In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold. It is a topological space (called the underlying space) with an orbifold structure (see below).
The underlying space locally looks like the quotient space of a Euclidean space under the linear action of a finite group. Definitions of orbifold have been given several times: by Satake in the context of automorphic forms in the 1950s under the name V-manifold; by Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the name orbifold, after a vote by his students; and by Haefliger in the 1980s in the context of Gromov's programme on CAT(k) spaces under the name orbihedron. The definition of Thurston will be described here: it is the most widely used and is applicable in all cases.
Mathematically, orbifolds arose first as surfaces with singular points long before they were formally defined. One of the first classical examples arose in the theory of modular forms with the action of the modular group SL(2,Z) on the upper half-plane: a version of the Riemann–Roch theorem holds after the quotient is compactified by the addition of two orbifold cusp points. In 3-manifold theory, the theory of Seifert fiber spaces, initiated by Seifert, can be phrased in terms of 2-dimensional orbifolds. In geometric group theory, post-Gromov, discrete groups have been studied in terms of the local curvature properties of orbihedra and their covering spaces.
In string theory, the word "orbifold" has a slightly different meaning, discussed in detail below. In conformal field theory, a mathematical part of string theory, it is often used to refer to the theory attached to the fixed point subalgebra of a vertex algebra under the action of a finite group of automorphisms.
The main example of underlying space is a quotient space of a manifold under the properly discontinuous action of a possibly infinite group of diffeomorphisms with finite isotropy subgroups. In particular this applies to any action of a finite group; thus a manifold with boundary carries a natural orbifold structure, since it is the quotient of its double by an action of Z2. Similarly the quotient space of a manifold by a smooth proper action of S1 carries the structure of an orbifold.
Orbifold structure gives a natural stratification by open manifolds on its underlying space, where one stratum corresponds to a set of singular points of the same type.
It should be noted that one topological space can carry many different orbifold structures. For example, consider the orbifold O associated with a factor space of the 2-sphere along a rotation by ; it is homeomorphic to the 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, the orbifold fundamental group of O is Z2 and its orbifold Euler characteristic is 1.