Complex analytic approach towards topology studies on the mapping class ganups for surfaces

We discovered a close relation between Stasheff associahedrons and (generalized) Magnus expansions of a free group. A certain part of the twisted Morita-Mumford classes can be extended to the automorphism group of a free group. It is parametrized by Stasheff associahedrons "infinitesimally" and "combinatorially" how the extended Johnson maps are far from true group homomorphisms. We extended our theory on harmonic Magnus expansions to the universal family of Riemann surfaces. This yields another series of canonical 1 forms on the universal family than what we have already obtained on the moduli space. As a corollary, we obtained a proof that the first Jonson map and the (0,3)-twisted Morita-Mumford class coincides with each other as differential forms on the moduli space. The Magus representation of the automorphism group of a free group was constructed in an intrinsic manner. Here 'intrinsic' means 'with no use of Fox' free differentials.' We co-organized a workshop entitled "Toward the future of the topological study of manifolds" in November 2004.