  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.
