Werner Meyer constructed a cocycle in $H^2(Sp(2g, \mathbb{Z}); \mathbb{Z})$ which computes the signature of a closed oriented surface bundle over a surface, with fibre a surface of genus g. 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 the 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 and 8, and based on a theorem of Deligne, we show that this is the best we can hope for using this method.

中文翻译：

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映射类群和辛群上的签名余环

Werner Meyer 在 $H^2(Sp(2g, \mathbb{Z}); \mathbb{Z})$ 中构建了一个cocycle，它计算了一个表面上的闭合定向表面丛的特征，其中纤维是 g 属的表面. 通过研究这个共环的性质，他还表明这种表面丛的特征是 4 的倍数。在本文中，我们从几何和代数的角度研究了特征共环。我们提出了与特征环相关的几何构造，并提供了 Meyer 对表面丛的分解的替代方法。此外，我们还讨论了 Meyer 和 Wall-Maslov 指数之间的精确关系。论文的主要定理定理 6.6 提供了必要的群上同调结果，以分析以任意整数 N 为模的表面丛的特征。使用这些结果，