#### Volume 6, issue 2 (2006)

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On realizing diagrams of $\Pi$–algebras

### David Blanc, Mark W Johnson and James M Turner

Algebraic & Geometric Topology 6 (2006) 763–807
 arXiv: math.AT/0604161
##### 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.

##### Keywords
realization of diagrams, (simplicial) $\Pi$–algebras, (resolution) model categories, cohomology
##### Mathematical Subject Classification 2000
Primary: 18G55
Secondary: 55Q05, 55P65