What is a principal refinement of a Postnikov system?

Question

I've been reading the book of Hilton, Mislin, and Roitberg on Localization of Nilpotent Groups and Spaces. In Section II.2 they define a principal refinement at stage $n$ of a Postnikov system $$\cdots \to X_n\overset{p_n}{\to} X_{n-1}\to \cdots$$ to be a factorization of $p_n$ into a finite sequence of fibrations $$X_n=Y_c \overset{q_c}{\to}\cdots\to Y_1\overset{q_1}{\to}Y_0=X_{n-1}$$ whose fibers are Eilenberg-MacLane spaces $K(G_i,n)$.

But isn't the point of a Postnikov system that $p_n$ is already a fibration whose fibers are Eilenberg-MacLane spaces? So I don't understand why the condition of the definition isn't satisfied trivially at every stage. Perhaps there's some subtlety involving the condition also given that each $q_i$ be induced by a map $g_i: Y_{i-1}\to K(G_i, n+1)$, but aren't all fibrations with fiber $K(G_i, n)$ induced this way since $K(G_i, n+1)$ is the base of a path-space fibration with fiber $K(G_i, n)$?

Answer 1

The key idea is that not all fibrations $E \to B$ with fibre an Eilenberg-MacLane space $K(\pi,n)$ can be constructed by pulling the principal path fibration $K(\pi,n) \to PK(\pi,n+1) \to K(\pi,n+1)$ along a classifying map $B \to K(\pi,n+1)$. If you can construct the fibration in this way then the classifying map is the Postnikov $k$-invariant. Clearly this at least requires that the group $\pi$ is abelian.

Now, it is a nice little exercise to check that existence of a principal refinement of the Postnikov tower is equivalent to $\pi_1$ being nilpotent and acting nilpotently on all of the higher homotopy groups.

(Recall that a group $G$ acts nilpotently on a group $H$ if $H$ has a finite sequence of $G$-invariant subgroups $H \supset H_1 \supset H_2 \supset \cdots H_k = 1$ such that $H_i/H_{i+1}$ is abelian and the action of $G$ on it is trivial.)

The general idea of Hilton, Mislin, and Roitberg is that is it obvious how to localise abelian groups, and nilpotent groups are those which can be assembled from abelian groups one layer at a time. So we can localise nilpotent groups by working one layer at a time, and then we can localise nilpotent spaces by working up the refined Postnikov tower one stage at a time.

Answer 2

A fibration with $K(G,n)$ fibers need not be principal. The base of the universal fibration with fiber $K(G,n)$ is not a $K(G,n+1)$, even if $G$ is abelian. For example, in the case $G=\mathbb Z$ and $n=1$ it is $BO(2)$, with $\pi_1=\mathbb Z/2$ and $\pi_2=\mathbb Z$. In fact, in general it has homotopy groups $\pi_1=Aut(G)$, $\pi_{n+1}=G$, all the rest trivial. That's if $n>1$; in the case $n=1$, $\pi_1$ is the outer automorphisms of $G$ (the automorphisms modulo inner automorphisms) and $\pi_2$ is the center of $G$.

You can see this by viewing that space (base of universal fibration with fiber $F$) as a delooping of the space of homotopy equivalences of $F$. The latter fibers over $F$ with fiber the pointed self-equivalences; in the case $F=K(G,n)$ the space of pointed self-equivalences is equivalent to the discrete space $Aut(G)$.

Answer 3

The existence of the maps $g_i$ is only possible if $G_i$ is abelian, and so what you are missing is that each $G_i$ must be abelian.

It's been a while since I looked at that book, but it very well may be that their definition of an Eilenberg-MacLane space included the assumption that $G$ is abelian.