In the mathematical disciplines oftopology andgeometry, anorbifold (for "orbit-manifold") is a generalization of amanifold. Roughly speaking, an orbifold is atopological space that is locally afinite group quotient of aEuclidean space.

Definitions of orbifold have been given several times: byIchirō Satake in the context ofautomorphic forms in the 1950s under the nameV-manifold;^[1] byWilliam Thurston in the context of the geometry of3-manifolds in the 1970s^[2] when he coined the nameorbifold, after a vote by his students; and byAndré Haefliger in the 1980s in the context ofMikhail Gromov's programme onCAT(k) spaces under the nameorbihedron.^[3]

Historically, orbifolds arose first as surfaces withsingular points long before they were formally defined.^[4] One of the first classical examples arose in the theory ofmodular forms^[5] with the action of themodular group${\displaystyle \mathrm{S}\mathrm{L}(2,\mathbb{Z})}$ on theupper half-plane: a version of theRiemann–Roch theorem holds after the quotient is compactified by the addition of two orbifold cusp points. In3-manifold theory, the theory ofSeifert fiber spaces, initiated byHerbert Seifert, can be phrased in terms of 2-dimensional orbifolds.^[6] Ingeometric group theory, post-Gromov, discrete groups have been studied in terms of the local curvature properties of orbihedra and their covering spaces.^[7]

Instring theory, the word "orbifold" has a slightly different meaning,^[8] discussed in detail below. Intwo-dimensional conformal field theory, it refers to the theory attached to the fixed point subalgebra of avertex algebra under the action of a finite group ofautomorphisms.

The main example of underlying space is a quotient space of a manifold under theproperly discontinuous action of a possibly infinitegroup ofdiffeomorphisms with finiteisotropy subgroups.^[9] In particular this applies to any action of afinite group; thus amanifold with boundary carries a natural orbifold structure, since it is the quotient of itsdouble by an action of${\displaystyle {\mathbb{Z}}_2}$.

One topological space can carry different orbifold structures. For example, consider the orbifold${\displaystyle O}$ associated with a quotient space of the 2-sphere along a rotation by${\displaystyle \pi }$ ; it ishomeomorphic 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, theorbifoldfundamental group of${\displaystyle O}$ is${\displaystyle {\mathbb{Z}}_2}$ and itsorbifoldEuler characteristic is 1.

Like a manifold, an orbifold is specified by local conditions; however, instead of being locally modelled onopen subsets of${\displaystyle {\mathbb{R}}^n}$, an orbifold is locally modelled on quotients of open subsets of${\displaystyle {\mathbb{R}}^n}$ 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 theisotropy subgroups.

An${\displaystyle n}$-dimensionalorbifold is aHausdorff topological space${\displaystyle X}$, called theunderlying space, with a covering by a collection of open sets${\displaystyle U_i}$, closed under finite intersection. For each${\displaystyle U_i}$, there is

• an open subset${\displaystyle V_i}$ of${\displaystyle {\mathbb{R}}^n}$, invariant under afaithful linear action of a finite group${\displaystyle {\mathrm{\Gamma }}_i}$;
• a continuous map${\displaystyle {\phi }_i}$ of${\displaystyle V_i}$ onto${\displaystyle U_i}$ invariant under${\displaystyle {\mathrm{\Gamma }}_i}$, called anorbifold chart, which defines a homeomorphism between${\displaystyle V_i/{\mathrm{\Gamma }}_i}$ and${\displaystyle U_i}$.

The collection of orbifold charts is called anorbifold atlas if the following properties are satisfied:

• for each inclusion${\displaystyle U_i\subset U_j}$ there is aninjectivegroup homomorphism${\displaystyle f_{ij}:{\mathrm{\Gamma }}_i\to {\mathrm{\Gamma }}_j}$.
• for each inclusion${\displaystyle U_i\subset U_j}$ there is a${\displaystyle {\mathrm{\Gamma }}_i}$-equivariant homeomorphism${\displaystyle {\psi }_{ij}}$, called agluing map, of${\displaystyle V_i}$ onto an open subset of${\displaystyle V_j}$.
• the gluing maps are compatible with the charts, i.e.${\displaystyle {\phi }_j\circ {\psi }_{ij}={\phi }_i}$.
• the gluing maps are unique up to composition with group elements, i.e. any other possible gluing map from${\displaystyle V_i}$ to${\displaystyle V_j}$ has the form${\displaystyle g\circ {\psi }_{ij}}$ for a unique${\displaystyle g\in {\mathrm{\Gamma }}_j}$.

As foratlases on manifolds, two orbifold atlases of${\displaystyle X}$ are equivalent if they can be consistently combined to give a larger orbifold atlas. Anorbifold 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. IfU_i${\displaystyle \subset }$U_j${\displaystyle \subset }$U_k, then there is a uniquetransition elementg_ijk in Γ_k such that

g_ijk·ψ_ik =ψ_jk·ψ_ij

These transition elements satisfy

(Adg_ijk)·f_ik =f_jk·f_ij

as well as thecocycle 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-calledcomplex 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 adifferentiable orbifold. It will be aRiemannian orbifold if in addition there are invariantRiemannian metrics on the orbifold charts and the gluing maps areisometries.

Recall that agroupoid consists of a set of objects${\displaystyle G_0}$, a set of arrows${\displaystyle G_1}$, and structural maps including the source and the target maps${\displaystyle s,t:G_1\to G_0}$ and other maps allowing arrows to be composed and inverted. It is called aLie groupoid if both${\displaystyle G_0}$ and${\displaystyle G_1}$ 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${\displaystyle x\in G_0}$, i.e. the set${\displaystyle (G_1{)}_x:=s^{-1}(x)\cap t^{-1}(x)}$, is theLie group called theisotropy group of${\displaystyle G_1}$ at${\displaystyle x}$. A Lie groupoid is calledproper if the map${\displaystyle (s,t):G_1\to G_0\times G_0}$ is aproper map, andétale if both the source and the target maps arelocal diffeomorphisms.

