This paper describes connected components of the strata of holomorphic abelian differentials on marked Riemann surfaces with prescribed degrees of zeros. Unlike the case for unmarked Riemann surfaces, we find there can be many connected components, distinguished by roots of the cotangent bundle of the surface. In the course of our investigation we also characterize the images of the fundamental groups of strata inside of the mapping class group. The main techniques of proof are mod r winding numbers and a mapping class group-theoretic analogue of the Euclidean algorithm.

中文翻译：

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Teichmüller 空间上阿贝尔微分层的连通分量

本文描述了具有规定零度的标记黎曼曲面上的全纯阿贝尔微分层的连通分量。与未标记黎曼曲面的情况不同，我们发现可以有许多连通分量，由曲面余切丛的根区分。在我们的调查过程中，我们还表征了映射类组内部的基本地层组的图像。证明的主要技术是 mod r 绕组数和欧几里得算法的映射类群论模拟。