Title:      On Realizing Diagrams of Pi-algebras
Authors:    David Blanc, Mark W. Johnson, and James M. Turner

E-mail addresses: blanc@math.haifa.ac.il, mwj3@psu.edu, jturner@calvin.edu

Addresses: 
       University of Haifa, 31905 Haifa, Israel
       Pennsylvania State Altoona, Altoona, PA 16601, USA
       Calvin College,  Grand Rapids, MI 49546, USA


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 by Dwyer, Kan, and Stover  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.