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 $f_G(-1) = \frac{1}{|G|}$?