This question was inspired by the question posed by John Baez here: https://mathoverflow.net/questions/67209?sort=votes#sort-top and Neil Strickland's answer to that question.

Let $X$ be a CW complex. If $X$ is finite, there are no problems defining its Euler characteristic whatsoever. However, when $X$ is infinite, there are at least two different ways which give similar but not identical results:

If the number $c(d)$ of cells of $X$ of dimension $d$ is finite for each $d$, one can form a generating function $f(t)=\sum_d c(d)t^d$ and define $\chi(X)=\lim_{t\to -1^+}f(t)$ (assuming the limit makes sense and exists).

If $X$ has a finite $d$-fold cover $\tilde X$ which is homotopy equivalent to a finite CW complex, then we can set $\chi(X)=\frac{1}{d}\sum_{i=1}^\infty (-1)^i\dim H^i(\tilde X,k)$ where $k$ is a field. This is the Wall characteristic (a.k.a. the rational Euler characteristic) of $X$ and it is a homotopy invariant and does not depend on $k$.

Now, if we try to calculate the characteristic of $\mathbb{R} P^\infty$ with its standard cell decomposition (one cell in each dimension from $0$ to $\infty$), then both definitions agree: the first one gives the generating function $\frac{1}{1-t}$, which gives $\frac{1}{2}$, when evaluated at $t=-1$; the second one also gives $\frac{1}{2}$ using the double cover $pt\cong S^\infty\to \mathbb{R} P^\infty$.

However, if we take a finite group $G$ and try to compute the characteristic of the Milnor construction of $BG$, then the first definition will give $\frac{1}{1+\# G}$, since the number of simplices of dimension $d$ is $(\# G)^d$, while the second one gives $\frac{1}{\# G}$.

I would like to ask: is there a conceptual reason the two definitions agree in the first case and almost (but not quite) agree in the second? This is, of course, a rather vague question, so here is a slightly more specific version: if the number of cells of $X$ of each dimension is finite, then is there a way to deduce the Wall characteristic of $X$ from the generating series for those numbers, assuming the Wall characteristic exists?

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