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Characteristic classes of surface bundles. (English) Zbl 0608.57020

Let \(\Sigma _ g\) be a closed orientable surface of genus g and let \(M_ g=\pi _ 0(Diff_ + \Sigma _ g)\) be its mapping class group, where \(Diff_ + \Sigma _ g\) is the group of all orientation preserving diffeomorphisms of \(\Sigma _ g\). Also let \(M_{g,*}\) be the mapping class group of \(\Sigma _ g\) relative to the base point. If \(g\geq 2\), then the extension \(\pi _ 1(\Sigma _ g)\to M_{g,*}\to M_ g\) ”classifies” all surface bundles of genus g, namely differentiable fibre bundles with fibre \(\Sigma _ g.\)
In this paper certain cohomology classes \(e\in H^ 2(M_{g,*}; {\mathbb{Z}})\), \(e_ i\in H^{2i}(M_ g; {\mathbb{Z}})\) \((i=1,2,...)\) are defined, and it is proved that the homomorphisms \({\mathbb{Q}}[e_ 1,e_ 2,...]\to H^ *(M_ g; {\mathbb{Q}})\) and \({\mathbb{Q}}[e,e_ 1,e_ 2,...]\to H^ *(M_{g,*}; {\mathbb{Q}})\) are injective up to degree g/3. Here the bound on the degree is based on a result of J. L. Harer [Ann. Math., II. Ser. 121, 215-249 (1985; Zbl 0579.57005)], and the same result for the former homomorphism was also proved independently by E. Y. Miller [J. Differ. Geom. 24, 1-14 (1986)]. A generalization of the above results to the case of surface bundles with cross sections is given and also several relations among characteristic classes of surface bundles are proved.
As a main application, a negative solution to the generalized Nielsen realization problem is given in the following form: the natural surjective homomorphism \(Diff_ + \Sigma _ g\to M_ g\) does not have a right inverse for all \(g\geq 18\).

MSC:

57R22 Topology of vector bundles and fiber bundles
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57R50 Differential topological aspects of diffeomorphisms
57R20 Characteristic classes and numbers in differential topology
57R30 Foliations in differential topology; geometric theory
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References:

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