Hyperbolic L-spaces from a contact geometry perspective

Nonino, Isacco (2026) Hyperbolic L-spaces from a contact geometry perspective. PhD thesis, University of Glasgow.

Abstract

Thurston’s seminal work on the geometrization conjecture shows that 3-manifolds admit a rich variety of geometric structures. In a different direction, the foundational result of Martinet [Mar71] establishes that every closed, orientable 3-manifold supports a contact structure. Within contact topology, there is a fundamental dichotomy between tight and overtwisted contact structures. While overtwisted structures are completely classified [Eli89], tight contact structures are considerably more rigid and remain far less understood. The classification of tight contact structures has therefore been a central problem in low-dimensional topology over the past decades. Early progress focused on model cases such as the 3-sphere [Eli92], solid tori and lens spaces [Hon00a, Hon00b], and has since expanded to include broad classes of Seifert fibered manifolds [EH01b, GS03, Wu06, GLS07, Tos20]. However, tight contact structures on hyperbolic 3-manifolds remain largely unexplored. In this thesis, we investigate the classification problem for tight contact structures on hyperbolic 3-manifolds. In particular, we present the first classification result for an infinite family of hyperbolic L-spaces, obtained in joint work with Hyunki Min [MN26]. We then turn to the study of symplectic fillability within this setting. It is known that not every tight contact structure is symplectically fillable [EH02], and understanding this distinction is a subtle and important problem. Building on previous constructions, we develop an algorithmic obstruction to (weak) symplectic fillability for families of contact structures arising from rational surgeries on L-space knots [Non24].

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Supervisor's Name: Wand, Professor Andy and Lecuona, Professor Ana
• Date of Award: 2026
• Depositing User: Theses Team
• Unique ID: glathesis:2026-86215
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 08 Sep 2026 15:24
• Last Modified: 08 Sep 2026 15:30
• Thesis DOI: 10.5525/gla.thesis.86215
• URI: https://theses.gla.ac.uk/id/eprint/86215