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 $|G|$.

See also

• Kawasaki's Riemann–Roch formula

References

• Dixon, L.; Harvey, J. A.; Vafa, C.; Witten, E. (1985). "Strings on orbifolds". Nuclear Physics B 261: 678–686. doi:10.1016/0550-3213(85)90593-0. http://theory.uchicago.edu/~harvey/pdf_files/orbiI.pdf. 
• 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. 
• Hirzebruch, Friedrich; Höfer, Thomas (1990). "On the Euler number of an orbifold". Mathematische Annalen 286 (1–3): 255–260. doi:10.1007/BF01453575. http://hirzebruch.mpim-bonn.mpg.de/135/1/78_On%20the%20Euler%20number%20of%20an%20orbifold.pdf. 
• Leinster, Tom (2008). "The Euler characteristic of a category". Documenta Mathematica 13: 21–49. http://emis.de/journals/DMJDMV/vol-13/02.pdf. 

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