CEU Electronic Theses and Dissertations, 2018
| Author | Matkovic, Irena |
|---|---|
| Title | Tight and Fillable Contact Structures on Seifert Fibered Manifolds |
| Summary | The fundamental question of contact topology is to classify contact structures. Since overtwisted structures respect homotopy principles, we restrict the question to tight structures, and among them to the fillable ones, which bound in symplectic or holomorphic category. First to be understood are prime atoroidal manifolds, which are either hyperbolic or small Seifert fibered. This thesis deals with small Seifert fibered manifolds, which are obtained from circle bundles over the sphere by performing Dehn surgery along (at most) three of its fibers. For many of these manifolds, the complete classification is known, and all tight structures are also fillable. We review these results from various perspectives: convex decompositions along with contact surgery, the dual presentation by open books, and in Heegaard Floer theory by means of the Ozsváth-Szabó contact invariant. Our main focus are zero-twisting structures on small Seifert fibered spaces of the form M(-1;r_1,r_2,r_3), which are special as they include non-fillable tight structures. We classify them by the Ozsváth-Szabó contact invariant and characterize which of them are (Stein) fillable. The crucial properties of these contact manifolds are the possibility to view the underlying manifold as the boundary of a negative definite plumbing and the planarity of the contact structures. For classification of tight structures, we use a specific description of Heegaard Floer homology by equivalence classes of characteristic cohomology elements on the bounded plumbing; we single out the elements which correspond to the contact invariants and give a contact interpretation to the equivalence relations between them. In order to characterize fillability, we study positive factorizations of the planar monodromy. |
| Supervisor | Stipsicz, András I. |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2018/matkovic_irena.pdf |
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