The realizability of operations on homotopy groups concentrated in two degrees

The homotopy groups of a space are endowed with homotopy operations which define the \Pi-algebra of the space. An Eilenberg-MacLane space is the realization of a \Pi-algebra concentrated in one degree. In this paper, we provide necessary and sufficient conditions for the realizability of a \Pi-algebra concentrated in two degrees. We then specialize to the stable case, and list infinite families of such \Pi-algebras that are not realizable.