STRUCTURE OF THE MAPPING CLASS GROUP AND GEOMETRY OF THE MODULI SPACE OF RIEMANN SURFACES

我们主要从拓扑学的角度研究了黎曼曲面的映射类群的结构和模空间的几何。得到的主要结果如下：(1)由Mumford-Morita类生成的曲面映射类群的有理上同调代数的子代数称为重言式代数。对这种重言式代数的研究有三种方法。第一种是基于Kawazumi提出的扭曲的Mumford-Morita类，第二种是基于三叶图的不变量，第三种是利用辛表示理论。综合前人的研究结果，我们发现上述三种方法是完全一致的。(Ii)映射类群的次要特征类的理论仍是一个很大的未知数。然而，在本研究中，我们证明了这些次特征类具有Torelli群的幂零补不能检测到的深层结构。Torelli群的可解或半单结构很有可能在未来变得越来越重要。(Iii)我们在理解Riemann曲面族的代数几何和拓扑结构方面取得了重大进展。特别地，我们得到了许多关于辛纤维的单边性的结果。

We have investigated the structure of the mapping class group of surfaces as well as the geometry of moduli space of Riemann surfaces mainly from the viewpoints of topology. The main results we obtained are as follows.(i) The subalgebra of the rational cohomology algebra of the mapping class group of surfaces generated by the Mumford-Morita classes is called the tautological algebra. There have been three approaches to the study of this tautological algebra. The first is based on the twisted Mumford-Morita classes introduced by Kawazumi, the second is in terms of invariants for trivalent graphs and the third is through symplectic representation theory. Summarizing previous results, we found that the above three approaches correspond exactly to each others.(ii) The theory of secondary characteristic classes of the mapping class group is still a largely unknown area. However, in this research, we proved that these secondary characteristic classes have deep structures that cannot be detected by the nilpotent completion of the Torelli group. It seems highly likely that the solvable or semi-simple structure of the Torelli group will become more and more important in the future.(iii) We made significant progress in understanding the algebro-geometrical as well as the topological structure of families of Riemann surfaces. In particular, we obtained many results concerning the monodromies of symplectic fibrations.