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.Template:R 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.)Template:R 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 $|G|$.Template:R

See also

• Kawasaki's Riemann–Roch formula

Further reading

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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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