Orbifold

An orbifold is a generalization of a manifold that allows for certain types of singularities. They were introduced by William Thurston in his 3-dimensional geometry and topology notes. Here is a technical definition:

• An orbifold is a topological space (denoted by $X$) with a chosen atlas, where each element of the atlas is equivalent to the quotient of an open subset of Euclidean space by the linear action of a finite group. In other words, locally around every point, an orbifold looks like the quotient of some Euclidean space by a group of isometries. These quotients are typically taken to be by finite groups, but more general group actions can be considered in the theory of V-manifolds or stacks.

Let's break that down a bit:

• An atlas is a collection of charts, where each chart is a homeomorphism from an open subset of the space to an open subset of a Euclidean space. For an orbifold, each chart is associated with a finite group that acts on the Euclidean space.

• A finite group is a group with a finite number of elements. The group action on the Euclidean space in the definition of an orbifold can include reflections, rotations, translations, and dilations.

• The quotient of an open subset of Euclidean space by the linear action of a finite group means that we identify points that are related by the group action. The result can have singularities where the group action has fixed points.

An important concept associated with an orbifold is the notion of an orbifold covering space, which is a generalization of the notion of a covering space in the theory of manifolds.

Orbifolds show up in various areas of mathematics and theoretical physics. They can be used to describe certain types of symmetries and have applications in string theory.

Please note that the full theory of orbifolds is quite extensive and goes well beyond this basic definition. There are various ways to define orbifolds more rigorously and abstractly, and there is a wealth of theory about their properties and classifications.