Orbifolds
Introduction
An orbifold is a concept in the field of mathematics, specifically within the domain of topology and geometry. It is a generalization of a manifold and shares several properties with manifolds, while also exhibiting unique characteristics that set it apart.
Definition
Formally, an orbifold is a topological space (or 'underlying space') with a groupoid of local homeomorphisms from open subsets of Euclidean spaces to the total space. This groupoid is required to be 'effective' in a certain sense, which means that the isotropy groups of the points in the underlying space are actually realized as local symmetries of the space.
Properties
Orbifolds exhibit a number of interesting properties that distinguish them from other topological spaces. These include:
- Local Euclidean Structure: Every point in an orbifold has a neighborhood that is homeomorphic to an open subset of Euclidean space divided by a finite group of isometries. This property is what gives orbifolds their unique structure and differentiates them from manifolds.
- Isotropy Groups: At each point in an orbifold, there is an associated 'isotropy group' or 'stabilizer group'. This is a finite group of local homeomorphisms of the orbifold that fix the point.
- Orbifold Cover: An orbifold has an associated 'orbifold cover', which is a manifold that 'covers' the orbifold in a certain sense. This cover is a useful tool for studying the structure and properties of the orbifold.
Examples
There are many examples of orbifolds that can be constructed in various ways. Some of the most common examples include:
- Circle Orbifolds: These are orbifolds that are homeomorphic to a circle. They can be constructed by taking a circle and identifying certain points.
- Torus Orbifolds: These are orbifolds that are homeomorphic to a torus. They can be constructed by taking a torus and identifying certain points.
- Sphere Orbifolds: These are orbifolds that are homeomorphic to a sphere. They can be constructed by taking a sphere and identifying certain points.
Applications
Orbifolds have a wide range of applications in various fields of mathematics and physics. Some of the most notable applications include:
- String Theory: In the field of string theory, orbifolds are used to model the compactified dimensions of space-time.
- Algebraic Geometry: In algebraic geometry, orbifolds are used to study the moduli spaces of algebraic curves.
- Differential Geometry: In differential geometry, orbifolds are used to study the geometry of spaces with singularities.