STUDY OF MODULI SPACE OF RIEMANN SURFACES

本文主要从拓扑学的角度研究了黎曼曲面模空间的结构。我们发现了我们的观点与代数几何和数论的观点之间的新关系，这些新关系具有不同于以往观点的本质上的新特点。因此，我们遇到了许多新的问题，值得进一步研究。我们得到的主要结果如下。(i)我们证明了我们在以前的工作中得到的模空间的任何群上圈都可以表示为已知稳定类上的多项式，并得到了它的显式公式（与N.Kawazumi的联合工作）。此外，通过分析这个公式在确定亏格后如何退化，证明了关于模空间上同调的Faber猜想的1/3。(ii)Torelli群是映射类群的一个子群，判定Torelli群的幂零完备性是一个非常重要的问题。在这方面，我们发现了新的障碍物的图像的约翰逊同态（iii）我们取得了进展，在了解家庭的黎曼曲面的拓扑结构。特别是，我们得到了有趣的结果关于单值周围的奇异纤维。(iv)我们还取得了进展，澄清之间的关系的结构的Torelli集团和外部表示的绝对伽罗瓦集团。

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.