2004 Fiscal Year Final Research Report Summary
Research on 3-manifolds by combinatorial and constructive methods
| Project/Area Number |
15540091
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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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 | Keio University |
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Principal Investigator |
ISHLI Ippei Keio University, Faculty of Science and Technology, Associate Professor, 理工学部, 助教授 (90051929)
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| Co-Investigator(Kenkyū-buntansha) |
MAEDA Yoshiaki Keio University, Faculty of Science and Technology, Professor, 理工学部, 教授 (40101076)
OTA Katsuhiro Keio University, Faculty of Science and Technology, Professor, 理工学部, 教授 (40213722)
MORIYOSHI Hitoshi Keio University, Faculty of Science and Technology, Associate Professor, 理工学部, 助教授 (00239708)
SHIMOMURA Shun Keio University, Faculty of Science and Technology, Professor, 理工学部, 教授 (00154328)
KAMETANI Yukio Keio University, Faculty of Science and Technology, Associate Professor, 理工学部, 助教授 (70253581)
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| Project Period (FY) |
2003 – 2004
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| Keywords | 3-manifold / spine / Heegaard splitting / Seifert fibered space / hyperbolic manifold / topological invariant / symplectic manifold |
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| Research Abstract |
In this research project, we have introduced a new topological invariant, called the "block number", for 3-manifolds, which estimates some kind of complexity of a 3-manifold just like as the Heegaard genus. The block number is defined by means of a flow-spine, and is enable us to classify 3-manifolds, and to parameterize 3-manifolds in each class by finitely many integers. Moreover the block number can be defined not only for a 3-manifold but also for a combed 3-manifold, a pair of a 3-manifold and a non-singular vector field on it. Using this invariant, we have gotten the following results :
1.The only combed 3-manifold having 0 as its block number is the canonical one on the product of the 2-sphere and the circle, and combed 3-manifolds with the block number 1 are canonical ones on lens spaces (including the 3-sphere).
2.The parameterization for 3-manifolds with the block number 2 was given. And, using the Reidemeister torsion, we have shown some results which imply that our parameterization uniformize the presentation of a combed 3-manifold.00
3.We have given a formula for calculating the value of the Thraev-Viro invariant for all Seifert fibered 3-manifolds.
On symplectic manifolds, we have gotten the following result :
4.If the universal covering space of a clsed symplectic manifold is contractible, the manifold does not admit any Riemannian metric with positive curvature.
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[Book] 量子的な微分幾何2004
Author(s)
大森英樹, 前田吉昭
Total Pages
370
Publisher
シュプリンガーフェアラーク社
Description
「研究成果報告書概要(和文)」より
