Orbifold

Sponsorship the way you would do it

In topology, an orbifold is a generalization of manifold. It is a topological space (called underlying space) with an orbifold structure (see below). The underlying space locally looks like a quotient of a Euclidean space under the action of a finite group of isometries.

The main example of underlying space is a quotient space of a manifold under the action of a finite group of diffeomorphisms, in particular manifold with boundary carries natural orbifold structure, since it is Z₂-factor of its double. A factor space of a manifold along smooth -action without fixed points cares structure of orbifold (this is not a partial case of the main example).

Orbifold structure gives a natural stratification by open manifolds on its underlying space, where one strata corresponds to a set of singular points of the same type.

It should be noted that one topological space can care many different orbifold structures. For example, consider the orbifold O associated with a factor space of 2-sphere along a rotation by , it is homeomorphic to 2-sphere, but the natural orbifold structure is different. It is possible to adopt most of the characteristics of manifolds to orbifolds and these characteristics are usually different from correspondent characteristics of underlying space. In the above example, its orbifold fundamental group of O is Z₂ and its orbifold Euler characteristic is 1.

Formal definition

The formal definition goes along the same lines as a definition of manifold, but instead of taking domains in Rⁿ as the target spaces of charts one should take domains of finite quations of Rⁿ.

A (topological) orbifold , is a Hausdorff topological space with countable base, called the underlying space, with an orbifold structure, which is defined by orbifold atlas (see below).

An orbifold chart is an open subset together with open set Rⁿ and a continuous map which satisfy the following property: there is a finite group acting linearly on and a homeomorphism such that , where denotes the projection .

A collection of orbifold charts is called orbifold atlas if it satisfy the following properties:

,
if then there is a neighborhood and and a homeomorphism such that .

The orbifold atlas defines the orbifold structure completely and we regard two orbifold atlases of to give the same orbifold structure if they can be combined to give a larger orbifold atlas.

One can add differentiability conditions on the gluing map in the above definition and get a definition of differentiable orbifolds on the same way as it was done for manifolds.

Hystory

The V-manifold of Ichiro Satake (1956) provided the first formal definition of what is now called orbifold. It was renamed this way and popularized by William Thurston.