Morita, Shigeyuki Mapping class groups of surfaces and three-dimensional manifolds. (English) Zbl 0743.57007 Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. I, 665-674 (1991). [For the entire collection see Zbl 0741.00019.]This paper gives a brief introduction to the study of the group \(M_ g\) of isotopy classes of orientation preserving diffeomorphisms of \(\Sigma_ g\), the closed orientable surface of genus \(g\). Particular attention is paid to the cohomology of \(M_ g\), the action of \(M_ g\) on the lower central series of \(\pi_ 1(\Sigma_ g)\), and the relationship between \(M_ g\) and certain topological invariants of three-manifolds. Reviewer: L.P.Neuwirth (Princeton) Cited in 6 Documents MSC: 57M60 Group actions on manifolds and cell complexes in low dimensions 55R40 Homology of classifying spaces and characteristic classes in algebraic topology 57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010) 57M50 General geometric structures on low-dimensional manifolds 20F99 Special aspects of infinite or finite groups 57M05 Fundamental group, presentations, free differential calculus Keywords:isotopy classes of orientation preserving diffeomorphisms; closed orientable surface; cohomology; lower central series; topological invariants of three-manifolds Citations:Zbl 0741.00019 PDFBibTeX XMLCite \textit{S. Morita}, in: Proceedings of the international congress of mathematicians (ICM), August 21--29, 1990, Kyoto, Japan. Volume I. Tokyo etc.: Springer-Verlag. 665--674 (1991; Zbl 0743.57007)