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 [math]\displaystyle{ M }[/math] quotiented by a finite group [math]\displaystyle{ G }[/math], the Euler characteristic of [math]\displaystyle{ M/G }[/math] is
- [math]\displaystyle{ \chi(M,G) = \frac{1}{|G|} \sum_{g_1 g_2 = g_2 g_1} \chi(M^{g_1, g_2}), }[/math]
where [math]\displaystyle{ |G| }[/math] is the order of the group [math]\displaystyle{ G }[/math], the sum runs over all pairs of commuting elements of [math]\displaystyle{ G }[/math], and [math]\displaystyle{ M^{g_1, g_2} }[/math] is the set of simultaneous fixed points of [math]\displaystyle{ g_1 }[/math] and [math]\displaystyle{ g_2 }[/math]. If the action is free, the sum has only a single term, and so this expression reduces to the topological Euler characteristic of [math]\displaystyle{ M }[/math] divided by [math]\displaystyle{ |G| }[/math].
See also
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
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