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

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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.