On realizing diagrams of $\Pi$–algebras

Abstract

Given a diagram of $\Pi$–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 $\Pi$–algebras. This extends a program begun in [J. Pure Appl. Alg. 103 (1995) 167-188] and [Topology 43 (2004) 857-892] to study the realization of a single $\Pi$–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.