2001 Fiscal Year Final Research Report Summary
Research for low-climensional manifolds with various geometric structures
| Project/Area Number |
12640068
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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 Univ., Integrated Human Studies, Ass. Professor, 総合人間学部, 助教授 (80134443)
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| Co-Investigator(Kenkyū-buntansha) |
FUJII Michihiko Kyoto Univ., Integrated Human Studies, Ass. Professor, 総合人間学部, 助教授 (60254231)
NISHIWADA Kimimasa Kyoto Univ., Integrated Human Studies, Professor, 総合人間学部, 教授 (60093291)
KATO Shinichi Kyoto Univ., Integrated Human Studies, Professor, 総合人間学部, 教授 (90114438)
ASANO Kiyoshi Kyoto Univ., Graduate School of Human and Environmental Studies, Professor, 大学院・人間・環境研究科, 教授 (90026774)
SAKURAGAWA Takashi Kyoto Univ., Integrated Human Studies, Ass. Professor, 総合人間学部, 助教授 (60196136)
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| Project Period (FY) |
2000 – 2001
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| Keywords | 3-manifold / 4-manifold / spin structure / Dirac operator / intersection form / Seiberg-Witten theory / V-manifold / cone manifold |
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| Research Abstract |
Ue studied the constraints on the diffeomorphism types of 3, 4-manifolds coming from the various geometric structures by certain invariants, in particular the one originated from Seiberg-Witten theory. First in the joint work with Mikio Furuta and Yoshihiro Fukumoto, he studied the W invariants of homology 3-spheres, and showed that in case of Seifert homology 3-spheres this invariant coincides with the Neumann-Siebenmann invariant, and that it is homology cobordism invariant under certain extra conditions. Saveliev used our key result about the estimates on the index of the Dirac operator over V 4-manifolds, and extended our results. Also Ue determined all the contributions from the isolated singularities to the index of the Dirac operator over V 4-manifolds. This contribution itself is an invariant for a pair of spherical 3-manifold and its spin structure. As its application, he gave certain constraints on the intersection forms of definite spin 4-manifolds bounded by spherical 3-manifolds, and also new estimates on the normal Euler number of the real projective plane embedded in 4-manifolds. These results are applicable to a wider class of 3-manifolds. Fujii studied 3-dimension cone manifolds with only simple loops as their singlar sets, and succeeded to give explicit solutions in terms of Gaussian hypergeometric functions to the system of ordinary differential equations for the harmonic vector field, which is a key to describe the harmonic 1-form on the tubular neighborhood of the singular set. Also in the joint work with Kazuhiko Fukui, Imanishi determined the first cohomology of the the group of Lipschitz homeomorphisms preserving the differentiable foliations of codimension 1 by utilizing several results about the group of Lipschitz homeomorphisms of the interval.
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Research Products
(12 results)
