Homology operations and vn-Bockstein operations

In this note, we determine the action of the first differential in the Atiyah–Hirzebruch spectral sequence for the nth Morava K-theory of the p-adic construction Dp(ℓ)(X)=Cℓ(p)∧ΣpX∧p of a pointed space X for n⩾1. Our formula gives us a way to compute the first differentials in the Atiyah–Hirzebruch spectral sequence for the Morava K-theory of iterated loop spaces. ous formulas are also proved for the Steenrod operations.