1999 Fiscal Year Final Research Report Summary
Heegaand spliffings and hyperbolic structures of 3-manifolds
| Project/Area Number |
09440033
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Geometry
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| Research Institution | Osaka University |
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Principal Investigator |
SAKUMA Makoto Grad. Sch. of Sci., Osaka University, Associate Professor, 大学院・理学研究科, 助教授 (30178602)
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| Co-Investigator(Kenkyū-buntansha) |
MUSAKAMI Jun Grad. Sch. of Sci., Osaka University, Associate Professor, 大学院・理学研究科, 助教授 (90157751)
EUOKI Tchiro Grad. Sch. of Sci., Osaka University, Associate Professor, 大学院・理学研究科, 助教授 (20146806)
MABUCHI Toshiki Grad. Sch. of Sci., Osaka University, Professor, 大学院・理学研究科, 教授 (80116102)
YAMASHITU Yasushi Faculty of Sci., Nava Women's University, Lecturer, 理学部, 講師 (70239987)
WADA Masaaki Faculty of Sci., Nova Women's University, Professor, 理学部, 教授 (80192821)
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| Project Period (FY) |
1997 – 1999
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| Keywords | Heegaard splitting / hyperbolic structure / cone-manifold / 2-bridge knot / quasi-Fuchsian group / once-punctured forus |
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| Research Abstract |
The study of Heegaard splittings of 3-manifolds has been one of the most important themes in 3-manifold theory, and we already have deep understanding of the Heegaard splittings of "non-hyperbolike" 3-manifolds. However, our understanding of those of hyperbolic manifolds is far from satisfaction. In particular, as far as we know, no relationship between the hyperbolic structures and the Heegaard splittigs had been known.
In this project we have proved that the hyperbolic structure of a 2-bridge knot complement is intimately related with its bridge structure, which is a kind of Heegaard splitting. In fact, we have given a concrete construction of the hyperbolic structure of a 2-bridge knot complement by using the 2-bridge structure. To be more precise, we have constructed a continuous family of hyperbolic cone-manifold structures on a 2-bridge knot complement which have singularities along the upper and lower tunnels, where the cone angle varies from 0 to 2π. The cone-manifold structure with cone angle 0 corresponds to a rational boundary group of the quasi-Fuchsian once-punctured torus space and that with cone angle 2π gives the hyperbolic structure of the 2-bridge knot complement. To establish this result, we have given an explicit formulation and a full proof to (a part of) the theory announced by Jorgensen on the quasi-Fuchsian once-punctured torus groups, and generalized the theory to that for the groups outside the quasi-Fuchsian once-punctured space. The computer program "OPTI" developed by Masaaki Wada for this project visualizes Jorgensen's theory and its generalization, and it has been an indispensable tool not only for this project but also for the study of Teichmuller spaces. We hope the result we have obtained in this project is the beginning of the study of the relationship between the hyperbolic structures and the Heegaard splittings of 3-manifolds.
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Research Products
(11 results)
