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.{{r|dhvw}} 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 space of simultaneous fixed points of $g_1$ and $g_2$ . (The appearance of $\chi$ 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 $M$ divided by $|G|$ .{{r|hirzebruch}}

References

{{reflist|refs=

{{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=678–686 |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 }}

{{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=255–260 |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}}

}}

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

Euler characteristic of an orbifold

See also

References

Further reading

External links