Anorbifold groupoid is given by one of the following equivalent definitions:

• a proper étale Lie groupoid;
• a proper Lie groupoid whose isotropies arediscrete spaces.

Since the isotropy groups of proper groupoids are automaticallycompact, the discreteness condition implies that the isotropies must be actuallyfinite groups.^[10]

Orbifold groupoids play the same role as orbifold atlases in the definition above. Indeed, anorbifold structure on a Hausdorff topological space${\displaystyle X}$ is defined as theMorita equivalence class of an orbifold groupoid${\displaystyle G\rightrightarrows M}$ together with a homeomorphism${\displaystyle |M/G|\simeq X}$, where${\displaystyle |M/G|}$ is the orbit space of the Lie groupoid${\displaystyle G}$ (i.e. the quotient of${\displaystyle M}$ by the equivalent relation when${\displaystyle x\sim y}$ if there is a${\displaystyle g\in G}$ with${\displaystyle s(g)=x}$ and${\displaystyle t(g)=y}$). This definition shows that orbifolds are a particular kind ofdifferentiable stack.

Given an orbifold atlas on a space${\displaystyle X}$, one can build apseudogroup made up by all diffeomorphisms between open sets of${\displaystyle X}$ which preserve the transition functions${\displaystyle {\phi }_i}$. In turn, the space${\displaystyle G_X}$ of germs of its elements is an orbifold groupoid. Moreover, since by definition of orbifold atlas each finite group${\displaystyle {\mathrm{\Gamma }}_i}$ acts faithfully on${\displaystyle V_i}$, the groupoid${\displaystyle G_X}$ is automatically effective, i.e. the map${\displaystyle g\in (G_X{)}_x\mapsto {\mathrm{g}\mathrm{e}\mathrm{r}\mathrm{m}}_x(t\circ s^{-1})}$ is injective for every${\displaystyle x\in X}$. 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 aclassical orbifold) is a special kind of orbifold structure according to the second definition.

Conversely, given an orbifold groupoid${\displaystyle G\rightrightarrows M}$, there is a canonical orbifold atlas over its orbit space, whose associated effective orbifold groupoid is Morita equivalent to${\displaystyle G}$. 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.^[11]

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.

• Any manifold without boundary is trivially an orbifold, where each of the groups Γ_i is thetrivial group. Equivalently, it corresponds to the Morita equivalence class of the unit groupoid.
• IfN is a compactmanifold with boundary, itsdoubleM can be formed by gluing together a copy ofN and its mirror image along their common boundary. There is naturalreflection action ofZ₂ on the manifoldM fixing the common boundary; the quotient space can be identified withN, so thatN has a natural orbifold structure.
• IfM is a Riemanniann-manifold with acocompactproper isometric action of a discrete group Γ, then the orbit spaceX =M/Γ has natural orbifold structure: for eachx inX take a representativem inM and an open neighbourhoodV_m ofm invariant under the stabiliser Γ_m, identified equivariantly with a Γ_m-subset ofT_mM under the exponential map atm; finitely many neighbourhoods coverX and each of their finite intersections, if non-empty, is covered by an intersection of Γ-translatesg_m·V_m with corresponding groupg_m Γg_m⁻¹. Orbifolds that arise in this way are calleddevelopable orgood.
• A classical theorem ofHenri Poincaré constructsFuchsian groups as hyperbolicreflection groups generated by reflections in the edges of ageodesic triangle in thehyperbolic plane for thePoincaré metric. If the triangle has anglesπ/n_i for positive integersn_i, the triangle is afundamental domain and naturally a 2-dimensional orbifold. The corresponding group is an example of a hyperbolictriangle group. Poincaré also gave a 3-dimensional version of this result forKleinian groups: in this case the Kleinian group Γ is generated by hyperbolic reflections and the orbifold isH³ / Γ.
• IfM is a closed 2-manifold, new orbifold structures can be defined onM by removing finitely many disjoint closed discs fromM and gluing back copies of discsD/ Γ_i whereD is the closedunit disc and Γ_i is a finite cyclic group of rotations. This generalises Poincaré's construction.

There are several ways to define theorbifold fundamental group. More sophisticated approaches use orbifoldcovering spaces orclassifying spaces ofgroupoids. The simplest approach (adopted by Haefliger and known also to Thurston) extends the usual notion ofloop used in the standard definition of thefundamental group.

Anorbifold 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 anorbifold 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 byhomotopy classes of orbifold loops.

If the orbifold arises as the quotient of asimply connected manifoldM by a proper rigid action of a discrete group Γ, the orbifold fundamental group can be identified with Γ. In general it is anextension of Γ byπ₁M.

The orbifold is said to bedevelopable orgood if it arises as the quotient by a group action; otherwise it is calledbad. Auniversal covering orbifold can be constructed for an orbifold by direct analogy with the construction of theuniversal 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 acontractible open subset corresponds to a group Γ, then there is a naturallocal 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 can be defined in the general framework ofdiffeology^[12] and have been proved to be equivalent^[13] toIchirô Satake's original definition:^[1]

Definition: An orbifold is a diffeological space locally diffeomorphic at each point to some${\displaystyle {\mathbb{R}}^n/G}$, where${\displaystyle n}$ is an integer and${\displaystyle G}$ is a finite linear group which may change from point to point.

