View source for Euler characteristic of an orbifold

In [[differential geometry]], the '''Euler characteristic of an orbifold''', or '''orbifold Euler characteristic''', is a generalization of the [[topology|topological]] [[Euler characteristic]] that includes contributions coming from nontrivial [[automorphism]]s. 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]].{{r|dhvw}} Given a compact manifold <math>M</math> quotiented by a finite group <math>G</math>, the Euler characteristic of <math>M/G</math> is
:<math>\chi(M,G) = \frac{1}{|G|} \sum_{g_1 g_2 = g_2 g_1} \chi(M^{g_1, g_2}), </math>
where <math>|G|</math> is the order of the group <math>G</math>, the sum runs over all pairs of commuting elements of <math>G</math>, and <math>M^{g_1, g_2}</math> is the space of simultaneous fixed points of <math>g_1</math> and <math>g_2</math>. (The appearance of <math>\chi</math> in the summation is the usual Euler characteristic.){{r|dhvw|hirzebruch}} If the action is free, the sum has only a single term, and so this expression reduces to the topological Euler characteristic of <math>M</math> divided by <math>|G|</math>.{{r|hirzebruch}}

== See also ==
*[[Kawasaki's Riemann–Roch formula]]

== References ==
{{reflist|refs=

<ref name=dhvw>{{cite journal |first1=L. |last1=Dixon |first2=J. A. |last2=Harvey |first3=C. |last3=Vafa |first4=E. |last4=Witten |author-link2=Jeffrey A. Harvey |author-link3=Cumrun Vafa |author-link4=Ed Witten |title=Strings on orbifolds |journal=[[Nuclear Physics B]] |volume=261 |year=1985 |pages=678&ndash;686 |url=http://theory.uchicago.edu/~harvey/pdf_files/orbiI.pdf |doi=10.1016/0550-3213(85)90593-0 |access-date=2018-03-22 |archive-date=2017-08-12 |archive-url=https://web.archive.org/web/20170812121732/http://theory.uchicago.edu/~harvey/pdf_files/orbiI.pdf |url-status=dead }}</ref>

<ref name=hirzebruch>{{cite journal|first1=Friedrich |last1=Hirzebruch |first2=Thomas |last2=Höfer |title=On the Euler number of an orbifold |journal=[[Mathematische Annalen]] |year=1990 |volume=286 |issue=1–3 |pages=255&ndash;260 |url=http://hirzebruch.mpim-bonn.mpg.de/135/1/78_On%20the%20Euler%20number%20of%20an%20orbifold.pdf |doi=10.1007/BF01453575 |s2cid=121791965 |author-link1=Friedrich Hirzebruch}}</ref>

}}

==Further reading==
* {{cite journal|first1=Michael |last1=Atiyah |first2=Graeme |last2=Segal |author-link1=Michael Atiyah |author-link2=Graeme Segal |title=On equivariant Euler characteristics |journal=[[Journal of Geometry and Physics]] |volume=6 |year=1989 |issue=4 |pages=671&ndash;677 |doi=10.1016/0393-0440(89)90032-6}}
* {{cite journal|first=Tom |last=Leinster |title=The Euler characteristic of a category |url=http://emis.de/journals/DMJDMV/vol-13/02.pdf |journal=[[Documenta Mathematica]] |volume=13 |year=2008 |pages=21&ndash;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

[[Category:Differential geometry]]
[[Category:String theory]]


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