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

Shigeyuki Morita

Geometry & Topology Monographs 2 (1999) 349–406

DOI: 10.2140/gtm.1999.2.349

arXiv: math.GT/9911258

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.

This paper is dedicated to Robion C Kirby on the occasion of his 60th birthday.

Keywords

mapping class group, Torelli group, Johnson homomorphism, moduli space of curves

Mathematical Subject Classification

Primary: 32G15, 57R20

Secondary: 14H10, 55R40, 57M99, 57N05

References
Publication

Received: 30 December 1998
Revised: 29 March 1999
Published: 21 November 1999

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