This definition calls a few remarks:

• This definition mimics the definition of a manifold in diffeology, which is a diffeological space locally diffeomorphic at each point to${\displaystyle {\mathbb{R}}^n}$.
• An orbifold is regarded first as a diffeological space, a set equipped with a diffeology. Then, the diffeology is tested to be locally diffeomorphic at each point to a quotient${\displaystyle {\mathbb{R}}^n/G}$ with${\displaystyle G}$ a finite linear group.
• This definition is equivalent^[14] with Haefliger orbifolds.^[15]
• {Orbifolds} makes a subcategory of the category {Diffeology} whose objects are diffeological spaces and morphisms smooth maps. A smooth map between orbifolds is any map which is smooth for their diffeologies. This resolves, in the context of Satake's definition, his remark:^[16] "The notion of${\displaystyle C^{\mathrm{\infty }}}$-map thus defined is inconvenient in the point that a composite of two${\displaystyle C^{\mathrm{\infty }}}$-map defined in a different choice of defining families is not always a${\displaystyle C^{\mathrm{\infty }}}$-map." Indeed, there are smooth maps between orbifolds that do not lift locally as equivariant maps.^[17]

Note that the fundamental group of an orbifold as a diffeological space is not the same as the fundamental group as defined above. That last one is related to the structure groupoid^[18] and its isotropy groups.

For applications ingeometric group theory, it is often convenient to have a slightly more general notion of orbifold, due to Haefliger. Anorbispace is to topological spaces what an orbifold is to manifolds. An orbispace is a topological generalization of the orbifold concept. It is defined by replacing the model for the orbifold charts by alocally compact space with arigid action of a finite group, i.e. one for which points with trivial isotropy are dense. (This condition is automatically satisfied by faithful linear actions, because the points fixed by any non-trivial group element form a properlinear subspace.) It is also useful to considermetric space structures on an orbispace, given by invariantmetrics on the orbispace charts for which the gluing maps preserve distance. In this case each orbispace chart is usually required to be alength space with uniquegeodesics connecting any two points.

LetX be an orbispace endowed with a metric space structure for which the charts are geodesic length spaces. The preceding definitions and results for orbifolds can be generalized to give definitions oforbispace fundamental group anduniversal covering orbispace, with analogous criteria for developability. The distance functions on the orbispace charts can be used to define the length of an orbispace path in the universal covering orbispace. If the distance function in each chart isnon-positively curved, then theBirkhoff curve shortening argument can be used to prove that any orbispace path with fixed endpoints is homotopic to a unique geodesic. Applying this to constant paths in an orbispace chart, it follows that each local homomorphism is injective and hence:

• every non-positively curved orbispace is developable (i.e.good).

Every orbifold has associated with it an additional combinatorial structure given by acomplex of groups.

Acomplex of groups (Y,f,g) on anabstract simplicial complexY is given by

• a finite group Γ_σ for each simplex σ ofY
• an injective homomorphismf_στ : Γ_τ${\displaystyle \to }$ Γ_σ whenever σ${\displaystyle \subset }$ τ
• for every inclusion ρ${\displaystyle \subset }$ σ${\displaystyle \subset }$ τ, a group elementg_ρστ in Γ_ρ such that (Adg_ρστ)·f_ρτ =f_ρσ·f_στ (here Ad denotes theadjoint action by conjugation)

The group elements must in addition satisfy the cocycle condition

f_πρ(g_ρστ)g_πρτ =g_πστg_πρσ

for every chain of simplices${\displaystyle \pi \subset \rho \subset \sigma \subset \tau .}$ (This condition is vacuous ifY has dimension 2 or less.)

Any choice of elementsh_στ in Γ_σ yields anequivalent complex of groups by defining

• f'_στ = (Adh_στ)·f_στ
• g'_ρστ =h_ρσ·f_ρσ(h_στ)·g_ρστ·h_ρτ⁻¹

A complex of groups is calledsimple wheneverg_ρστ = 1 everywhere.

• An easy inductive argument shows that every complex of groups on asimplex is equivalent to a complex of groups withg_ρστ = 1 everywhere.

It is often more convenient and conceptually appealing to pass to thebarycentric subdivision ofY. The vertices of this subdivision correspond to the simplices ofY, so that each vertex has a group attached to it. The edges of the barycentric subdivision are naturally oriented (corresponding to inclusions of simplices) and each directed edge gives an inclusion of groups. Each triangle has a transition element attached to it belonging to the group of exactly one vertex; and the tetrahedra, if there are any, give cocycle relations for the transition elements. Thus a complex of groups involves only the 3-skeleton of the barycentric subdivision; and only the 2-skeleton if it is simple.

IfX is an orbifold (or orbispace), choose a covering by open subsets from amongst the orbifold chartsf_i:V_i${\displaystyle \to }$U_i. LetY be the abstract simplicial complex given by thenerve of the covering: its vertices are the sets of the cover and itsn-simplices correspond tonon-empty intersectionsU_α =U_i1${\displaystyle \cap }$ ···${\displaystyle \cap }$U_in. For each such simplex there is an associated group Γ_α and the homomorphismsf_ij become the homomorphismsf_στ. For every triple ρ${\displaystyle \subset }$ σ${\displaystyle \subset }$ τ corresponding to intersections

${\displaystyle U_i\supset U_i\cap U_j\supset U_i\cap U_j\cap U_k}$

there are chartsφ_i :V_i${\displaystyle \to }$U_i,φ_ij :V_ij${\displaystyle \to }$U_i${\displaystyle \cap }$U_j and φ_ijk :V_ijk${\displaystyle \to }$U_i${\displaystyle \cap }$U_j${\displaystyle \cap }$U_k and gluing maps ψ :V_ij${\displaystyle \to }$V_i, ψ' :V_ijk${\displaystyle \to }$V_ij and ψ" :V_ijk${\displaystyle \to }$V_i.

