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 $|G|$ .

References

Dixon, L.; Harvey, J. A.; Vafa, C.; Witten, E. (1985). "Strings on orbifolds" (PDF). Nuclear Physics B. 261: 678&ndash, 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&ndash, 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&ndash, 260. doi:10.1007/BF01453575.
Leinster, Tom (2008). "The Euler characteristic of a category" (PDF). Documenta Mathematica. 13: 21&ndash, 49.

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Euler characteristic of an orbifold

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References

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