reference request: Postnikov towers for non-simply-connected spaces

Question

I've read that for a space $X$ which is connected but not necessarily simply-connected, we can no longer obtain the $n^{\rm th}$ layer $P_nX$ of the Postnikov tower for $X$ as the pullback of a path-loops fibration (with contractible total space). Instead, there is a pullback diagram $$ \begin{array}{ccc} P_nX & \rightarrow & B\pi_1 X \\ \downarrow & & \downarrow \\ P_{n-1}X & \rightarrow & K(\pi_nX,n+1) \times_{\pi_1 X} E \pi_1X \end{array} $$ (where the fiber is of course still $K(\pi_nX,n)$). Does anyone know of a reference for this? I checked the two books I have on hand -- Hatcher and Switzer -- but neither of them covers this.

Answer 1

I think your $B\pi_1X$ should be something else... What is the right-hand vertical map supposed to be?

In any case, this should do:

Link

C. Robinson "Moore-Postnikov Systems for non-simply connected spaces"

In particular the beginning of section 4 (his $\hat{K}$ is the same as the Borel construction that you have, I think.)