Euler characteristic of an orbifold

In differential geometry, the Euler characteristic of an orbifold, or orbifold Euler characteristic, is a generalization of the topological Euler characteristic that includes contributions coming from nontrivial automorphisms. In particular, unlike a topological Euler characteristic, it is not restricted to integer values and is in general a rational number. It is of interest in mathematical physics, specifically in string theory. Given a compact manifold $$M$$ quotiented by a finite group $$G$$, the Euler characteristic of $$M/G$$ is
 * $$\chi(M,G) = \frac{1}{|G|} \sum_{g_1 g_2 = g_2 g_1} \chi(M^{g_1, g_2}), $$

where $$|G|$$ is the order of the group $$G$$, the sum runs over all pairs of commuting elements of $$G$$, and $$M^{g_1, g_2}$$ is the space of simultaneous fixed points of $$g_1$$ and $$g_2$$. (The appearance of $$\chi$$ in the summation is the usual Euler characteristic.) If the action is free, the sum has only a single term, and so this expression reduces to the topological Euler characteristic of $$M$$ divided by $$|G|$$.