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Characteristic classes of surface bundles. (English) Zbl 0579.55006
Consider the mapping class group M(g) of a surface of genus $$g\geq 2$$. The author constructs certain cohomology classes $$e_ i\in H^{2i}(M(g),{\mathbb{Q}})$$ and announces some non-triviality results. For each natural number k and for all g sufficiently large (depending on k) the elements $$e_ 1,...,e_ k\in H^*(M(g),{\mathbb{Q}})$$ are all non- trivial. Again for all k and g sufficiently large the 2i-th Betti number of M(g) is at least i for $$i=1,...,k$$. The proofs, which are sketched in the article, apply a method of M. F. Atiyah [Global Analysis, Papers in Honor of K. Kodaira, 73-84 (1969; Zbl 0193.523)] iteratively to construct sufficiently many surface bundles with non-trivial characteristic classes.
Reviewer: S.Jekel

##### MSC:
 55R40 Homology of classifying spaces and characteristic classes in algebraic topology 20F34 Fundamental groups and their automorphisms (group-theoretic aspects) 57N05 Topology of the Euclidean $$2$$-space, $$2$$-manifolds (MSC2010) 55R10 Fiber bundles in algebraic topology
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##### References:
 [1] M. F. Atiyah, The signature of fibre-bundles, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 73 – 84. [2] A. Borel and F. Hirzebruch, Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458 – 538. · Zbl 0097.36401 · doi:10.2307/2372795 · doi.org [3] Raoul Bott, On a topological obstruction to integrability, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 127 – 131. [4] C. J. Earle and J. Eells, The diffeomorphism group of a compact Riemann surface, Bull. Amer. Math. Soc. 73 (1967), 557 – 559. · Zbl 0196.09402 [5] John Harer, The second homology group of the mapping class group of an orientable surface, Invent. Math. 72 (1983), no. 2, 221 – 239. · Zbl 0533.57003 · doi:10.1007/BF01389321 · doi.org [6] W. J. Harvey, Geometric structure of surface mapping class groups, Homological group theory (Proc. Sympos., Durham, 1977) London Math. Soc. Lecture Note Ser., vol. 36, Cambridge Univ. Press, Cambridge-New York, 1979, pp. 255 – 269. [7] Steven P. Kerckhoff, The Nielsen realization problem, Ann. of Math. (2) 117 (1983), no. 2, 235 – 265. · Zbl 0528.57008 · doi:10.2307/2007076 · doi.org [8] K. Kodaira, A certain type of irregular algebraic surfaces, J. Analyse Math. 19 (1967), 207 – 215. · Zbl 0172.37901 · doi:10.1007/BF02788717 · doi.org
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