There is a unique transition elementg_ρστ in Γ_i such thatg_ρστ·ψ" =ψ·ψ′. The relations satisfied by the transition elements of an orbifold imply those required for a complex of groups. In this way a complex of groups can be canonically associated to the nerve of an open covering by orbifold (or orbispace) charts. In the language of non-commutativesheaf theory andgerbes, the complex of groups in this case arises as asheaf of groups associated to the coveringU_i; the datag_ρστ is a 2-cocycle in non-commutativesheaf cohomology and the datah_στ gives a 2-coboundary perturbation.

Theedge-path group of a complex of groups can be defined as a natural generalisation of theedge path group of a simplicial complex. In the barycentric subdivision ofY, take generatorse_ij corresponding to edges fromi toj wherei${\displaystyle \to }$j, so that there is an injection ψ_ij : Γ_i${\displaystyle \to }$ Γ_j. Let Γ be the group generated by thee_ij and Γ_k with relations

e_ij⁻¹ ·g ·e_ij = ψ_ij(g)

forg in Γ_i and

e_ik =e_jk·e_ij·g_ijk

ifi${\displaystyle \to }$j${\displaystyle \to }$k.

For a fixed vertexi₀, the edge-path group Γ(i₀) is defined to be the subgroup of Γ generated by all products

g₀ · e_i0i1 ·g₁ · e_i1i2 · ··· ·g_n · e_{ini 0}

wherei₀,i₁, ...,i_n,i₀is an edge-path,g_k lies in Γ_ik ande_ji=e_ij⁻¹ ifi${\displaystyle \to }$j.

A simplicialproper action of a discrete group Γ on asimplicial complexX with finite quotient is said to beregular if itsatisfies one of the following equivalent conditions:^[9]

• X admits a finite subcomplex asfundamental domain;
• the quotientY =X/Γ has a natural simplicial structure;
• the quotient simplicial structure on orbit-representatives of vertices is consistent;
• if (v₀, ...,v_k) and (g₀·v₀, ...,g_k·v_k) are simplices, theng·v_i =g_i·v_i for someg in Γ.

The fundamental domain and quotientY =X / Γ can naturally be identified as simplicial complexes in this case, given by the stabilisers of the simplices in the fundamental domain. A complex of groupsY is said to bedevelopable if it arises in this way.

• A complex of groups is developable if and only if the homomorphisms of Γ_σ into the edge-path group are injective.
• A complex of groups is developable if and only if for each simplex σ there is an injective homomorphism θ_σ from Γ_σ into a fixed discrete group Γ such that θ_τ·f_στ = θ_σ. In this case the simplicial complexX is canonically defined: it hask-simplices (σ, xΓ_σ) where σ is ak-simplex ofY andx runs over Γ / Γ_σ. Consistency can be checked using the fact that the restriction of the complex of groups to asimplex is equivalent to one with trivial cocycleg_ρστ.

The action of Γ on the barycentric subdivisionX ' ofX always satisfies the following condition, weaker than regularity:

• whenever σ andg·σ are subsimplices of some simplex τ, they are equal, i.e. σ =g·σ

Indeed, simplices inX ' correspond to chains of simplices inX, so that a subsimplices, given by subchains of simplices, is uniquely determined by thesizes of the simplices in the subchain. When an action satisfies this condition, theng necessarily fixes all the vertices of σ. A straightforward inductive argument shows that such an action becomes regular on the barycentric subdivision; in particular

• the action on the second barycentric subdivisionX" is regular;
• Γ is naturally isomorphic to the edge-path group defined using edge-paths and vertex stabilisers for the barycentric subdivision of the fundamental domain inX".

There is in fact no need to pass to athird barycentric subdivision: as Haefliger observes using the language ofcategory theory, in this case the 3-skeleton of the fundamental domain ofX" already carries all the necessary data – including transition elements for triangles – to define an edge-path group isomorphic to Γ.

In two dimensions this is particularly simple to describe. The fundamental domain ofX" has the same structure as the barycentric subdivisionY ' of a complex of groupsY, namely:

• a finite 2-dimensional simplicial complexZ;
• an orientation for all edgesi${\displaystyle \to }$j;
• ifi${\displaystyle \to }$j andj${\displaystyle \to }$k are edges, theni${\displaystyle \to }$k is an edge and (i,j,k) is a triangle;
• finite groups attached to vertices, inclusions to edges and transition elements, describing compatibility, to triangles.

An edge-path group can then be defined. A similar structure is inherited by the barycentric subdivisionZ ' and its edge-path group is isomorphic to that ofZ.

If a countable discrete group acts by aregularsimplicialproper action on asimplicial complex, the quotient can be given not only the structure of a complex of groups, but also that of an orbispace. This leads more generally to the definition of "orbihedron", the simplicial analogue of an orbifold.

LetX be a finite simplicial complex with barycentric subdivisionX '. Anorbihedron structure consists of:

• for each vertexi ofX ', a simplicial complexL_i' endowed with a rigid simplicial action of a finite group Γ_i.
• a simplicial map φ_i ofL_i' onto thelinkL_i ofi inX ', identifying the quotientL_i' / Γ_i withL_i.

This action of Γ_i onL_i' extends to a simplicial action on the simplicial coneC_i overL_i' (the simplicial join ofi andL_i'), fixing the centrei of the cone. The map φ_i extends to a simplicial map ofC_i onto thestar St(i) ofi, carrying the centre ontoi; thus φ_i identifiesC_i / Γ_i, the quotient of the star ofi inC_i, with St(i) and gives anorbihedron chart ati.

• for each directed edgei${\displaystyle \to }$j ofX ', an injective homomorphismf_ij of Γ_i into Γ_j.
• for each directed edgei${\displaystyle \to }$j, a Γ_i equivariant simplicialgluing map ψ_ij ofC_i intoC_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 fromV_i toV_j has the formg·ψ_ij for a uniqueg in Γ_j.

