when a space is a product of eilenberg mclane spaces for $\pi_1(X)$ is not abelian

In When is $\Pi_{i\leq n} K(\pi_i(X), i)$ the nth base for a postnikov tower on $X$. , I discuss in my answer when an Abelian cw complex $X$ is a product of Eilenberg-Maclane spaces, and show that it happens when the hurewicz maps splits in every dimension.

What is the analagous result for $\pi_1(X)$ not abelian? Here we cannot have a splitting of the hurewicz map so the theorem is not literally true.

Edit 5/30 night: Some stuff I have thought about. and a plan to solve the problem. Professor Fedor Bogomolov mentioned when I asked him a construction of McQuillan. Could anyone point me to a relevant construction by him? I think to solve this problem I need a functorial way of constructing the abelianization of a space, given by $F$. The hope is that $F(X)$ will be a product of Eilenberg Maclane spaces iff $X$ is, which would reduce the theorem to my answer to When is $\Pi_{i\leq n} K(\pi_i(X), i)$ the nth base for a postnikov tower on $X$..

The first stab at such a construction would be to attach the generators of $\pi_1(X)$ to $X$. When $X$ is a product of eilenberg maclane spaces, this is satisfactory: I can devise a refinement of this construction that constructs $X$ up to weak homotopy equivalence. Namely I can assume that I am only attaching cells to 1-skeleton of $X$ which is $K(\pi_1,1) \times \{pt\} \times \{pt\} \times \{pt\} ....$. Since the cells will only be attached to the $K(\pi_1,1) \times \{pt\}....$, attaching the cells only to $K(\pi_1,1) \sqcup e^2_{\alpha \text{ abelian }} \times \{pt\} \times \{pt\} \times \{pt\} ....$ will get me a space that has the abelianized homotopy groups of $X$ that also does not depend on the choice of $e^2_{\alpha \text{ abelian }}$, up to homotopy equivalence of the constructed space.

To generalize this, and to try to get a similar functor for spaces that don't fiber trivially into products of eilenberg maclane spaces, I will need to read a paper by robinson that constructs a postnikov tower for nonsimply connected spaces. I will abelianize $X_1$(its an eilenberg maclane space so it will be unique up to homotopy) and use the postnikov invariant(which might not be well defined) $k_1$ to construct $X_2$ as the pullback of the path fibration, and use $k_2$ to construct $X_3$... and then we get $X=lim_{\xleftarrow{i} } X_i$.

I will need to check in the paper that the fundamental class of the fundamental class of the fiber $F \hookrightarrow X_2 \to X_1$ is transgressive. It probably won't be. If it is not i will need to see how else to construct $k_1$.

• Why don't you ask your professor Bogomolov for the relevant McQuillan construction? He probably knows what he had in mind! Commented May 31, 2016 at 3:43
• I think he was referring to the Quillen Plus construction. I stumbled on this while I was reading about the J-homomorphism, saw that there was a topologist who was named Quillen on the wikipedia page. I then looked up Quillen construction and found it. Commented Jun 17, 2016 at 3:40