Tag - Contact geometry

I will introduce a new version of symplectic homology that resembles the relative symplectic homology and that is related to the symplectic homology of a Liouville sector. This version, called selective symplectic homology, is associated with a Liouville domain and an open subset of its boundary. The selective symplectic homology is obtained as the direct limit of the Floer homology groups for Hamiltonians whose slopes tend to infinity on the open subset but remain close to 0 and positive on the rest of the boundary. As an application, I will prove a contact non-squeezing phenomenon on homotopy spheres that are fillable by Liouville domains with infinite dimensional symplectic homology: there exists a smoothly embedded closed ball in such a sphere that cannot be made arbitrarily small by a contact isotopy. These homotopy spheres include examples that are diffeomorphic to standard spheres and whose contact structures are homotopic to standard contact structures.

Homeomorphism is called contact if it can be written as C⁰-limit of contactomorphisms. The contact version of Eliashberg-Gromov rigidity theorem states that smooth contact homeomorphisms preserve contact structure. Submanifold L of a contact manifold (Y, ξ) is called isotropic if ξ|_TL =0. Isotropic submanifolds of maximal dimension are called Legendrian, otherwise we call them subcritical isotropic. In this talk, we will try to answer whether the isotropic property is preserved by contact homeomorphisms. It is expected that subcritical isotropic submanifolds are flexible, while we expect that Legendrians are rigid. We show that subcritical isotropic curves are flexible, and we give a new proof of the rigidity of Legendrians in dimension 3. Moreover, we provide a certain type of rigidity of Legendrians in higher dimensions.

Given a trivalent plane graph embedded in the Euclidean plane (up to isotopy), Treumann and Zaslow constructed and studied a certain associated Legendrian surface embedded in standard contact ℝ⁵, nowadays referred to as a Legendrian 2-weave. Using gradient flow trees, Casals and Murphy computed its Legendrian contact dg-algebra with commutative coefficients (i.e. working over the group ring of the first homology group). We extend their computation to the non-commutative setting (i.e. working over the group ring of the fundamental group). For these Legendrian 2-weaves, we further verify the well-known conjecture that the moduli space of representations of this fully non-commutative version of the Legendrian contact dg-algebra are in bijective correspondence with a certain moduli space of sheaves.

I will explain a construction of a Legendrian version of embedded contact homology (ECH) for a sutured contact manifold Y along with a collection of Legendrians L contained in the boundary. The chain complex is generated by sets of Reeb orbits and Reeb chords, and the differential counts certain embedded curves of 'relative ECH index' 1. This version of ECH and its PFH analogue will help to categorify the zeta function of gradient flows of circle valued Morse functions discussed by Hutchings in his thesis. It also provides a unification between the cylindrical formulation of Heegaard-Floer theory given by Lipschitz and standard ECH.

I will discuss some quantitative aspects for Legendrians in a (more or less) general contact manifold. These include lower bounds on the number of Reeb chords between a Legendrian and its contact Hamiltonian image, the non-degeneracy of the Chekanov/Hofer/Shelukhin Legendrian metric, and some 3-dimensional non-squeezing results. The main tool is the barcode of a relative Rabinowitz Floer theory.

The 2011 PhD thesis of Farris outlined a scheme to show that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a ℤ/2ℤ-graded group to the exterior algebra of the homology of its base. In addition to providing an overview of the proof, I will explain how Morgan Weiler and I extended this result by computing the ℤ-grading on the chain complex, permitting our computation of the unstable and stable portions of ECH.

30 years ago I proved that any tight contact structure on the 3-sphere is diffeomorphic to the standard one. I also optimistically claimed at the same paper that similar methods could be used to prove a multi-parametric version: the space of tight contact structures on the 3-sphere, fixed at a point, is contractible. In our recent joint with N. Mishachev paper we proved this result. While the proof indeed roughly follows the strategy of my 1991 paper, it is much more involved. In particular, it uses a new criterion for tightness of a characteristic foliation on the 2-sphere, which is valid without any contact convexity assumptions.

Knots associated to overtwisted manifolds are less explored. There are two types of knots in an overtwisted manifold – loose and non-loose. Non-loose knots are knots with tight complements whereas loose knots have overtwisted complements. While we understand loose knots, non-loose knots remain a mystery. The classification and structure problems of these knots vary greatly compared to the knots in tight manifolds. Especially we are interested in how satellite operations on a knot in overtwisted manifold changes the geometric property of the knot. In this talk, I will discuss under what conditions cabling operation on a non-loose knot preserves non-looseness.