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

Connected components of strata of Abelian differentials over Teichm\"uller space