In differential geometry, the Euler characteristic of an orbifold, or orbifold Euler characteristic, is a generalization of the

topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological spac ...

that includes contributions coming from nontrivial automorphisms. In particular, unlike a topological Euler characteristic, it is not restricted to

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

values and is in general a

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...

. It is of interest in mathematical physics, specifically in string theory. Given a compact manifold

M

quotiented by a finite group

G

, the Euler characteristic of

M/G

is :

\chi(M,G) = \frac \sum_ \chi(M^),

where

, G,

is the order of the group

G

, the sum runs over all pairs of commuting elements of

G

, and

M^

is the set of simultaneous fixed points of

g_1

and

g_2

. 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,

References

* * * *

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 {{geometry-stub Differential geometry String theory

See also

References

External links