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
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.
Mapping class group, Torelli group, Johnson homomorphism, moduli space of curves
AMS subject classification.
Primary: 57R20, 32G15. Secondary: 14H10, 57N05, 55R40,57M99.
Submitted: 30 December 1998.
(Revised: 29 March 1999.)
Published: 21 November 1999.
Notes on file formats
Department of Mathematical Sciences, University of Tokyo
Komaba, Tokyo 153-8914, Japan
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