III.65 Orbifolds

If you take a QUOTIENT [I.3 §3.3] of the plane ² by a group of symmetries, then you may obtain a MANIFOLD [I.3 §6.9]. For instance, if the group consists of all translations by an integer vector, then two points (x, y) and (z, w) are equivalent if and only if z - x and w - y are both integers, and the quotient space is a torus. However, if you take instead the group of all rotations about the origin through a multiple of π/3, then every point apart from the origin is equivalent to exactly five others, while the origin is equivalent only to itself. The result in this case is not a manifold, because the exceptional behavior at the origin results in a singularity. However, it is a well-understood kind of singularity. An orbifold is, roughly speaking, just like a manifold, except that whereas manifolds are locally like ⁿ, orbifolds are locally like quotients of ⁿ by groups of symmetries, and can therefore have a few singularities. See ALGEBRAIC GEOMETRY [IV.4 §7] and also MIRROR SYMMETRY [IV.16 §7].