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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 Π–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 Π–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 Π–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
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Publication
Received: 20 October 2005
Revised: 5 April 2006
Accepted: 5 April 2006
Published: 21 June 2006
Authors
David Blanc
Department of Mathematics
University of Haifa
31905 Haifa
Israel
Mark W Johnson
Department of Mathematics
Penn State Altoona
Altoona PA 16601
USA
James M Turner
Department of Mathematics
Calvin College
Grand Rapids MI 49546
USA