Higher homotopy operations and the realizability of homotopy groups

We describe an obstruction theory for the realization of a Π-algebra-that is, a graded group G_*with a prescribed action of the primary homotopy operations-as the homotopy groups of some space. The obstructions consist of higher homotopy operations, for which we provide an explicit definition in terms of certain sequences of polyhedra. There is a similar theory for realizing morphisms between Π-algebras, and thus, in particular, for distinguishing different realizations of a fixed Π-algebra. As an application we show that, for all primes p, the Π-algebra π_*S^r⊗Z/p cannot be realized.