2004 Fiscal Year Final Research Report Summary
Complex analytic approach towards topology studies on the mapping class ganups for surfaces
| Project/Area Number |
14540065
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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 The University of Tokyo, Graduate School 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) |
MATSUMOTO Yukio The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (20011637)
MORITA Shigeyuki The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (70011674)
HASHIMOTO Yoshitake Osaka City University, Graduate School of Sciences, Associate Professor, 大学院・理学研究科, 助教授 (20271182)
SHIBUKAWA Youichi Hokkaido University, Graduate School of Sciences, Assistant Professor, 大学院・理学研究科, 助手 (90241299)
AKITA Toshiyuki Hokkaido University, Graduate School of Sciences, Associate Professor, 大学院・理学研究科, 助教授 (30279252)
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| Project Period (FY) |
2002 – 2004
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| Keywords | Riemann surface / moduli space / mapping class group / Magnus expansion / Morita-Mumford class / Stasheff associahedron / automorphism group of a free group / Johnson homomorphism |
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| Research Abstract |
We discovered a close relation between Stasheff associahedrons and (generalized) Magnus expansions of a free group. A certain part of the twisted Morita-Mumford classes can be extended to the automorphism group of a free group. It is parametrized by Stasheff associahedrons "infinitesimally" and "combinatorially" how the extended Johnson maps are far from true group homomorphisms.
We extended our theory on harmonic Magnus expansions to the universal family of Riemann surfaces. This yields another series of canonical 1 forms on the universal family than what we have already obtained on the moduli space. As a corollary, we obtained a proof that the first Jonson map and the (0,3)-twisted Morita-Mumford class coincides with each other as differential forms on the moduli space.
The Magus representation of the automorphism group of a free group was constructed in an intrinsic manner. Here 'intrinsic' means 'with no use of Fox' free differentials.'
We co-organized a workshop entitled "Toward the future of the topological study of manifolds" in November 2004.
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