Ifi${\displaystyle \to }$j${\displaystyle \to }$k, then there is a uniquetransition elementg_ijk in Γ_k such that

g_ijk·ψ_ik = ψ_jk·ψ_ij

These transition elements satisfy

(Adg_ijk)·f_ik =f_jk·f_ij

as well as the cocycle relation

ψ_km(g_ijk)·g_ikm =g_ijm·g_jkm.

• The group theoretic data of an orbihedron gives a complex of groups onX, because the verticesi of the barycentric subdivisionX ' correspond to the simplices inX.
• Every complex of groups onX is associated with an essentially unique orbihedron structure onX. This key fact follows by noting that the star and link of a vertexi ofX ', corresponding to a simplex σ ofX, have natural decompositions: the star is isomorphic to the abstract simplicial complex given by the join of σ and the barycentric subdivision σ' of σ; and the link is isomorphic to join of the link of σ inX and the link of the barycentre of σ in σ'. Restricting the complex of groups to the link of σ inX, all the groups Γ_τ come with injective homomorphisms into Γ_σ. Since the link ofi inX ' is canonically covered by a simplicial complex on which Γ_σ acts, this defines an orbihedron structure onX.
• The orbihedron fundamental group is (tautologically) just the edge-path group of the associated complex of groups.
• Every orbihedron is also naturally an orbispace: indeed in the geometric realization of the simplicial complex, orbispace charts can be defined using the interiors of stars.
• The orbihedron fundamental group can be naturally identified with the orbispace fundamental group of the associated orbispace. This follows by applying thesimplicial approximation theorem to segments of an orbispace path lying in an orbispace chart: it is a straightforward variant of the classical proof that thefundamental group of apolyhedron can be identified with itsedge-path group.
• The orbispace associated to an orbihedron has acanonical metric structure, coming locally from the length metric in the standard geometric realization in Euclidean space, with vertices mapped to an orthonormal basis. Other metric structures are also used, involving length metrics obtained by realizing the simplices inhyperbolic space, with simplices identified isometrically along common boundaries.
• The orbispace associated to an orbihedron isnon-positively curved if and only if the link in each orbihedron chart hasgirth greater than or equal to 6, i.e. any closed circuit in the link has length at least 6. This condition, well known from the theory ofHadamard spaces, depends only on the underlying complex of groups.
• When the universal covering orbihedron is non-positively curved the fundamental group is infinite and is generated by isomorphic copies of the isotropy groups. This follows from the corresponding result for orbispaces.

Historically one of the most important applications of orbifolds ingeometric group theory has been totriangles of groups. This is the simplest 2-dimensional example generalising the 1-dimensional "interval of groups" discussed inSerre's lectures on trees, whereamalgamated free products are studied in terms of actions on trees. Such triangles of groups arise any time a discrete group acts simply transitively on the triangles in theaffine Bruhat–Tits building forSL₃(Q_p); in 1979Mumford discovered the first example forp = 2 (see below) as a step in producing analgebraic surface not isomorphic toprojective space, but having the sameBetti numbers. Triangles of groups were worked out in detail by Gersten and Stallings, while the more general case of complexes of groups, described above, was developed independently by Haefliger. The underlying geometric method of analysing finitely presented groups in terms of metric spaces of non-positive curvature is due to Gromov. In this context triangles of groups correspond to non-positively curved 2-dimensional simplicial complexes with the regular action of a group,transitive on triangles.

Atriangle of groups is asimple complex of groups consisting of a triangle with verticesA,B,C. There are groups

• Γ_A, Γ_B, Γ_C at each vertex
• Γ_BC, Γ_CA, Γ_AB for each edge
• Γ_ABC for the triangle itself.

There are injective homomorphisms of Γ_ABC into all the other groups and of an edge group Γ_XY into Γ_X and Γ_Y. The three ways of mapping Γ_ABC into a vertex group all agree. (Often Γ_ABC is the trivial group.) The Euclidean metric structure on the corresponding orbispace is non-positively curved if and only if the link of each of the vertices in the orbihedron chart has girth at least 6.

This girth at each vertex is always even and, as observed by Stallings, can be described at a vertexA, say, as the length of the smallest word in the kernel of the natural homomorphism into Γ_A of theamalgamated free product over Γ_ABC of the edge groups Γ_AB and Γ_AC:

${\displaystyle {\mathrm{\Gamma }}_{AB}{\star }_{{\mathrm{\Gamma }}_{ABC}}{\mathrm{\Gamma }}_{AC}\to {\mathrm{\Gamma }}_A.}$

The result using the Euclidean metric structure is not optimal. Angles α, β, γ at the verticesA,B andC were defined by Stallings as 2π divided by the girth. In the Euclidean case α, β, γ ≤ π/3. However, if it is only required that α + β + γ ≤ π, it is possible to identify thetriangle with the corresponding geodesic triangle in thehyperbolic plane with thePoincaré metric (or the Euclidean plane if equality holds). It is a classical result from hyperbolic geometry that the hyperbolic medians intersect in the hyperbolic barycentre,^[19] just as in the familiar Euclidean case. The barycentric subdivision and metric from this model yield a non-positively curved metric structure on the corresponding orbispace. Thus, if α+β+γ≤π,

• the orbispace of the triangle of groups is developable;
• the corresponding edge-path group, which can also be described as thecolimit of the triangle of groups, is infinite;
• the homomorphisms of the vertex groups into the edge-path group are injections.

Letα =${\displaystyle \sqrt{-7}}$ be given by thebinomial expansion of (1 − 8)^1/2 inQ₂ and setK =Q(α)${\displaystyle \subset }$Q₂. Let

ζ = exp 2πi/7

λ = (α − 1)/2 =ζ +ζ² +ζ⁴

μ =λ/λ*.

