Euler characteristic of an orbifold
# 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 set of simultaneous fixed points of $g_{1}$ and $g_{2}$. 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 $|G|$.

