natural fibration over the 2-sphere whose total space is the space of contact structures on the 3-manifold, and from this we are able to detect the existence of exotic loops of contact structures on contact 3-manifolds with convex sphere boundary.

Contact invariants of [formula omitted]-Gorenstein toric contact manifolds, the Ehrhart polynomial and Chen-Ruan cohomology

Q-Gorenstein toric contact manifolds provide an interesting class of examples of contact manifolds with torsion first Chern class. They are completely determined by certain rational convex polytopes, called toric diagrams, and arise both as links of toric isolated singularities and as prequantizations of monotone toric symplectic orbifolds. In this paper we show how the cylindrical contact homology invariants of a Q-Gorenstein toric contact manifold are related to•the Ehrhart (quasi-)polynomial of its toric diagram;•the Chen-Ruan cohomology of any crepant toric orbifold resolution of its corresponding toric isolated singularity;•the Chen-Ruan cohomology of any monotone toric symplectic orbifold base that gives rise to it through prequantization.

Positive topological entropy of positive contactomorphisms

A positive contactomorphism of a contact manifold $M$ is the end point of a contact isotopy on $M$ that is always positively transverse to the contact structure. Assume that $M$ contains a Legendrian sphere $\Lambda$, and that $(M,\Lambda)$ is fillable by a Liouville domain $(W,\omega)$ with exact Lagrangian $L$ such that $\omega|_{\pi_2(W,L)}=0$. We show that if the exponential growth of the action filtered wrapped Floer homology of $(W,L)$ is positive, then every positive contactomorphism of $M$ has positive topological entropy. This result generalizes the result of Alves and Meiwes from Reeb flows to positive contactomorphisms, and it yields many examples of contact manifolds on which every positive contactomorphism has positive topological entropy, among them the exotic contact spheres found by Alves and Meiwes. A main step in the proof is to show that wrapped Floer homology is isomorphic to the positive part of Lagrangian Rabinowitz-Floer homology.

Contact manifolds with symplectomorphic symplectizations

We provide examples of contact manifolds of any odd dimension $\geq 5$ which are not diffeomorphic but have exact symplectomorphic symplectizations.

Moment-Angle Manifolds, Intersection of Quadrics and Higher Dimensional Contact Manifolds

We construct new examples of contact manifolds in arbitrarily large dimensions. These manifolds which we call quasi moment-angle manifolds, are closely related to the classical moment-angle manifolds.

A hierarchy of local symplectic filling obstructions for contact 3-manifolds

We generalize the familiar notions of overtwistedness and Giroux torsion in 3-dimensional contact manifolds, defining an infinite hierarchy of local filling obstructions called planar torsion, whose integer-valued order $k \ge 0$ can be interpreted as measuring a gradation in "degrees of tightness" of contact manifolds. We show in particular that any contact manifold with planar torsion admits no contact type embeddings into any closed symplectic 4-manifold, and has vanishing contact invariant in Embedded Contact Homology, and we give examples of contact manifolds that have planar k-torsion for any $k \ge 2$ but no Giroux torsion. We also show that the complement of the binding of a supporting open book never has planar torsion. The unifying idea in the background is a decomposition of contact manifolds in terms of contact fiber sums of open books along their binding. As the technical basis of these results, we establish existence, uniqueness and compactness theorems for certain classes of J-holomorphic curves in blown up summed open books; these also imply algebraic obstructions to planarity and embeddings of partially planar domains. The results are applied further in followup papers on weak symplectic fillings (arXiv:1003.3923, joint with K. Niederkrueger), non-exact symplectic cobordisms (arXiv:1008.2456) and Symplectic Field Theory (arXiv:1009.3262, joint with J. Latschev).

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.

Weak symplectic fillings and holomorphic curves

We prove several results on weak symplectic fillings of contact 3-manifolds, including: (1) Every weak filling of any planar contact manifold can be deformed to a blow up of a Stein filling. (2) Contact manifolds that have fully separating planar torsion are not weakly fillable - this gives many new examples of contact manifolds without Giroux torsion that have no weak fillings. (3) Weak fillability is preserved under splicing of contact manifolds along symplectic pre-Lagrangian tori - this gives many new examples of contact manifolds without Giroux torsion that are weakly but not strongly fillable. We establish the obstructions to weak fillings via two parallel approaches using holomorphic curves. In the first approach, we generalize the original Gromov-Eliashberg "Bishop disk" argument to study the special case of Giroux torsion via a Bishop family of holomorphic annuli with boundary on an "anchored overtwisted annulus". The second approach uses punctured holomorphic curves, and is based on the observation that every weak filling can be deformed in a collar neighborhood so as to induce a stable Hamiltonian structure on the boundary. This also makes it possible to apply the techniques of Symplectic Field Theory, which we demonstrate in a test case by showing that the distinction between weakly and strongly fillable translates into contact homology as the distinction between twisted and untwisted coefficients.

Vassiliev invariants of Legendrian, transverse, and framed knots in contact three-manifolds

We show that for a large class of contact three-manifolds the groups of Vassiliev invariants of Legendrian and of framed knots are canonically isomorphic. As a corollary, we obtain that the group of finite order Arnold's J +-type invariants of wave fronts on a surface F is isomorphic to the group of Vassiliev invariants of framed knots in the spherical cotangent bundle ST ∗F of F. On the other hand, we construct the first examples of contact manifolds for which Vassiliev invariants of Legendrian knots can distinguish Legendrian knots that realize isotopic framed knots and are homotopic as Legendrian immersions.