In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space. Definitions of orbifold have been given several times: by Ichirō Satake in the context of automorphic forms in the 1950s under the name V-manifold; by William 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 André Haefliger in the 1980s in the context of Mikhail Gromov's programme on CAT(k) spaces under the name orbihedron. Historically, 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 S L ( 2 , Z ) {\displaystyle \mathrm {SL} (2,\mathbb {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 Herbert 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 two-dimensional conformal field theory, it refers 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 Z 2 {\displaystyle \mathbb {Z} _{2}} . One topological space can carry different orbifold structures. For example, consider the orbifold O {\displaystyle O} associated with a quotient space of the 2-sphere along a rotation by π {\displaystyle \pi } ; 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 {\displaystyle O} is Z 2 {\displaystyle \mathbb {Z} _{2}} and its orbifold Euler characteristic is 1.
Formal definitions
Definition using orbifold atlas Like a manifold, an orbifold is specified by local conditions; however, instead of being locally modelled on open subsets of R n {\displaystyle \mathbb {R} ^{n}} , an orbifold is locally modelled on quotients of open subsets of R n {\displaystyle \mathbb {R} ^{n}} by finite group actions. The structure of an orbifold encodes not only that of the underlying quotient space, which need not be a manifold, but also that of the isotropy subgroups. An n {\displaystyle n} -dimensional orbifold is a Hausdorff topological space X {\displaystyle X} , called the underlying space, with a covering by a collection of open sets U i {\displaystyle U_{i}} , closed under finite intersection. For each U i {\displaystyle U_{i}} , there is
an open subset V i {\displaystyle V_{i}} of R n {\displaystyle \mathbb {R} ^{n}} , invariant under a faithful linear action of a finite group Γ i {\displaystyle \Gamma _{i}} ; a continuous map φ i {\displaystyle \varphi _{i}} of V i {\displaystyle V_{i}} onto U i {\displaystyle U_{i}} invariant under Γ i {\displaystyle \Gamma _{i}} , called an orbifold chart, which defines a homeomorphism between V i / Γ i {\displaystyle V_{i}/\Gamma _{i}} and U i {\displaystyle U_{i}} . The collection of orbifold charts is called an orbifold atlas if the following properties are satisfied:
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