STUDY OF MODULI SPACE OF RIEMANN SURFACES

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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.