We consider the universal family E n d of superelliptic curves: each curve Σ n d in the family is a d -fold covering of the unit disk, totally ramified over aset P of n distinct points; $$\Sigma _n^d \hookrightarrow E_n^d \to {{\rm{C}}_n}$$ Σ n d ↪ E n d → C n is a fiber bundle, where C n is the configuration space of n distinct points. We find that E n d is the classifying space for the complex braid group of type B( d, d, n ) and we compute a big part of the integral homology of E n d , including a complete calculation of the stable groups over finite fields by means of Poincaré series. The computation of the main part of the above homology reduces to the computation of the homology of the classical braid group with coefficients in the first homology group of Σ n d , endowed with the monodromy action. While giving a geometric description of such monodromy of the above bundle, we introduce generalized $${1 \over d}$$ 1 d -twists, associated to each standard generator of the braid group, which reduce to standard Dehn twists for d = 2.

中文翻译：

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超椭圆曲线族、复杂编织群和广义 Dehn 扭曲

我们考虑超椭圆曲线的通用族 E nd ：族中的每条曲线 Σ nd 都是单位圆盘的 ad 折叠覆盖，在 n 个不同点的集合 P 上完全分支；$$\Sigma _n^d \hookrightarrow E_n^d \to {{\rm{C}}_n}$$ Σ nd ↪ E nd → C n 是一个纤维丛，其中 C n 是 n 个不同点的配置空间. 我们发现 E nd 是类型 B( d, d, n ) 的复杂编织群的分类空间，我们计算了 E nd 的大部分积分同调，包括对有限域上稳定群的完整计算：庞加莱级数的方法。上述同调的主要部分的计算简化为经典辫群的同调计算，其系数在Σ nd 的第一同调群中，具有单向作用。