Constructing $\pi_1$ actions on higher homotopy groups.

Solution 1:

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.

Solution 2:

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.