Constructing $pi_1$ actions on higher homotopy groups.

Question

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 $pi_1Xcong mathbb{Z}$, 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.

Qiaochu Yuan · Accepted Answer

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.

Ronnie Brown · Answer

To give a broader context, what Qiaochu is building includes a notion of $X= K((G,M),n)$ where $G$ is a group and $M$ is a $G$-module. Such construction can be seen as   a special case of a classifying space $B(C)$ where $C$ is a crossed complex: this has  structure of a groupoid $(C_1,C_0)$, a crossed module,  $(C_2, C_1)$, modules $(C_n,C_1), n>2$, and boundary maps $C_n to C_{n-1}. n>2$, all modelling fundamental groupoids and relative homotopy groups. With different terminology,  a simplicial definition of $B(C)$ when $C_0$ is a singleton was introduced by A.L. Blakers in 1948 (Annals of Math.), and  homotopical   applications of crossed complexes  were developed by J.H.C. Whitehead in his 1949 paper "Combinatorial Homotopy II". For more information, see,    the joint book Nonabelian Algebraic Topology, which also relates crossed complexes to chain complexes with operators, and $B(C)$ to the Dold-Kan construction.
December 19, 2020
In 1957, I overheard Henry Whitehead tellihg John Minor that early workers in homotopy theory were fascinated by these operations.
I  should also give a reference to the substantial generalisation of this  operation of the fundamental group to an operation of a groupoid of homotopies  which is given in Chapter 7 of the book "Topology and Groupoids" advertised  here;  this  operation has implications such as a gluing theorem for homotopy equivalences, which were in the 1968 edition; this  operation, and its dual, are called "transport" in tom Dieck's substantial  book on Algebraic Topology (EMS $2008$). P.R. Heath has several nice papers on "Groupoid operations and fibre homotopy equivalences" and related topics.

Constructing $pi_1$ actions on higher homotopy groups.

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