On symplectic fillings

In this note we make several observations concerning symplectic fillings. In particular we show that a (strongly or weakly) semi-fillable contact structure is fillable and any filling embeds as a symplectic domain in a closed symplectic manifold. We also relate properties of the open book decomposition of a contact manifold to its possible fillings. These results are also useful in proving property P for knots [P Kronheimer and T Mrowka, Geometry and Topology, 8 (2004) 295-310, math.GT/0311489] and in showing the contact Heegaard Floer invariant of a fillable contact structure does not vanish [P Ozsvath and Z Szabo, Geometry and Topology, 8 (2004) 311-334, math.GT/0311496].Comment: Published electronically at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-5.abs.htm

Generating function polynomials for legendrian links

It is shown that, in the 1-jet space of the circle, the swapping and the flyping procedures, which produce topologically equivalent links, can produce nonequivalent legendrian links. Each component of the links considered is legendrian isotopic to the 1-jet of the 0-function, and thus cannot be distinguished by the classical rotation number or Thurston-Bennequin invariants. The links are distinguished by calculating invariant polynomials defined via homology groups associated to the links through the theory of generating functions. The many calculations of these generating function polynomials support the belief that these polynomials carry the same information as a refined version of Chekanov's first order polynomials which are defined via the theory of holomorphic curves.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol5/paper23.abs.htm

Overtwisted open books from sobering arcs

We study open books on three manifolds which are compatible with an overtwisted contact structure. We show that the existence of certain arcs, called sobering arcs, is a sufficient condition for an open book to be overtwisted, and is necessary up to stabilization by positive Hopf-bands. Using these techniques we prove that some open books arising as the boundary of symplectic configurations are overtwisted, answering a question of Gay in Algebr. Geom. Topol. 3 (2003) 569--586.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-47.abs.htm

Transverse contact structures on Seifert 3-manifolds

We characterize the oriented Seifert-fibered three-manifolds which admit positive, transverse contact structures.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-49.abs.htm

Transversal torus knots

We classify positive transversal torus knots in tight contact structures up to transversal isotopy.Comment: 16 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol3/paper11.abs.htm

Effect of Legendrian Surgery

The paper is a summary of the results of the authors concerning computations of symplectic invariants of Weinstein manifolds and contains some examples and applications. Proofs are sketched. The detailed proofs will appear in our forthcoming paper. In the Appendix written by S. Ganatra and M. Maydanskiy it is shown that the results of this paper imply P. Seidel's conjecture equating symplectic homology with Hochschild homology of a certain Fukaya category.Comment: v.4 is significantly extended, especially Sections 6 and 8. Several other sections, including Appendix are rewritte

Strongly fillable contact 3-manifolds without Stein fillings

We use the Ozsvath-Szabo contact invariant to produce examples of strongly symplectically fillable contact 3-manifolds which are not Stein fillable.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper38.abs.htm

Bordism groups of solutions to differential relations

In terms of category theory, the Gromov homotopy principle for a set valued functor $F$ asserts that the functor $F$ can be induced from a homotopy functor. Similarly, we say that the bordism principle for an abelian group valued functor $F$ holds if the functor $F$ can be induced from a (co)homology functor. We examine the bordism principle in the case of functors given by (co)bordism groups of maps with prescribed singularities. Our main result implies that if a family $R$ of prescribed singularity types satisfies certain mild conditions, then there exists an infinite loop space $B(R)$ such that for each smooth manifold $N$ the cobordism group of maps into $N$ with only $R$-singularities is isomorphic to the group of homotopy classes of maps $[N, B(R)]$.Comment: 38 page

Knotted Legendrian surfaces with few Reeb chords

For $g>0$, we construct $g+1$ Legendrian embeddings of a surface of genus $g$ into $J^1(R^2)=R^5$ which lie in pairwise distinct Legendrian isotopy classes and which all have $g+1$ transverse Reeb chords ($g+1$ is the conjecturally minimal number of chords). Furthermore, for $g$ of the $g+1$ embeddings the Legendrian contact homology DGA does not admit any augmentation over $Z/2Z$, and hence cannot be linearized. We also investigate these surfaces from the point of view of the theory of generating families. Finally, we consider Legendrian spheres and planes in $J^1(S^2)$ from a similar perspective.Comment: 30 pages, 11 figure