Characteristic classes of surface bundles. (English) Zbl 0579.55006

Morita, Shigeyuki

Bull. Am. Math. Soc., New Ser. 11, 386-388 (1984).

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

Keywords:

mapping class group of a surface; cohomology groups of the mapping; class group; surface bundles with non-trivial characteristic classes

Citations:

Zbl 0193.523

References:

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