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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*Sr⊗Z/p cannot be realized

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Given a diagram of …-algebras (graded groups equipped with an action of the primary homotopy operations), we ask whether it can be realized as the homotopy groups of a diagram of spaces. The answer given here is in the form of an obstruction theory, of somewhat wider application, formulated in terms of generalized …-algebras. This extends a program begun by Dwyer, Kan, Stover, Blanc and Goerss [21; 10] to study the realization of a single …-algebra.In particular, we explicitly analyze the simple case of a single map, and provide a detailed example, illustrating the connections to higher homotopy operations.18G55; 55Q05, 55P65