Project/Area Number  08454014 
Research Category 
GrantinAid for Scientific Research (B)

Section  一般 
Research Field 
Geometry

Research Institution  University of Tokyo 
Principal Investigator 
MORITA Shigeyuki University of Tokyo Professor, 大学院数理科学研究科, 教授 (70011674)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Fiscal Year 
1996 – 1997

Project Status 
Completed(Fiscal Year 1997)

Budget Amount *help 
¥4,800,000 (Direct Cost : ¥4,800,000)
Fiscal Year 1997 : ¥2,400,000 (Direct Cost : ¥2,400,000)
Fiscal Year 1996 : ¥2,400,000 (Direct Cost : ¥2,400,000)

Keywords  Riemann surface / moduli space / mapping class group / Torelli group / family of Riemann surfaces / monodromy / absolute Galois group / Johnson homomorphism / リーマン面 / モデュライ空間 / 写像類群 / トレリ群 / リーマン面の族 / モノドロミ / 絶対ガロア群 / ジョンソン準同型 
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. 4.トレリ群の構造と絶対ガロア群の表現との関連について,整数論および位相幾何学の立場からの研究が進み,両者の深い関連が視野に入ってきた.
