2001 Fiscal Year Final Research Report Summary
A topological study of the moduli space of Riemann surface
| Project/Area Number |
11640054
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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 | The University of Tokyo |
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Principal Investigator |
KAWAZUMI Nariya University of Tokyo, Department of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (30214646)
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| Co-Investigator(Kenkyū-buntansha) |
OHBA Kiyoshi Ochanomizu University, Faculty of Science, Assistant, 理学部, 助手 (80242337)
MORITA Shigeyuki University of Tokyo, Department of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70011674)
MATSUMOTO Yukio University of Tokyo, Department of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (20011637)
AKITA Toshiyuki Hokkaido University, Faculty of Science, Associate Professor, 大学院・数学研究科, 助教授 (30279252)
SHIBUKAWA Youichi Hokkaido University, Faculty of Science, Assistant, 大学院・数学研究科, 助手 (90241299)
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| Project Period (FY) |
1999 – 2001
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| Keywords | Riemann surface / moduli space / Morita-Mumford classes / surface symmetry / Bruschi-Calogero equation / hyperelliptic mapping class group / Magnus expansion / mapping class group |
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| Research Abstract |
Rational cohomology of the mapping class group (Morita and Kawazumi) : We proved the Morita-Mumford classes generate the primary approximation to the cohomology of the moduli of Riemann surfaces induced by the Johnson homomorphisms even in the unstable range. We gave a complete description how the corresponding twisted Morita-Mumford classes behave when a finite graph degenerates.
Differential geometry of the moduli and Magnus expansions (Kawazumi) : Using Magnus expansions of a free group, we obtain an alternative proof of the IH-relation among the Johnson homomorphims. The notion of the harmonic Magnus expansion, which is canonicaly given by a complex structure of a surface, gives an interpretation of differential forms representing Morita-Mumford classes. We compute the quasi-conformal variation of the harmonic Magnus expansions as an explicit quadratic differential. We expect that the quadratic differential would give the key to the Johnson images of the mapping class groups.
Torsion cohomology of the mapping class groups (Akita and Kawazumi) : Akita has given fascinating conjectures related to Morita-Mumford classes on the whole mapping class groups. We proved them for any semi-free cyclic subgroup. Kawazumi proved them for the hyperelliptic mapping class groups. Akita proved the twice of the odd Morita-Mumford classes are functions of G-signatures for any finite subgroup G of the mapping class groups.
Bruschi-Calogero equation (Shibukawa and Kawazumi) : We gave all the meromorphic solutions of the equation. Shibukawa gave the complete classification of the R-matrices acting on the germs of meromorphic functions.
The details and other results are reported in the official booklet written in Japanese.
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