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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.

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

Citations:

Zbl 0741.00019
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