The study of low-dimensional manifolds with various geometric structures
| Project/Area Number |
16540063
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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 | Kyoto University |
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Principal Investigator |
UE Masaaki Kyoto University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (80134443)
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| Co-Investigator(Kenkyū-buntansha) |
FUJII Michihiko Kyoto University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (60254231)
KATO Shin'ichi Kyoto University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90114438)
NISHIWADA Kimimasa Kyoto University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (60093291)
USHIKI Shigehiro Kyoto University, Graduate School of Human and Environmental Studies, Professor, 大学院・人間・環境学研究科, 教授 (10093197)
IMANISHI Hideki Kyoto University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (90025411)
山内 正敏 京都大学, 大学院・理学研究科, 教授 (30022651)
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| Project Period (FY) |
2004 – 2005
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| Project Status |
Completed (Fiscal Year 2005)
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| Budget Amount *help |
¥2,200,000 (Direct Cost: ¥2,200,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
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| Keywords | four-dimensinal manifold / homology 3-sphere / Seifert manifold / Dirac operator / V manifold / Seiberg-Witten theory / Dehn surgery / intersection form / ディラック作用素 / 幾何構造 / ホモロジー球面 / Fukumoto-Furuta不変量 / 双曲構造 |
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| Research Abstract |
The head investigator continued the research on the structures of 3 and 4-manifolds, in particular the diffeomorphism types of them. For the research of the structures of 4-manifolds with boundary, he generalized the Fukumoto-Furuta invariants to apply them to rational homology 3-spheres, which have been originally defined by using the index of the Dirac operator on V-manifolds based on the Seiberg-Witten theory. In particular he proved that the Fukumoto-Furuta invariant for Siefert 3-manifolds coincides with the Neumann-Siebenmann invariant, and also proved its spin rational homology cobordism invariance. He applied these results to the constraints for the intersection forms of 4-manifolds whose boundaries are Seifert manifolds, and to the conditions for the Seifert 3-manifolds to be obtained by Dehn surgery on knots in the 3-sphere. The constraints for the intersection forms of 4-manifolds with boundary or the conditions for the 3-manifolds to be obtained by Dehn surgery on knots also have been studied by 3-manifold invariants derived from the Heegaard Floer homology by Oszvath and Szabo, which are based on the different principle from ours. We investigated the relation between Oszvath-Szabo's invariant and the Fukumoto-Furuta invariant for the lens spaces via the eta invariant, but their relationship for more general cases is still open for the research in the future.
The investigator Fujii found the relation between the deformation of the hyperbolic structure of the complement of a hyperbolic knot and the rational points of the elliptic curves. The investigator Kato, Ushiki, and Nishiwada studied the representation theory of p-adic symmetric spaces, two dimensional complex dynamical systems, Theorema elegantissimum by Gauss respectively, and Imanishi continued the study of foliations.
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Report
(3 results)
Research Products
(13 results)