LetE =Q(ζ), a 3-dimensional vector space overK with basis 1,ζ, andζ². DefineK-linear operators onE as follows:

• σ is the generator of theGalois group ofE overK, an element of order 3 given by σ(ζ) = ζ²
• τ is the operator of multiplication byζ onE, an element of order 7
• ρ is the operator given byρ(ζ) = 1,ρ(ζ²) =ζ andρ(1) =μ·ζ², so thatρ³ is scalar multiplication by μ.

The elementsρ,σ, andτ generate a discrete subgroup ofGL₃(K) which actsproperly on theaffine Bruhat–Tits building corresponding toSL₃(Q₂). This group actstransitively on all vertices, edges and triangles in the building. Let

σ₁ =σ,σ₂ =ρσρ⁻¹,σ₃ =ρ²σρ⁻².

Then

• σ₁,σ₂ andσ₃ generate a subgroup Γ ofSL₃(K).
• Γ is the smallest subgroup generated byσ andτ, invariant under conjugation byρ.
• Γ actssimply transitively on the triangles in the building.
• There is a triangle Δ such that the stabiliser of its edges are the subgroups of order 3 generated by theσ_i's.
• The stabiliser of a vertices of Δ is theFrobenius group of order 21 generated by the two order 3 elements stabilising the edges meeting at the vertex.
• The stabiliser of Δ is trivial.

The elementsσ andτ generate the stabiliser of a vertex. Thelink of this vertex can be identified with the spherical building ofSL₃(F₂) and the stabiliser can be identified with thecollineation group of theFano plane generated by a 3-fold symmetry σ fixing a point and a cyclic permutation τ of all 7 points, satisfyingστ =τ²σ. IdentifyingF₈* with the Fano plane, σ can be taken to be the restriction of theFrobenius automorphismσ(x) =x²² ofF₈ and τ to be multiplication by any element not in theprime fieldF₂, i.e. an order 7 generator of thecyclic multiplicative group ofF₈. This Frobenius group acts simply transitively on the 21 flags in the Fano plane, i.e. lines with marked points. The formulas for σ and τ onE thus "lift" the formulas onF₈.

Mumford also obtains an actionsimply transitive on the vertices of the building by passing to a subgroup of Γ₁ = <ρ,σ,τ, −I>. The group Γ₁ preserves theQ(α)-valued Hermitian form

f(x,y) =xy* +σ(xy*) +σ²(xy*)

onQ(ζ) and can be identified withU₃(f)${\displaystyle \cap }$GL₃(S) whereS =Z[α,⁠1/2⁠]. SinceS/(α) =F₇, there is a homomorphism of the group Γ₁ intoGL₃(F₇). This action leaves invariant a 2-dimensional subspace inF₇³ and hence gives rise to a homomorphismΨ of Γ₁ intoSL₂(F₇), a group of order 16·3·7. On the other hand, the stabiliser of a vertex is a subgroup of order 21 andΨ is injective on this subgroup. Thus if thecongruence subgroup Γ₀ is defined as theinverse image underΨ of the 2-Sylow subgroup ofSL₂(F₇), the action ofΓ₀ on vertices must be simply transitive.

Other examples of triangles or 2-dimensional complexes of groups can be constructed by variations of the above example.

Cartwright et al. consider actions on buildings that aresimply transitive on vertices. Each such action produces a bijection (or modified duality) between the pointsx and linesx* in theflag complex of a finiteprojective plane and a collection of oriented triangles of points (x,y,z), invariant under cyclic permutation, such thatx lies onz*,y lies onx* andz lies ony* and any two points uniquely determine the third. The groups produced have generatorsx, labelled by points, and relationsxyz = 1 for each triangle. Generically this construction will not correspond to an action on a classical affine building.

More generally, as shown by Ballmann and Brin, similar algebraic data encodes all actions that are simply transitively on the vertices of a non-positively curved 2-dimensional simplicial complex, provided the link of each vertex has girth at least 6. This data consists of:

• a generating setS containing inverses, but not the identity;
• a set of relationsghk = 1, invariant under cyclic permutation.

The elementsg inS label the verticesg·v in the link of a fixed vertexv; and the relations correspond to edges (g⁻¹·v,h·v) in that link. The graph with verticesS and edges (g,h), forg⁻¹h inS, must have girth at least 6. The original simplicial complex can be reconstructed using complexes of groups and the second barycentric subdivision.

Further examples of non-positively curved 2-dimensional complexes of groups have been constructed by Swiatkowski based on actionssimply transitive on oriented edges and inducing a 3-fold symmetry on each triangle; in this case too the complex of groups is obtained from the regular action on the second barycentric subdivision. The simplest example, discovered earlier with Ballmann, starts from a finite groupH with a symmetric set of generatorsS, not containing the identity, such that the correspondingCayley graph has girth at least 6. The associated group is generated byH and an involution τ subject to (τg)³ = 1 for eachg inS.

In fact, if Γ acts in this way, fixing an edge (v,w), there is an involution τ interchangingv andw. The link ofv is made up of verticesg·w forg in a symmetric subsetS ofH = Γ_v, generatingH if the link is connected. The assumption on triangles implies that

τ·(g·w) =g⁻¹·w

forg inS. Thus, if σ = τg andu =g⁻¹·w, then

σ(v) =w, σ(w) =u, σ(u) =w.

By simple transitivity on the triangle (v,w,u), it follows that σ³ = 1.

