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.^[1] Given a compact manifold ${\displaystyle M}${\displaystyle M} quotiented by a finite group ${\displaystyle G}${\displaystyle G}, the Euler characteristic of ${\displaystyle M/G}${\displaystyle M/G} is

${\displaystyle \chi (M,G)={\frac {1}{|G|}}\sum _{g_{1}g_{2}=g_{2}g_{1}}\chi (M^{g_{1},g_{2}}),}${\displaystyle \chi (M,G)={\frac {1}{|G|}}\sum _{g_{1}g_{2}=g_{2}g_{1}}\chi (M^{g_{1},g_{2}}),}

where ${\displaystyle |G|}${\displaystyle |G|} is the order of the group ${\displaystyle G}${\displaystyle G}, the sum runs over all pairs of commuting elements of ${\displaystyle G}${\displaystyle G}, and ${\displaystyle M^{g_{1},g_{2}}}${\displaystyle M^{g_{1},g_{2}}} is the space of simultaneous fixed points of ${\displaystyle g_{1}}${\displaystyle g_{1}} and ${\displaystyle g_{2}}${\displaystyle g_{2}}. (The appearance of ${\displaystyle \chi }${\displaystyle \chi } in the summation is the usual Euler characteristic.)^{[1] [2]} If the action is free, the sum has only a single term, and so this expression reduces to the topological Euler characteristic of ${\displaystyle M}${\displaystyle M} divided by ${\displaystyle |G|}${\displaystyle |G|}.^[2]

• Kawasaki's Riemann–Roch formula

• Atiyah, Michael; Segal, Graeme (1989). "On equivariant Euler characteristics". Journal of Geometry and Physics . 6 (4): 671–677. doi:10.1016/0393-0440(89)90032-6.
• Leinster, Tom (2008). "The Euler characteristic of a category" (PDF). Documenta Mathematica . 13: 21–49.

• https://mathoverflow.net/questions/51993/euler-characteristic-of-orbifolds
• https://mathoverflow.net/questions/267055/is-every-rational-realized-as-the-euler-characteristic-of-some-manifold-or-orbif