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 quotiented by a finite group, the Euler characteristic of is
where is the order of the group, the sum runs over all pairs of commuting elements of, and is the set of simultaneous fixed points of and. If the action is free, the sum has only a single term, and so this expression reduces to the topological Euler characteristic of divided by.
