×

zbMATH — the first resource for mathematics

Secondary characteristic classes of surface bundles. (English) Zbl 1166.55004
A surface bundle is a smooth fibre bundle \(\pi :E \rightarrow B\) with closed oriented two-dimensional fibres. The Miller-Morita-Mumford characteristic classes associate to such a bundle classes \(\kappa_{i} \;\in \;H^{2i}(B; {\mathbb Z})\). The author shows that \(k_{i}\) is divisible by \(p^{2}\) if and only if \(i+1\) is divisible by \(p(p-1)\). The proof introduces new characteristic classes \(\lambda_{i}\) with the property that \(p \lambda_{i}(E) = \kappa_{i(p-1)-1}(E) \;\in \;H^*(B; {\mathbb Z}/p^{2})\).

MSC:
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] S Galatius, I Madsen, U Tillmann, Divisibility of the stable Miller-Morita-Mumford classes, J. Amer. Math. Soc. 19 (2006) 759 · Zbl 1102.57015
[2] I Madsen, U Tillmann, The stable mapping class group and \(Q(\mathbbC P^\infty_+)\), Invent. Math. 145 (2001) 509 · Zbl 1050.55007
[3] H Toda, Composition methods in homotopy groups of spheres, Annals of Math. Studies 49, Princeton Univ. Press (1962) · Zbl 0101.40703
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.