The Eilenberg-Maclane space $K(\mathbb{Z}/2\mathbb{Z}, 1)$ has a particularly simple cell structure: it has exactly one cell of each dimension. This means that its "Euler characteristic" should be equal to
$$1 – 1 + 1 – 1 \pm …,$$
or Grandi's series. Now, we "know" (for example by analytic continuation) that this sum is morally equal to $\frac{1}{2}$. One way to see this is to think of $K(\mathbb{Z}/2\mathbb{Z}, 1)$ as infinite projective space, e.g. the quotient of the infinite sphere $S^{\infty}$ by antipodes. Since $S^{\infty}$ is contractible, the "orbifold Euler characteristic" of the quotient by the action of a group of order two should be $\frac{1}{2}$.
More generally, following John Baez $K(G, 1)$ for a finite group $G$ should be "the same" (I'm really unclear about what notion of sameness is being used here) as $G$ thought of as a one-object category, which has groupoid cardinality $\frac{1}{|G|}$. In particular, $K(\mathbb{Z}/n\mathbb{Z}, 1)$ should have groupoid cardinality $\frac{1}{n}$. I suspect that $K(\mathbb{Z}/n\mathbb{Z}, 1)$ has $1, n-1, (n-1)^2, …$ cells of each dimension, hence orbifold Euler characteristic
$$\frac{1}{n} = 1 – (n-1) + (n-1)^2 \mp ….$$
Unfortunately, I don't actually know how to construct Eilenberg-Maclane spaces…
Question 1a: How do I construct $K(\mathbb{Z}/n\mathbb{Z}, 1)$, and does it have the cell structure I think it has? (I've been told that one can write down the cell structure of $K(G, 1)$ for a finitely presented group $G$ explicitly, but I would really appreciate a reference for this construction.)
Question 2: $K(\mathbb{Z}/2\mathbb{Z}, 1)$ turns out to be "the same" as the set of all finite subsets of $(0, 1)$, suitably interpreted; the finite subsets of size $n$ form the cell of dimension $n$. Jim Propp and other people who think about combinatorial Euler characteristic would write this as $\chi(2^{(0, 1)}) = 2^{\chi(0, 1)}$. Is it true more generally that $K(\mathbb{Z}/n\mathbb{Z}, 1)$ is "the same" as the set of all functions $(0, 1) \to [n]$, suitably interpreted?
Question 3: What notion of "sameness" makes the above things I said actually true?
Question 4: Let $G$ be a finite group and let $K(G, 1)$ be constructed using the standard construction I asked about in Question 1. If $c_n$ denotes the number of cells of dimension $n$, let $f_G(z) = \sum_{n \ge 0} c_n z^n$. Can $f_G$ always be analytically continued to $z = -1$ so that $f_G(-1) = \frac{1}{|G|}$?
Best Answer
There are multiple possible cell structures on K(Z/n,1).
One is generic. For any finite group G there is a model for BG that has (|G|-1)k new simplices in each nonzero degree k. This is the standard simplicial bar construction of K(G,1). This gives you that BG has Euler characteristic 1/|G|, if you like.
One is more specific. There is another cell structure on K(Z/n,1), viewing it as a union of generalized lens spaces, that has exactly one cell in each degree. This is a topological avatar of the "simple" resolution of Z by free Z[Z/n]-modules. Obviously this doesn't give you the Euler characteristic argument you're seeking - one needs to keep track of more intricate information about the cell attachments in order to extract something.