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



What's the cell structure of K(Z/nZ, 1)?  Does it let me sum this divergent series?  What about other finite groups?
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.


