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Signature cocycles on the mapping class group and symplectic groups. (English) Zbl 1528.20097

Summary: W. Meyer [Math. Ann. 201, 239–264 (1973; Zbl 0241.55019)] constructed a cocycle in \(H^2(\mathsf{Sp}(2g, \mathbb{Z}); \mathbb{Z})\) which computes the signature of a closed oriented surface bundle over a surface. By studying properties of this cocycle, he also showed that the signature of such a surface bundle is a multiple of 4. In this paper, we study signature cocycles both from the geometric and algebraic points of view. We present geometric constructions which are relevant to the signature cocycle and provide an alternative to Meyer’s decomposition of a surface bundle. Furthermore, we discuss the precise relation between the Meyer and Wall-Maslov index. The main theorem of the paper, Theorem 6.6, provides the necessary group cohomology results to analyze the signature of a surface bundle modulo any integer \(N\). Using these results, we are able to give a complete answer for \(N = 2, 4, \text{ and } 8\), and based on a theorem of P. Deligne [C. R. Acad. Sci., Paris, Sér. A 287, 203–208 (1978; Zbl 0416.20042)], we show that this is the best we can hope for using this method.

MSC:

20J06 Cohomology of groups
55R10 Fiber bundles in algebraic topology
20C33 Representations of finite groups of Lie type
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