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The structure of the mapping class group and characteristic classes of surface bundles. (English) Zbl 0791.57018

Bödigheimer, Carl-Friedrich (ed.) et al., Mapping class groups and moduli spaces of Riemann surfaces. Proceedings of workshops held June 24-28, 1991, in Göttingen, Germany, and August 6-10, 1991, in Seattle, WA (USA). Providence, RI: American Mathematical Society. Contemp. Math. 150, 303-315 (1993).
Let \(\Sigma_ g\) be a closed oriented surface of genus \(g\). Denote by \({\mathcal M}_ g\) the mapping class group of \(\Sigma_ g\) which consists of all the isotopy classes of orientation preserving diffeomorphisms of \(\Sigma_ g\). The group \({\mathcal M}_ g\) acts on the abelianization of \(\pi_ 1(\Sigma_ g)\) which is just the first homology group \(H = H_ 1 (\Sigma_ g;\mathbb{Z})\) of \(\Sigma_ g\) and induces a representation \(\rho_ 2: {\mathcal M}_ g \to \text{Aut }H\).
If we fix the symplectic basis of \(H\) then \(\text{lm }\rho_ 2\) can be canonically identified with the Siegel modular group \(\text{Sp} (2g;\mathbb{Z})\). Now let \(\Gamma\) be any characteristic subgroup of \(\Gamma_ 1= \pi_ 1(\Sigma_ g)\). Then \({\mathcal M}_ g\) acts on the quotient group \(\Gamma_ 1/\Gamma\) by outer automorphisms and we obtain a representation \(\rho_ \Gamma: {\mathcal M}_ g \to \text{Out }(\Gamma_ 1/\Gamma)\). Let \(\{\Gamma_ k\}_{k \in \mathbb{N}}\) be the lower central series of \(\Gamma_ 1,\Gamma_{k + 1} = [\Gamma_ k,\Gamma_ 1]\) and \(\rho_ k = \rho_{\Gamma_ k}\), \(k \geq 1\).
In the present paper the properties of \(\rho_ k\), \(k \in \mathbb{N}\), and some related homomorphisms are studied and their applications to characteristic classes of surface bundles are considered.
For the entire collection see [Zbl 0777.00025].

MSC:

57R20 Characteristic classes and numbers in differential topology
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
57N10 Topology of general \(3\)-manifolds (MSC2010)
57M99 General low-dimensional topology
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