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I am working on exercise 4.2.7 of Hatcher, which is to construct a CW complex $X$ with arbitrary homotopy groups and a prescribed action of the fundamental group on these homotopy groups (so making the higher homotopy groups each be a specified $\mathbb{Z}\pi_1(X)$-module), and am having trouble with constructing a specified action of the fundamental group on the higher homotopy groups.

Here is what I have tried in the special case that we want $\pi_1X\cong \mathbb{Z}$, and we want all except the first and $n^{th}$ groups to be trivial. If $Y=S^1 \vee_{\alpha} S^n_{\alpha}$, I know that $\pi_n(Y)$ is a rank $\alpha$ free $\pi_1(Y)$ module, with generators the inclusions $S^n \to S^1 \vee_{\alpha} S^n_{\alpha}$. So, we could get an arbitrary $\pi_1$-module structure on a space by attaching $n+1$ cells to $Y$ according to the relations we want between our generators. I think this construction works for $\pi_1$ being a free group of any rank, since we could take a wedge of $n$-spheres with a wedge of a bunch of $S^1$s.

So, my main questions are: how can I make a space with arbitrary fundamental group and a specified action of the fundamental group on the $n^{th}$ homotopy group, and how can I get a specifed $\pi_1$ action on different higher homotopy groups at the same time?

I believe this question is answered here on Mathoverflow, but I don't understand what it means to attach free orbits along the action, and I'm not familiar with the Borel construction.

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up vote 5 down vote accepted

One way to think about the action of $\pi_1(X)$ on the higher homotopy groups of $X$ is to think of it as being induced by the action of $\pi_1(X)$ on the universal cover $\widetilde{X}$, which has the same higher homotopy groups as $X$. We can try to reverse this argument, and to construct the desired space by first constructing its universal cover with the desired homotopy groups, then constructing the desired action of $\pi_1(X)$ on it, and finally quotienting by this action appropriately.

Constructing $\widetilde{X}$ is the easiest part: we can just take it to be a product $\prod_{n \ge 2} B^n \pi_n(X)$ of Eilenberg-MacLane spaces (where by $B^n A$ I mean $K(A, n)$). If you believe that the construction of Eilenberg-MacLane spaces is functorial then any desired action of $\pi_1(X)$ on each $\pi_n(X)$ induces an action on $B^n \pi_n(X)$ and hence we get the desired action of $\pi_1(X)$ on $\widetilde{X}$.

The tricky part now to make sure that quotienting $\widetilde{X}$ by $\pi_1(X)$ actually gives a covering map, so that the quotient $X$ actually has the correct homotopy groups. This is what the Borel construction is for; it's a distinguished way to modify $\widetilde{X}$ in a way that preserves both its (weak) homotopy type and the action of $\pi_1(X)$ on it, but so that the action of $\pi_1(X)$ is free.

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