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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'';〔Satake (1956).〕 by Thurston in the context of the geometry of 3-manifolds in the 1970s〔Thurston (1978), Chapter 13.〕 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''.〔Haefliger (1990).〕 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.〔Poincaré (1985).〕 One of the first classical examples arose in the theory of modular forms〔Serre (1970).〕 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.〔Scott (1983).〕 In geometric group theory, post-Gromov, discrete groups have been studied in terms of the local curvature properties of orbihedra and their covering spaces.〔Bridson and Haefliger (1999).〕 In string theory, the word "orbifold" has a slightly different meaning,〔Di Francesco, Mathieu & Sénéchal (1997)〕 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.〔Bredon (1972).〕 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 ''S''1 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. ==Formal definitions== Like a manifold, an orbifold is specified by local conditions; however, instead of being locally modelled on open subsets of R''n'', an orbifold is locally modelled on quotients of open subsets of 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''-dimensional orbifold is a faithful linear action of a finite group Γ''i'' * a continuous map φ''i'' of ''V''''i'' onto ''U''''i'' invariant under Γ''i'', called an orbifold chart, which defines a homeomorphism between ''V''''i'' / Γ''i'' and ''U''''i''. The collection of orbifold charts is called an orbifold atlas if the following properties are satisfied: * for each inclusion ''U''''i'' ''U''''j'' there is an injective group homomorphism ''f''''ij'' : Γ''i'' Γ''j'' * for each inclusion ''U''''i'' ''U''''j'' there is a Γ''i''-equivariant homeomorphism ψ''ij'', called a gluing map, of ''V''''i'' onto an open subset of ''V''''j'' * the gluing maps are compatible with the charts, i.e. φ''j''·ψ''ij'' = φ''i'' * the gluing maps are unique up to composition with group elements, i.e. any other possible gluing map from ''V''''i'' to ''V''''j'' has the form ''g''·ψ''ij'' for a unique ''g'' in Γ''j'' The orbifold atlas defines the orbifold structure completely: two orbifold atlases of ''X'' give the same orbifold structure if they can be consistently combined to give a larger orbifold atlas. 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 ''U''''i'' ''U''''j'' ''U''''k'', then there is a unique ''transition element'' ''g''ijk in Γ''k'' such that :''g''''ijk''·ψ''ik'' = ψ''jk''·ψ''ij'' These transition elements satisfy :(Ad ''g''''ijk'')·''f''''ik'' = ''f''''jk''·''f''''ij'' as well as the ''cocycle relation'' (guaranteeing associativity) :f''km''(''g''''ijk'')·''g''''ikm'' = ''g''''ijm''·''g''''jkm''. 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 Riemannian manifold 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Orbifold」の詳細全文を読む スポンサード リンク
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