arXiv research

We review Giroux's contact handles and contact handle attachments in dimension three and show that a bypass attachment consists of a pair of contact 1 and 2-handles. As an application we describe explicit contact handle decompositions of infinitely many pairwise non-isotopic overtwisted 3-spheres. We also give an alter…

We describe various handle moves in contact surgery diagrams, notably contact analogues of the Kirby moves. As an application of these handle moves, we discuss the respective classifications of long and loose Legendrian knots.

A standard convexity condition on the boundary of a symplectic manifold involves an induced positive contact form (and contact structure) on the boundary; the corresponding concavity condition involves an induced negative contact form. We present two methods of symplectically attaching 2-handles to convex boundaries of…

In this paper we give explicit, handle-by-handle constructions of concave symplectic fillings of all closed, oriented contact 3-manifolds. These constructions combine recent results of Giroux relating contact structures and open book decompositions of 3-manifolds, earlier results of the author on attaching 4-dimensiona…

Innovates contact structure invariant, non-decreasing under operations.

Holomorphic handle attaching proves complex surface properties.

We use contact handle decompositions and a stabilization process to compute the cylindrical contact homology of a subcritical Stein-fillable contact manifold with vanishing first Chern class, and show that it is completely determined by the homology of a subcritical Stein-filling of the contact manifold.

Researchers construct gluing maps and prove duality in sutured Floer homology.

New contact Kirby moves complete the set for contact surgery diagrams.

Higher-dimensional contact manifolds lack certain properties.

We construct four-dimensional symplectic cobordisms between contact three-manifolds generalizing an example of Eliashberg. One key feature is that any handlebody decomposition of one of these cobordisms must involve three-handles. The other key feature is that these cobordisms contain chains of symplectically embedded …

This article sketches various ideas in contact geometry that have become useful in low-dimensional topology. Specifically we (1) outline the proof of Eliashberg and Thurston's results concerning perturbations of foliatoins into contact structures, (2) discuss Eliashberg and Weinstein's symplectic handle attachments, an…

According to Giroux, contact manifolds can be described as open books whose pages are Stein manifolds. For 5-dimensional contact manifolds the pages are Stein surfaces, which permit a description via Kirby diagrams. We introduce handle moves on such diagrams that do not change the corresponding contact manifold. As an …

We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured monopole Floer homology theory (SHM). Our invariant can be viewed as a generalization of Kronheimer and Mrowka's contact invariant for closed contact 3-manifolds and as the monopole Floer analogue of Honda, Kazez, a…

We give a combinatorial description of the Legendrian contact homology algebra associated to a Legendrian link in $S^1\times S^2$ or any connected sum $\#^k(S^1\times S^2)$, viewed as the contact boundary of the Weinstein manifold obtained by attaching 1-handles to the 4-ball. In view of the surgery formula for symplec…

We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to…

The topology of Stein surfaces and contact 3-manifolds is studied by means of handle decompositions. A simple characterization of homeomorphism types of Stein surfaces is obtained --- they correspond to open handlebodies with all handles of index lessthan or = 2. An uncountable collection of exotic R^4's is shown to ad…

We present an approach to constructing broken Lefschetz fibrations (BLFs) $f:X\rightarrow S^2$ from a handle decomposition of a 4-manifold $X$. Given a handle decomposition as input these techniques yield explicit descriptions of the BLFs, and do not rely on classification results from contact topology or the choice of…

The abstract discusses applications of Menke's JSJ decomposition to symplectic fillings of various 3-manifolds.

The purpose of this paper is to introduce Liouville hypersurfaces in contact manifolds, which generalize ribbons of Legendrian graphs and pages of supporting open books. Liouville hypersurfaces are used to define a gluing operation for contact manifolds called the Liouville connect sum. Performing this operation on a c…

Given two open books with equal pages we show the existence of an exact symplectic cobordism whose negative end equals the disjoint union of the contact manifolds associated to the given open books, and whose positive end induces the contact manifold associated to the open book with the same page and concatenated monod…

It has been a central open problem in Heegaard Floer theory whether cobordisms of links induce homomorphisms on the associated link Floer homology groups. We provide an affirmative answer by introducing a natural notion of cobordism between sutured manifolds, and showing that such a cobordism induces a map on sutured F…

From any 4-dimensional oriented handlebody X without 3- and 4-handles and with b_2>0, we construct arbitrary many compact Stein 4-manifolds which are mutually homeomorphic but not diffeomorphic to each other, so that their topological invariants (their fundamental groups, homology groups, boundary homology groups, and …

We define relative Gromov-Witten invariants and establish a general gluing theory of pseudo-holomorphic curves for symplectic cutting and contact surgery. Then, we use our general gluing theory to study the change of GW-invariants of Calabi-Yau 3-folds tranform under flops and extremal transitions. We prove a complete …

Researchers develop methods to construct Lagrangian cobordisms between Legendrian knots.

We give a method for constructing a Legendrian representative of a knot in $S^3$ which realizes its maximal Thurston-Bennequin number under a certain condition. The method utilizes Stein handle decompositions of $D^4$, and the resulting Legendrian representative is often very complicated (relative to the complexity of …

Develops scalable differentiable physics for complex object interactions.

Study tackles causal effects of close contact on MRSA infections from entangled treatment data.

The study explores embedding closed contact manifolds in higher dimensions.

Study on contact Hamiltonian functions for singular contact structures.

It is known that for coprime integers $p>q\geq 1$, the lens space $L(p^2,pq-1)$ bounds a rational ball, $B_{p,q}$, arising as the 2-fold branched cover of a (smooth) slice disk in $B^4$ bounding the associated 2-bridge knot. Lekilli and Maydanskiy give handle decompositions for each $B_{p,q}$. Whereas, Yamada gives an …

The paper defines and studies contact surgery numbers for contact 3-manifolds.

Study weak quasi contact metric manifolds to generalize K-contact and Sasakian manifolds criteria.

Simplified construction of contact triad connection for analysis.

The paper examines conditions for contact surgeries on rational homology 3-spheres.

We study the contact equivalence problem for toric contact structures on $S^3$-bundles over $S^2$. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To f…

Contact surgeries transform manifolds, and all admit symplectic caps.

Introduces contact dual pairs using line bundles.

Abstract: Survey on contact submanifolds.

New contact structures on folded sums of contact mapping tori are tight under certain conditions.

Embeds all contact 3-manifolds into specific 5-manifolds.

Study explores weak generalized K-contact structures in contact metric spaces.

New contact structures detected by contact homology.

Local flatness theorem for paraquaternionic contact structures.

New examples show contact invariants can be non-zero even with half Giroux torsion.

New result on contact surgeries and overtwistedness.

Survey of contact homology for contact manifolds.

We consider certain type of fiber bundles with odd dimensional compact contact base, exact symplectic fibers, and the structure group contained in the group of exact symplectomorphisms of the fiber. We call such fibrations "contact symplectic fibrations". By a result of Hajduk-Walczak, some of these admit contact struc…