The second barycentric subdivision gives a complex of groups consisting of singletons or pairs of barycentrically subdivided triangles joined along their large sides: these pairs are indexed by the quotient spaceS/~ obtained by identifying inverses inS. The single or "coupled" triangles are in turn joined along one common "spine". All stabilisers of simplices are trivial except for the two vertices at the ends of the spine, with stabilisersH and <τ>, and the remaining vertices of the large triangles, with stabiliser generated by an appropriate σ. Three of the smaller triangles in each large triangle contain transition elements.

When all the elements ofS are involutions, none of the triangles need to be doubled. IfH is taken to be thedihedral groupD₇ of order 14, generated by an involutiona and an elementb of order 7 such that

ab=b⁻¹a,

thenH is generated by the 3 involutionsa,ab andab⁵. The link of each vertex is given by the corresponding Cayley graph, so is just thebipartite Heawood graph, i.e. exactly the same as in the affine building forSL₃(Q₂). This link structure implies that the corresponding simplicial complex is necessarily aEuclidean building. At present, however, it seems to be unknown whether any of these types of action can in fact be realised on a classical affine building: Mumford's group Γ₁ (modulo scalars) is only simply transitive on edges, not on oriented edges.

Two-dimensional orbifolds have the following three types of singular points:

• A boundary point
• An elliptic point orgyration point of ordern, such as the origin ofR² quotiented out by a cyclic group of ordern of rotations.
• A corner reflector of ordern: the origin ofR² quotiented out by a dihedral group of order 2n.

A compact 2-dimensional orbifold has anEuler characteristic${\displaystyle \chi }$given by

${\displaystyle \chi =\chi (X_0)-\sum _i(1-1/n_i)/2-\sum _i(1-1/m_i)}$,

where${\displaystyle \chi (X_0)}$ is the Euler characteristic of the underlying topological manifold${\displaystyle X_0}$, and${\displaystyle n_i}$ are the orders of the corner reflectors, and${\displaystyle m_i}$ are the orders of the elliptic points.

A 2-dimensional compact connected orbifold has a hyperbolic structure if its Euler characteristic is less than 0, a Euclidean structure if it is 0, and if its Euler characteristic is positive it is eitherbad or has an elliptic structure (an orbifold is called bad if it does not have a manifold as a covering space). In other words, its universal covering space has a hyperbolic, Euclidean, or spherical structure.

The compact 2-dimensional connected orbifolds that are not hyperbolic are listed in the table below. The 17 parabolic orbifolds are the quotients of the plane by the 17wallpaper groups.

This sectionneeds expansion. You can help byadding missing information.(July 2008)

A 3-manifold is said to besmall if it is closed, irreducible and does not contain any incompressible surfaces.

Orbifold Theorem. LetM be a small 3-manifold. Let φ be a non-trivial periodic orientation-preserving diffeomorphism ofM. ThenM admits a φ-invariant hyperbolic or Seifert fibered structure.

This theorem is a special case of Thurston'sorbifold theorem, announced without proof in 1981; it forms part ofhis geometrization conjecture for 3-manifolds. In particular it implies that ifX is a compact, connected, orientable, irreducible, atoroidal 3-orbifold with non-empty singular locus, thenM has a geometric structure (in the sense of orbifolds). A complete proof of the theorem was published by Boileau, Leeb & Porti in 2005.^[20]

Instring theory, the word "orbifold" has a slightly new meaning. For mathematicians, an orbifold is a generalization of the notion ofmanifold that allows the presence of the points whose neighborhood isdiffeomorphic to a quotient ofRⁿ by a finite group, i.e.Rⁿ/Γ. In physics, the notion of an orbifold usually describes an object that can be globally written as an orbit spaceM/G whereM is a manifold (or a theory), andG is a group of its isometries (or symmetries) — not necessarily all of them. In string theory, these symmetries do not have to have a geometric interpretation.

Aquantum field theory defined on an orbifold becomes singular near the fixed points ofG. However string theory requires us to add new parts of theclosed stringHilbert space — namely the twisted sectors where the fields defined on the closed strings are periodic up to an action fromG. Orbifolding is therefore a general procedure of string theory to derive a new string theory from an old string theory in which the elements ofG have been identified with the identity. Such a procedure reduces the number of states because the states must be invariant underG, but it also increases the number of states because of the extra twisted sectors. The result is usually a perfectly smooth, new string theory.

D-branes propagating on the orbifolds are described, at low energies, by gauge theories defined by thequiver diagrams. Open strings attached to theseD-branes have no twisted sector, and so the number of open string states is reduced by the orbifolding procedure.

More specifically, when the orbifold groupG is a discrete subgroup of spacetime isometries, then if it has no fixed point, the result is usually a compact smooth space; the twisted sector consists of closed strings wound around the compact dimension, which are calledwinding states.

When the orbifold group G is a discrete subgroup of spacetime isometries, and it has fixed points, then these usually haveconical singularities, becauseRⁿ/Z_k has such a singularity at the fixed point ofZ_k. In string theory, gravitational singularities are usually a sign of extradegrees of freedom which are located at a locus point in spacetime. In the case of the orbifold thesedegrees of freedom are the twisted states, which are strings "stuck" at the fixed points. When the fields related with these twisted states acquire a non-zerovacuum expectation value, the singularity is deformed, i.e. the metric is changed and becomes regular at this point and around it. An example for a resulting geometry is theEguchi–Hanson spacetime.

From the point of view of D-branes in the vicinity of the fixed points, the effective theory of the open strings attached to these D-branes is a supersymmetric field theory, whose space of vacua has a singular point, where additional massless degrees of freedom exist. The fields related with the closed string twisted sector couple to the open strings in such a way as to add a Fayet–Iliopoulos term to the supersymmetric field theory Lagrangian, so that when such a field acquires a non-zerovacuum expectation value, the Fayet–Iliopoulos term is non-zero, and thereby deforms the theory (i.e. changes it) so that the singularity no longer exists[1],[2].

