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 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 .
One topological space can carry different orbifold structures. For example, consider the orbifold associated with a quotient 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 is 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 , an orbifold is locally modelled on quotients of open subsets of 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 -dimensional orbifold is a Hausdorff topological space , called the underlying space, with a covering by a collection of open sets , closed under finite intersection. For each , there is an open subset of , invariant under a faithful linear action of a finite group ; a continuous map of onto invariant under , called an orbifold chart, which defines a homeomorphism between and .
The collection of orbifold charts is called an orbifold atlas if the following properties are satisfied: for each inclusion there is an injective group homomorphism . for each inclusion there is a -equivariant homeomorphism , called a gluing map, of onto an open subset of . the gluing maps are compatible with the charts, i.e. . the gluing maps are unique up to composition with group elements, i.e. any other possible gluing map from to has the form for a unique .
As for atlases on manifolds, two orbifold atlases of are equivalent if they can be consistently combined to give a larger orbifold atlas. An orbifold structure is therefore an equivalence class of orbifold atlases.
Note that the orbifold structure determines the isotropy subgroup of any point of the orbifold up to isomorphism: it can be computed as the stabilizer of the point in any orbifold chart. If Ui Uj Uk, then there is a unique transition element gijk in Γk such that
gijk·ψik = ψjk·ψij
These transition elements satisfy
(Ad gijk)·fik = fjk·fij
as well as the cocycle relation (guaranteeing associativity)
fkm(gijk)·gikm = gijm·gjkm.
More generally, attached to an open covering of an orbifold by orbifold charts, there is the combinatorial data of a so-called complex of groups (see below).
Exactly as in the case of manifolds, differentiability conditions can be imposed on the gluing maps to give a definition of a differentiable orbifold. It will be a Riemannian orbifold if in addition there are invariant Riemannian metrics on the orbifold charts and the gluing maps are isometries.
Definition using Lie groupoids
Recall that a groupoid consists of a set of objects , a set of arrows , and structural maps including the source and the target maps and other maps allowing arrows to be composed and inverted. It is called a Lie groupoid if both and are smooth manifolds, all structural maps are smooth, and both the source and the target maps are submersions. The intersection of the source and the target fiber at a given point , i.e. the set , is the Lie group called the isotropy group of at . A Lie groupoid is called proper if the map is a proper map, and étale if both the source and the target maps are local diffeomorphisms.
An orbifold groupoid is given by one of the following equivalent definitions:
a proper étale Lie groupoid; a proper Lie groupoid whose isotropies are discrete spaces.
Since the isotropy groups of proper groupoids are automatically compact, the discreteness condition implies that the isotropies must be actually finite groups.
Orbifold groupoids play the same role as orbifold atlases in the definition above. Indeed, an orbifold structure on a Hausdorff topological space is defined as the Morita equivalence class of an orbifold groupoid together with a homeomorphism , where is the orbit space of the Lie groupoid (i.e. the quotient of by the equivalent relation when if there is a with and ). This definition shows that orbifolds are a particular kind of differentiable stack.
Relation between the two definitions Given an orbifold atlas on a space , one can build a pseudogroup made up by all diffeomorphisms between open sets of which preserve the transition functions . In turn, the space of germs of its elements is an orbifold groupoid. Moreover, since by definition of orbifold atlas each finite group acts faithfully on , the groupoid is automatically effective, i.e. the map is injective for every . Two different orbifold atlases give rise to the same orbifold structure if and only if their associated orbifold groupoids are Morita equivalent. Therefore, any orbifold structure according to the first definition (also called a classical orbifold) is a special kind of orbifold structure according to the second definition.
Conversely, given an orbifold groupoid , there is a canonical orbifold atlas over its orbit space, whose associated effective orbifold groupoid is Morita equivalent to . Since the orbit spaces of Morita equivalent groupoids are homeomorphic, an orbifold structure according to the second definition reduces an orbifold structure according to the first definition in the effective case.
Accordingly, while the notion of orbifold atlas is simpler and more commonly present in the literature, the notion of orbifold groupoid is particularly useful when discussing non-effective orbifolds and maps between orbifolds. For example, a map between orbifolds can be described by a homomorphism between groupoids, which carries more information than the underlying continuous map between the underlying topological spaces.
Examples Any manifold without boundary is trivially an orbifold, where each of the groups Γi is the trivial group. Equivalently, it corresponds to the Morita equivalence class of the unit groupoid. If N is a compact manifold with boundary, its double M can be formed by gluing together a copy of N and its mirror image along their common boundary. There is natural reflection action of Z2 on the manifold M fixing the common boundary; the quotient space can be identified with N, so that N has a natural orbifold structure. If M is a Riemannian n-manifold with a cocompact proper isometric action of a discrete group Γ, then the orbit space X = M/Γ has natural orbifold structure: for each x in X take a representative m in M and an open neighbourhood Vm of m invariant under the stabiliser Γm, identified equivariantly with a Γm-subset of TmM under the exponential map at m; finitely many neighbourhoods cover X and each of their finite intersections, if non-empty, is covered by an intersection of Γ-translates gm·Vm with corresponding group gm Γ gm−1. Orbifolds that arise in this way are called developable or good. A classical theorem of Henri Poincaré constructs Fuchsian groups as hyperbolic reflection groups generated by reflections in the edges of a geodesic triangle in the hyperbolic plane for the Poincaré metric. If the triangle has angles /ni for positive integers ni, the triangle is a fundamental domain and naturally a 2-dimensional orbifold. The corresponding group is an example of a hyperbolic triangle group. Poincaré also gave a 3-dimensional version of this result for Kleinian groups: in this case the Kleinian group Γ is generated by hyperbolic reflections and the orbifold is H3 / Γ. If M is a closed 2-manifold, new orbifold structures can be defined on M by removing finitely many disjoint closed discs from M and gluing back copies of discs D/ Γi where D is the closed unit disc and Γi is a finite cyclic group of rotations. This generalises Poincaré's construction.
Orbifold fundamental group There are several ways to define the orbifold fundamental group. More sophisticated approaches use orbifold covering spaces or classifying spaces of groupoids. The simplest approach (adopted by Haefliger and known also to Thurston) extends the usual notion of loop used in the standard definition of the fundamental group.
An orbifold path is a path in the underlying space provided with an explicit piecewise lift of path segments to orbifold charts and explicit group elements identifying paths in overlapping charts; if the underlying path is a loop, it is called an orbifold loop. Two orbifold paths are identified if they are related through multiplication by group elements in orbifold charts. The orbifold fundamental group is the group formed by homotopy classes of orbifold loops.
If the orbifold arises as the quotient of a simply connected manifold M by a proper rigid action of a discrete group Γ, the orbifold fundamental group can be identified with Γ. In general it is an extension of Γ by 1 M.
The orbifold is said to be developable or good if it arises as the quotient by a group action; otherwise it is called bad. A universal covering orbifold can be constructed for an orbifold by direct analogy with the construction of the universal covering space of a topological space, namely as the space of pairs consisting of points of the orbifold and homotopy classes of orbifold paths joining them to the basepoint. This space is naturally an orbifold.
Note that if an orbifold chart on a contractible open subset corresponds to a group Γ, then there is a natural local homomorphism of Γ into the orbifold fundamental group.
In fact the following conditions are equivalent: The orbifold is developable. The orbifold structure on the universal covering orbifold is trivial. The local homomorphisms are all injective for a covering by contractible open sets.
Orbifolds as diffeologies