2006 Fiscal Year Final Research Report Summary
Geometry of Groups and Moduli Spaces
| Project/Area Number |
16204005
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| Research Category |
Grant-in-Aid for Scientific Research (A)
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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 | UNIVERSITY OF TOKYO |
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Principal Investigator |
MORITA Shigeyuki University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院数理学研究科, 教授 (70011674)
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| Co-Investigator(Kenkyū-buntansha) |
FURUTA Mikio University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院数理学研究科, 教授 (50181459)
TSUBOI Takashi University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院数理学研究科, 教授 (40114566)
KAWAZUMI Nariya University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院数理学研究科, 助教授 (30214646)
FUJIWARA Koji Tohoku University, Graduate School of Science, Associate Professor, 大学院理学研究科, 助教授 (60229078)
MITSUMATSU Yoshihiko Chuo University, Department of Science & Technology, Professor, 理工学部, 教授 (70190725)
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| Project Period (FY) |
2004 – 2006
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| Keywords | mapping class group / Riemann surface / moduli space / Floer homotopy type / diffeomorphism group / Teichmuller space / hyperbolic group / Morita-Mumford class |
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| Research Abstract |
We investigated mapping class group of surfaces and moduli space of Riemann surfaces as well as related groups and moduli spaces mainly from the viewpoint of topology. In a joint work with Penner, the representative constructed a canonical 1-cocycle on the Teichmuller space. Kawazumi, together with Penner and Bene, realized higher Johnson homomorphisms combinatorially on the Teichmuller space. Furuta developed theory of Floer homotopy type associated to the Seiberg-Witten theory and Tsuboi obtained a remarkable result concerning the structure of real analytic diffeomorphism group of manifolds. Fujiwara, Kohno, and Matsumoto obtained deep results in combinatorial structure of hyperbolic as well as the mapping class group, study of configuration spaces and arithmetic mapping class group, respectively. Mitsumatsu, Kitano, Akita with-Kawazumi, Hirose, and Murakami obtained interesting results, respectively, in the studies of 3-dimensional contact geometry, twisted Alexander polynomials, integral Morita-Mumford classes, mapping class group of surfaces in 4-manifolds, and volumes in hyperbolic geometry. Also the representative studied the structure of the Lie algebra consisting of symplectic derivations of the tensor algebra, without unit, generated by the homology of surfaces and, as an application, constructed unstable homology classes of genus 1 moduli spaces. Based on these results, further study of our theme has come into view.
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