1997 Fiscal Year Final Research Report Summary
STUDY OF MODULI SPACE OF RIEMANN SURFACES
| Project/Area Number |
08454014
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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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 Professor, 大学院数理科学研究科, 教授 (70011674)
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| Co-Investigator(Kenkyū-buntansha) |
SHISHIKURA Mitsuhiro Univ.of Tokyo Ass.Prof., 大学院数理科学研究科, 教授 (70192606)
ODA Takayuki University of Tokyo Professor, 大学院数理科学研究科, 教授 (10109415)
TSUBOI Takashi University of Tokyo Professor, 大学院数理科学研究科, 教授 (40114566)
KOHNO Toshitake University of Tokyo Professor, 大学院数理科学研究科, 教授 (80144111)
MATSUMOTO Yukio University of Tokyo Professor, 大学院数理科学研究科, 教授 (20011637)
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| Project Period (FY) |
1996 – 1997
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| Keywords | Riemann surface / moduli space / mapping class group / Torelli group / family of Riemann surfaces / monodromy / absolute Galois group / Johnson homomorphism |
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| Research Abstract |
We have investigated the structure of the moduli space of Riemann surfaces mainly from the viewpoints of topology. We found new relations between our vewpoints with those of algebraic geometry and number theory which have essentially new features different from the former ones. Thereby we meet many new problems which deserve future investigations. The main results we obtained are as follows.
(i) We proved that any group cocycle of the moduli space which we obtained in our earlier works can be represented as a polynomial on the known stable classes and we obtained explicit formula for it (joint work with N.Kawazumi). Moreover, by analizing closely how this formula degenerates once we fix the genus, we proved 1/3 of the Faber conjecture concerning the cohomology of the moduli space.
(ii) It is a very important problem to determine the nilpotent completion of the Torelli group which is a certain subgroup of the mapping class group. In connection to this, we found new obstructions for the images of the Johnson homomorphisms
(iii) We made progress in understanding the topological structure of families of Riemann surfaces. In particular, we obtained interesting results concerning the monodromies around singular fibers.
(iv) We also made progress in clarifying the relation between the structure of the Torelli group and outer representations of the absolute Galois group.
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