Gordana Matic

Right-veering diffeomorphisms of compact surfaces with boundary

We initiate the study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary. The monoid strictly contains the monoid of products of positive Dehn twists. We explain the relationship to tight contact structures and open book decompositions.

Convex decomposition theory

We use convex decomposition theory to (1) reprove the existence of a universally tight contact structure on every irreducible 3-manifold with nonempty boundary, and (2) prove that every toroidal 3-manifold carries infinitely many nonisotopic, nonisomorphic tight contact structures.

Stein 4-manifolds with boundary and contact structures

Abstract We discuss several applications of Seiberg-Witten theory in conjunction with an embedding theorem (proved elsewhere) for complex 2-dimensional Stein manifolds with boundary. We show that a closed, real 2-dimensional surface smoothly embedded in the interior of such a manifold satisfies an adjunction inequality, regardless of the sign of its self-intersection. This inequality gives constraints on the minimum genus of a smooth surface representing a given 2-homology class. We also discuss consequences for the contact structures existing on the boundaries of these Stein manifolds. We prove a slice version of the Bennequin-Eliashberg inequality for holomorphically fillable contact structures, and we show that there exist families of homology 3-spheres with arbitrarily large numbers of homotopic, nonisomorphic tight contact structures. Another result we mention is that the canonical class of a complex 2-dimensional Stein manifold with boundary is invariant under self-diffeomorphisms fixing the boundary.

Transverse contact structures on Seifert 3–manifolds

We characterize the oriented Seifert-fibered three-manifolds which admit positive, transverse contact structures.

Pinwheels and bypasses

We give a necessary and sufficient condition for the addition of a collection of disjoint bypasses to a convex surface to be universally tight - namely the nonexistence of a polygonal region which we call a virtual pinwheel. AMS Classification 57M50; 53C15

Contact Invariants in Floer Homology

In a pair of seminal papers Peter Ozsvath and Zoltan Szabo defined a collection of homology groups associated to a 3-manifold they named Heegaard-Floer homologies. Soon after, they associated to a contact structure ξ on a 3-manifold, an element of its Heegaard-Floer homology, the contact invariant c(ξ). This invariant has been used to prove a plethora of results in contact topology of 3-manifolds. In this series of lectures we introduce and review some basic facts about Heegaard Floer Homology and its generalization to manifolds with boundary due to Andras Juhasz, the Sutured Floer Homology. We use the open book decompositions in the case of closed manifolds, and partial open book decompositions in the case of contact manifolds with convex boundary to define contact invariants in both settings, and show some applications to fillability questions.