Euler characteristic of an orbifold explained
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
quotiented by a finite group
, the Euler characteristic of
is
where
is the order of the group
, the sum runs over all pairs of commuting elements of
, and
is the space of simultaneous fixed points of
and
. (The appearance of
in the summation is the usual Euler characteristic.) If the action is free, the sum has only a single term, and so this expression reduces to the topological Euler characteristic of
divided by
.
See also
Further reading
- Michael . Atiyah . Graeme . Segal . Michael Atiyah . Graeme Segal . On equivariant Euler characteristics . . 6 . 1989 . 4 . 671 - 677 . 10.1016/0393-0440(89)90032-6.
- Tom . Leinster . The Euler characteristic of a category . . 13 . 2008 . 21 - 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
