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New sheaf theoretic methods in differential topology. (English) Zbl 1212.57008
Summary: The Mumford conjecture predicts the ring of rational characteristic classes for surface bundles with oriented connected fibers of large genus. The first proof in [I. Madsen and M. Weiss, Ann. Math. (2) 165, No. 3, 843–941 (2007; Zbl 1156.14021)] relied on a number of well known but difficult theorems in differential topology. Most of these difficult ingredients have been eliminated in the years since then. This can be seen particularly in the paper of S. Galatius, I. Madsen, U. Tillmann and M. Weiss [Acta Math. 202, No. 2, 195–239 (2009; Zbl 1221.57039)] which has a second proof of the Mumford conjecture, and in the work of S. Galatius [Ann. Math. (2) 173, No. 2, 705–768 (2010; Zbl 1268.20057)] which is concerned mainly with a “graph” analogue of the Mumford conjecture. The newer proofs emphasize Tillmann’s theorem [cf. U. Tillmann, Invent. Math. 130, No. 2, 257–275 (1997; Zbl 0891.55019)] as well as some sheaf-theoretic concepts and their relations with classifying spaces of categories. These notes are an overview of the shortest known proof, or more precisely, the shortest known reduction of the Mumford conjecture to the Harer-Ivanov stability theorems for the homology of mapping class groups. Some digressions on the theme of classifying spaces and sheaf theory are included for motivation.
MSC:
57R19 Algebraic topology on manifolds and differential topology
57R20 Characteristic classes and numbers in differential topology
57R22 Topology of vector bundles and fiber bundles
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