Splitting and nonsplitting in the enumeration degrees

This paper continues the project, initiated in (Arslanov, Cooper and Kalimullin 2003) [3], of describing general conditions under which relative splittings are derivable in the local structure of the enumeration degrees, for which the Ershov hierarchy provides an informative setting.The main results below include a proof that any high total e-degree below is splittable over any low e-degree below it, a non-cupping result in the high enumeration degrees which occurs at a low level of the Ershov hierarchy, and a 0̸‴-priority construction of a e-degree unsplittable over a 3-c.e. e-degree below it.