Insuperstring theory,^[21][22]the construction of realisticphenomenological models requiresdimensional reduction because the strings naturally propagate in a 10-dimensional space whilst the observed dimension ofspace-time of the universe is 4. Formal constraints on the theories nevertheless place restrictions on thecompactified space in which the extra "hidden" variables live: when looking for realistic 4-dimensional models withsupersymmetry, the auxiliary compactified space must be a 6-dimensionalCalabi–Yau manifold.^[23]

There are a large number of possible Calabi–Yau manifolds (tens of thousands), hence the use of the term "landscape" in the current theoretical physics literature to describe the baffling choice. The general study of Calabi–Yau manifolds is mathematically complex and for a long time examples have been hard to construct explicitly. Orbifolds have therefore proved very useful since they automatically satisfy the constraints imposed by supersymmetry. They provide degenerate examples of Calabi–Yau manifolds due to theirsingular points,^[24] but this is completely acceptable from the point of view of theoretical physics. Such orbifolds are called "supersymmetric": they are technically easier to study than general Calabi–Yau manifolds. It is very often possible to associate a continuous family of non-singular Calabi–Yau manifolds to a singular supersymmetric orbifold. In 4 dimensions this can be illustrated using complexK3 surfaces:

• Every K3 surface admits 16 cycles of dimension 2 that are topologically equivalent to usual 2-spheres. Making the surface of these spheres tend to zero, the K3 surface develops 16 singularities. This limit represents a point on the boundary of themoduli space of K3 surfaces and corresponds to the orbifold${\displaystyle T^4/{\mathbb{Z}}_2}$ obtained by taking the quotient of the torus by the symmetry of inversion.

The study of Calabi–Yau manifolds in string theory and the duality between different models of string theory (type IIA and IIB) led to the idea ofmirror symmetry in 1988. The role of orbifolds was first pointed out by Dixon, Harvey, Vafa and Witten around the same time.^[25]

Beyond their manifold and various applications in mathematics and physics, orbifolds have been applied tomusic theory at least as early as 1985 in the work ofGuerino Mazzola^[26][27] and later byDmitri Tymoczko and collaborators.^{[28][29][30][31]} One of the papers of Tymoczko was the first music theory paper published by the journalScience.^[32][33][34] Mazzola and Tymoczko have participated in debate regarding their theories documented in a series of commentaries available at their respective web sites.^[35][36]

Tymoczko models musical chords consisting ofn notes, which are not necessarily distinct, as points in the orbifold${\displaystyle T^n/S_n}$ – the space ofn unordered points (not necessarily distinct) in the circle, realized as the quotient of then-torus${\displaystyle T^n}$ (the space ofnordered points on the circle) by thesymmetric group${\displaystyle S_n}$ (corresponding from moving from an ordered set to an unordered set).

Musically, this is explained as follows:

• Musical tones depend on the frequency (pitch) of their fundamental, and thus are parametrized by the positive real numbers,R⁺.
• Musical tones that differ by an octave (a doubling of frequency) are considered the same tone – this corresponds to taking thelogarithm base 2 of frequencies (yielding the real numbers, as${\displaystyle \mathbf{R}={\log }_2{\mathbf{R}}^{+}}$), then quotienting by the integers (corresponding to differing by some number of octaves), yielding a circle (as${\displaystyle S^1=\mathbf{R}/\mathbf{Z}}$).
• Chords correspond to multiple tones without respect to order – thust notes (with order) correspond tot ordered points on the circle, or equivalently a single point on thet-torus${\displaystyle T^t:=S^1\times \cdots \times S^1,}$ and omitting order corresponds to taking the quotient by${\displaystyle S_t,}$ yielding an orbifold.

Fordyads (two tones), this yields the closedMöbius strip; fortriads (three tones), this yields an orbifold that can be described as a triangular prism with the top and bottom triangular faces identified with a 120° twist (a⁠1/3⁠ twist) – equivalently, as a solid torus in 3 dimensions with a cross-section an equilateral triangle and such a twist.

The resulting orbifold is naturally stratified by repeated tones (properly, by integer partitions oft) – the open set consists of distinct tones (the partition${\displaystyle t=1+1+\cdots +1}$), while there is a 1-dimensional singular set consisting of all tones being the same (the partition${\displaystyle t=t}$), which topologically is a circle, and various intermediate partitions. There is also a notable circle which runs through the center of the open set consisting of equally spaced points. In the case of triads, the three side faces of the prism correspond to two tones being the same and the third different (the partition${\displaystyle 3=2+1}$), while the three edges of the prism correspond to the 1-dimensional singular set. The top and bottom faces are part of the open set, and only appear because the orbifold has been cut – if viewed as a triangular torus with a twist, these artifacts disappear.

Tymoczko argues that chords close to the center (with tones equally or almost equally spaced) form the basis of much of traditional Western harmony, and that visualizing them in this way assists in analysis. There are 4 chords on the center (equally spaced underequal temperament – spacing of 4/4/4 between tones), corresponding to theaugmented triads (thought of asmusical sets) C♯FA, DF♯A♯, D♯GB, and EG♯C (then they cycle: FAC♯ = C♯FA), with the 12major chords and 12minor chords being the points next to but not on the center – almost evenly spaced but not quite. Major chords correspond to 4/3/5 (or equivalently, 5/4/3) spacing, while minor chords correspond to 3/4/5 spacing. Key changes then correspond to movement between these points in the orbifold, with smoother changes effected by movement between nearby points.

• Branched covering
• Euler characteristic of an orbifold
• Geometric quotient
• Kawasaki's Riemann–Roch formula
• Orbifold notation
• Orientifold
• Ring of modular forms
• Stack (mathematics)

• Differential topology
• Generalized manifolds
• Group actions

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