I'm looking for a good reference that has a detailed treatment of obstruction theory in the case where the target space is not simple. The specific situation I am interested in involves lifting a map of 3-skeletons from a $K(G, 1)$ to an arbitrary homotopy 3-type $X$ to the 4-skeletons (and hence to a true map between the spaces); the obstruction to this "should" live in $H^4(G, \pi\_3(X))$ (with the appropriate action of $G$ on $\pi\_3(X)$) via a local coefficient system, but I've been having trouble hashing out the details. Experts I've asked have given answers ranging from saying that it's impossible to saying that they're certain that it's possible but they don't know a reference that does it. Books I've looked at tend to gloss over the details, but they seem to indicate that this should work. Can anybody set me straight on this?



Obstruction theory for non-simple spaces
Paul Olum developed some obstruction theory for maps into non-simple spaces back in the 1940-ies and 50-ies. You may want to check out his paper "Obstructions to extensions and homotopies", Annals of Mathematics, Vol 52, 1950, pp 1-50, if you have not looked at it yet.


