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
χ ( M , G ) = 1 | G | ∑ g 1 g 2 = g 2 g 1 χ ( 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 | 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 space of simultaneous fixed points of g 1 {\displaystyle g_{1}} and g 2 {\displaystyle g_{2}} . (The appearance of χ {\displaystyle \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 {\displaystyle M} divided by | G | {\displaystyle |G|} .
See also Kawasaki's Riemann–Roch formula
References
Further reading 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.
External links 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