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.^[1] Given a compact manifold ${\displaystyle M}$ quotiented by a finite group ${\displaystyle G}$, the Euler characteristic of ${\displaystyle M/G}$ is

${\displaystyle \chi (M,G)={\frac {1}{|G|}}\sum _{g_{1}g_{2}=g_{2}g_{1}}\chi (M^{g_{1},g_{2}}),}$

where ${\displaystyle |G|}$ is the order of the group ${\displaystyle G}$, the sum runs over all pairs of commuting elements of ${\displaystyle G}$, and ${\displaystyle M^{g_{1},g_{2}}}$ is the space of simultaneous fixed points of ${\displaystyle g_{1}}$ and ${\displaystyle g_{2}}$. (The appearance of ${\displaystyle \chi }$ in the summation is the usual Euler characteristic.)^[1][2] If the action is free, the sum has only a single term, and so this expression reduces to the topological Euler characteristic of ${\displaystyle M}$ divided by ${\displaystyle |G|}$.^[2]

See also

• Kawasaki's Riemann–Roch formula