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 ${\displaystyle M}$ quotiented by a finite group ${\displaystyle G}$, the Euler characteristic of ${\displaystyle M/G}$ is

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

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

See also

• Kawasaki's Riemann–Roch formula

References

• Dixon, L.; Harvey, J. A.; Vafa, C.; Witten, E. (1985). "Strings on orbifolds" (PDF). Nuclear Physics B. 261: 678–686. doi:10.1016/0550-3213(85)90593-0.
• Atiyah, Michael; Segal, Graeme (1989). "On equivariant Euler characteristics". Journal of Geometry and Physics. 6: 671–677. doi:10.1016/0393-0440(89)90032-6.
• Hirzebruch, Friedrich; Höfer, Thomas (1990). "On the Euler number of an orbifold" (PDF). Mathematische Annalen. 286: 255–260. doi:10.1007/BF01453575.
• 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