Hello,

Suppose $M$ is a compact oriented smooth manifold and $G$ is a finite group acting on it. Then it is well-known, although I have yet to find a proof or derivation of it, that the (normal topological) Euler characteristic of the orbifold $M/G$ is $\chi(M/G) = \frac1{|G|}\sum_i\sum_{g\in G} (-1)^i\mathrm{Tr}_{H^i(M)}(g^*)$, where $H^i(M)$ is the $i$-th (De Rham) cohomology group and $g^*$ is the map on it induced by $g$.

Everywhere where I have looked, this is then said to equal $\frac1{|G|}\sum_{g\in G}\chi(M^g)$ (where $M^g$ is the set of points that $g$ leaves fixed) because of the Lefschetz formula. Not being familiar with this formula, I've looked up several versions of it. Especially the one for compact oriented manifolds seems very useful, but it (along with a number of other versions) holds only when all the fixed points are isolated. In particular, the set of fixed points should be countable. However, I have seen this formula being used used in situations in which this does not hold. For example, let the permuation group $S_n$ act on $M\times\dots\times M = M^n$ by permuting the factors. The set of fixed points of this action is certainly not countable.

So how does this work? Is this Lefschetz formula applicable to this situation after all, or is there another usable version of it that should be used here? Also, is there perhaps a book or arXiv document that shows how to calculate the first expression of the Euler characteristic?