Related papers: Generalized Chain Surgeries and Applications

This is an announcement of results proved in [GGS1], [GGS2], [C], and [CG] where methods from Lie theory were used as new tools for the study of symplectic Lefschetz fibrations.

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-K\"ahler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a…

We construct, somewhat non-standard, Legendrian surgery diagrams for some Stein fillable contact structures on some plumbing trees of circle bundles over spheres. We then show how to put such a surgery diagram on the pages of an open book…

We describe explicit horizontal open books on some Seifert fibered 3--manifolds. We show that the contact structures compatible with these horizontal open books are Stein fillable and horizontal as well. Moreover we draw surgery diagrams…

The distribution of a given sequence in the set of all sequences with n ones and m = M - n zeros are found by relating the problem to the partitions of a natural number in m natural summands, taking into account the order. The formulas…

This brief survey of some singularity invariants related to Milnor fibers should serve as a quick guide to references. We attempt to place things into a wide geometric context while leaving technicalities aside. We focus on relations among…

In this note we complete the discussion of minimality of symplectic fiber sums. We find, that for fiber sums along spheres the minimality of the sum is determined by the cases discussed by M. Usher and one additional case: If the sum is the…

From a fibered link in the 3-sphere may be constructed a field of not everywhere tangent 2-planes; when the fibered link is the link of an isolated critical point of a map from 4-space to the plane, the plane field is essentially the field…

In a recent paper of Akhmedov, Etnyre, Mark and Smith, it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic)…

We study holomorphic spheres in certain symplectic cobordisms and derive information about periodic Reeb orbits in the concave end of these cobordisms from the non-compactness of the relevant moduli spaces. We use this to confirm the strong…

We prove bilinear inequalities for differential operators in $\mathbb{R}^2$. Such type inequalities turned out to be useful for anisotropic embedding theorems for overdetermined systems and the limiting order summation exponent. However,…

We give a short proof of a conjecture of Stipsicz on the minimality of fiber sums of Lefschetz fibrations, which was proved earlier by Usher. We then construct the first examples of genus g > 1 Lefschetz fibrations on minimal symplectic…

Let $G$ be a group which admits the structure of an iterated semidirect product of finitely generated free groups. We construct a finite, free resolution of the integers over the group ring of $G$. This resolution is used to define…

We consider symplectic singularities in the sense of A. Beauville as examples of Poisson schemes. Using Poisson methods, we prove that a symplectic singularity admits a finite stratification with smooth symplectic strata. We also prove that…

We consider product 4--manifolds S^1 X M, where M is a closed, connected and oriented 3-manifold. We prove that if S^1 X M admits a complex structure or a Lefschetz or Seifert fibration, then the following statement is true: S^1 X M admits…

We introduce an idea of constructing Lefschetz fibrations of Weinstein manifolds from Weinstein handle decompositions on them. We prove theorems that formulate the idea for the cases of cotangent bundles and some plumbings. As a corollary,…

A bifibration structure on a $6$-manifold is a map to either the complex projective plane $\mathbb{P}^2$ or a $\mathbb{P}^1$-bundle over $\mathbb{P}^1$, such that its composition with the projection to $\mathbb{P}^1$ is a ($6$-dimensional)…

A study of symplectic actions of a finite group $G$ on smooth 4-manifolds is initiated. The central new idea is the use of $G$-equivariant Seiberg-Witten-Taubes theory in studying the structure of the fixed-point set of these symmetries.…

Let $H\subseteq S^3$ be the two-component Hopf link. After choosing a Legendrian representative of $H$ with respect to the standard tight contact structure on $S^3$ we perform contact $(-1)$-surgery on the link itself. The manifold we get…

This research announcement continues the study of the symplectic homology of Weinstein manifolds undertaken in [BEE1] where the symplectic homology, as a vector space, was expressed in terms of the Legendrian homology algebra of the…