Geometry & Topology Monographs 2 (1999), Proceedings of the Kirbyfest, paper no. 20, pages 349-406.

Structure of the mapping class groups of surfaces: a survey and a prospect

Shigeyuki Morita

Abstract. In this paper, we survey recent works on the structure of the mapping class groups of surfaces mainly from the point of view of topology. We then discuss several possible directions for future research. These include the relation between the structure of the mapping class group and invariants of 3--manifolds, the unstable cohomology of the moduli space of curves and Faber's conjecture, cokernel of the Johnson homomorphisms and the Galois as well as other new obstructions, cohomology of certain infinite dimensional Lie algebra and characteristic classes of outer automorphism groups of free groups and the secondary characteristic classes of surface bundles. We give some experimental results concerning each of them and, partly based on them, we formulate several conjectures and problems.

Keywords. Mapping class group, Torelli group, Johnson homomorphism, moduli space of curves

AMS subject classification. Primary: 57R20, 32G15. Secondary: 14H10, 57N05, 55R40,57M99.

E-print: arXiv:math.GT/9911258

Submitted: 30 December 1998. (Revised: 29 March 1999.) Published: 21 November 1999.

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Shigeyuki Morita
Department of Mathematical Sciences, University of Tokyo
Komaba, Tokyo 153-8914, Japan

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