Let $G$ be the free group generated by two elements. Then, it is not hard to see that the classifying space of $G$ is given by gluing two circles at a point. A CW-decomposition of this space consists of one $0$-cell, and two $1$-cells.
Now consider the free Abelian group generated by two elements. In this case, the classifying space is given by the torus $\mathbb{T}^{2}$. A CW-decomposition of this space consists of one $0$-cell, two $1$-cells, and one $2$-cell.
This generalizes to $n$ elements. The classifying space for the free group is always given by glueing $n$ circles to a common point. The classifying space for the free abelian group is given by $\mathbb{T}^{n}$.
This leads me to the following question.
Given a topological space $X$, which is the classifying space of a discrete group $G$, is there a "geometric" interpretation of the classifying space of the Abelianization, $G/[G,G]$, of the group $G$?
Any restrictions that make the problem more manageable or easier to interpret are acceptable, as long as they do not render the problem trivial$^{1}$. I am a bit flexible about what I accept as a "geometric" interpretation$^{2}$, since I am not sure what would work.
I have an idea what a "geometric" interpretation would be in terms of CW-decompositions. It goes something along the lines of:
- Take a certain CW-decomposition of your space $X$.
- Add some cells.
This works well for the free group on $2$ elements. But it is not so clear to me how this works for the free group on $n$ elements. (If someone could explain this I would appreciate it.)
In absence of(/addition to?) a nice description of the procedure outlined above, I would be happy with a list of some of its properties, for example functoriality. Or; how does it behave under quotients? How about wedge/smash products?
(1) For example, "suppose that $G$ is Abelian", or "suppose that $G$ is the free group on $n$ elements", would be deemed trivial.
(2) It should be clear that I do not want the description, take the fundamental group of $X$, abelianize, and then take the classifying space.
Related is this list of classifying spaces.
There is a standard procedure in homotopy theory for constructing a classifying space for every group. One starts by constructing a 2-complex with the given fundamental group, and then one inductively attaches higher dimensional cells to kill all higher homotopy groups. At each stage, you simply attach zillions of cells, enough to kill everything that needs to die (sorry for the gruesome terminology, it seems to be standard).
A slight modification of this procedure gives a procedure for converting a classifying space of a group $G$ into a classifying space for $G/[G,G]$. Here's the details.
Start with $X=X_0$ which is a CW complex that is a classifying space for $G$.
Step 1. Each element of the commutator subgroup $c \in [G,G]$ is represented by some loop $\gamma_c$ in the 1-skeleton. Let $X_1$ be obtained from $X_0$ by attaching a 2-cell along the loop $\gamma_c$, for each $c \in [G,G]$. By Van Kampen's theorem one now has $\pi_1(X_1) \approx G/[G,G]$. But by attaching 2-cells, we may have created some $\pi_2$.
Step 2. Each element $c \in \pi_2(X_1)$ is represented by some continuous function $\gamma_c : S^2 \to X_1$ with image contained in the 2-skeleton. Let $X_2$ be obtained from $X_1$ by attaching a 3-cell along $\gamma_c$ for each $c \in \pi_2(X_1)$. One now has $\pi_1(X_2) \approx G/[G,G]$ and $\pi_2(X_2)$ is trivial. But by attaching 3-cells, we may have created some $\pi_3$.
...
Assume by induction that, by successively attaching cells of dimensions $2$, ..., $n$, we have CW complexes $X_1 \subset ... \subset X_{n-1}$ such that $\pi_1(X_{n-1}) \approx G/[G,G]$ and $\pi_2(X_{n-1}),...,\pi_{n-1}(X_{n-1})$ are trivial. By attaching $n$-cells we may have created some $\pi_n$.
Step n. Each element $c \in \pi_n(X_{n-1})$ is represented by some continuous map $\gamma_c : S^n \to X_{n-1}$ with image in the $n$-skeleton. Let $X_n$ be obtained from $X_{n-1}$ by attaching an $(n+1)$-cell along $\gamma_c$, for each $c \in \pi_n(X_{n-1})$.
...
By induction, we have a nested sequence of CW complexes $X_0 \subset X_1 \subset X_2 \subset \cdots$, and the direct limit of this sequence is the desired classifying space for $G/[G,G]$.
As a final comment, let me point out that this construction uses only that $\pi_1(X_0)=G$, it never uses that $X_0$ is a classifying space for $G$. If $X_0$ starts out possessing nontrivial higher homotopy groups, they will get killed at some stage of the procedure.