| _version_ | 1866901742501756928 |
|---|---|
| author | Revista, Zen MATH, 10 |
| author_facet | Revista, Zen MATH, 10 |
| contents | Dehn surgery is a fundamental construction in 3-manifold topology that allows for the generation of new 3-manifolds from existing ones by "drilling" out a tubular neighborhood of a link and "filling" it back in with solid tori. William Thurston's groundbreaking work on the geometrization conjecture revealed that "almost all" 3-manifolds admit a hyperbolic structure, making hyperbolic 3-manifolds a central object of study. Consequently, the vast majority of Dehn surgeries performed on hyperbolic knot or link complements also result in hyperbolic 3-manifolds. However, a finite number of "exceptional" surgeries can yield manifolds with other geometric structures, such as Seifert fibered spaces, reducible manifolds, or toroidal manifolds. This paper investigates the phenomenon of the "hyperbolic obstruction" to Seifert fibered Dehn surgeries. Specifically, it explores the mathematical conditions under which a hyperbolic 3-manifold, upon Dehn filling, cannot become a Seifert fibered space. We delve into the theoretical frameworks, including Thurston's Hyperbolic Dehn Surgery Theorem, Culler-Shalen theory, and the classification of Seifert fibered spaces, to understand the constraints imposed by hyperbolicity on the resulting manifold's topology. The existence of such an obstruction highlights the rigidity of hyperbolic geometry and its profound influence on the landscape of 3-manifold topology. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17806448 |
| institution | Zenodo |
| language | |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Hyperbolic Obstruction to Seifert Fibered Dehn Surgeries Revista, Zen MATH, 10 Dehn surgery is a fundamental construction in 3-manifold topology that allows for the generation of new 3-manifolds from existing ones by "drilling" out a tubular neighborhood of a link and "filling" it back in with solid tori. William Thurston's groundbreaking work on the geometrization conjecture revealed that "almost all" 3-manifolds admit a hyperbolic structure, making hyperbolic 3-manifolds a central object of study. Consequently, the vast majority of Dehn surgeries performed on hyperbolic knot or link complements also result in hyperbolic 3-manifolds. However, a finite number of "exceptional" surgeries can yield manifolds with other geometric structures, such as Seifert fibered spaces, reducible manifolds, or toroidal manifolds. This paper investigates the phenomenon of the "hyperbolic obstruction" to Seifert fibered Dehn surgeries. Specifically, it explores the mathematical conditions under which a hyperbolic 3-manifold, upon Dehn filling, cannot become a Seifert fibered space. We delve into the theoretical frameworks, including Thurston's Hyperbolic Dehn Surgery Theorem, Culler-Shalen theory, and the classification of Seifert fibered spaces, to understand the constraints imposed by hyperbolicity on the resulting manifold's topology. The existence of such an obstruction highlights the rigidity of hyperbolic geometry and its profound influence on the landscape of 3-manifold topology. |
| title | The Hyperbolic Obstruction to Seifert Fibered Dehn Surgeries |
| url | https://doi.org/10.5281/zenodo.17806448 |