Tight Contact Structures on Hyperbolic Three- Manifolds

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Tezin Türü: Doktora

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Bilimleri Enstitüsü, Fen Bilimleri Enstitüsü, Türkiye

Tezin Onay Tarihi: 2018

Öğrenci: Merve Secgin

Asıl Danışman (Eş Danışmanlı Tezler İçin): MEHMET FIRAT ARIKAN

Özet:

In this dissertation, we study tight contact structures on hyperbolic 3-manifolds and homology spheres. We build a family of infinitely many hyperbolic 3-manifolds admitting tight contact structures. To put it more explicitly, we consider a certain infinite family of surface bundles over the circle whose monodromies are taken from some collection of pseudo-Anosov diffeomorphisms. We show the existence of tight contact structure on every closed 3-manifold obtained via rational r-surgery along a section of any member of the family except one r. Consequently, we obtain infinitely many hyperbolic closed 3-manifolds admitting tight contact structures. Moreover, we construct infinitely many contractible 4-manifolds bounded by a homology sphere as generalized Mazur type manifolds built by Akbulut and Kirby. Specifically, the construction is formed by a 4-dimensional 2-handlebody where infinitely many of them have hyperbolic Stein fillable boundaries.