Let $\mathrm{\Sigma }$ be a surface with either boundary or marked points, equipped with an arbitrary framing. In this note we determine the action of the associated ‘framed mapping class group’ on the homology of $\mathrm{\Sigma }$ relative to its boundary (respectively, marked points), describing the image as the kernel of a certain crossed homomorphism related to classical spin structures. Applying recent work of the authors, we use this to describe the monodromy action of the orbifold fundamental group of a stratum of abelian differentials on the relative periods.

中文翻译：

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框架映射类组的相对同源表示

让 $\mathrm{\Sigma }$是具有边界或标记点的表面，并具有任意框架。在本说明中，我们确定关联的“框架映射类组”对$\mathrm{\Sigma }$相对于其边界（分别是标记点）的距离，将图像描述为与经典自旋结构有关的某种交叉同态的核。应用作者最近的工作，我们用它来描述阿贝尔微分层的球面基本群对相对周期的单峰作用。