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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 {\displaystyle M} quotiented by a finite group G {\displaystyle G} , the Euler characteristic of M / G {\displaystyle M/G} is
where | G | {\displaystyle |G|} is the order of the group G {\displaystyle G} , the sum runs over all pairs of commuting elements of G {\displaystyle G} , and M g 1 , g 2 {\displaystyle M^{g_{1},g_{2}}} is the set of simultaneous fixed points of g 1 {\displaystyle g_{1}} and g 2 {\displaystyle 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 {\displaystyle M} divided by | G | {\displaystyle |G|}.