Orbifold - Knowledge

42: 3955:, 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. 9254: 8860: 7599: 6393:. 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, 6403: 6657: 9149:" 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 their 9153:, 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 complex 6633:(or the Euclidean plane if equality holds). It is a classical result from hyperbolic geometry that the hyperbolic medians intersect in the hyperbolic barycentre, 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, 9765:), 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 9821:

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

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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

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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

9311:(lime green and navy blue). The white regions are degenerate trichords (one-note repeated three times), with the three lines (representing two note chords) connecting their centers forming the walls of the twisted triangular prism, 2D planes perpendicular to plane of the image acting as mirrors. 23:

This terminology should not be blamed on me. It was obtained by a democratic process in my course of 1976–77. An orbifold is something with many folds; unfortunately, the word "manifold" already has a different definition. I tried "foldamani", which was quickly displaced by the suggestion of

4620:. 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-commutative 3332:{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: " 7510:

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:

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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

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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 under

9797:), 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. 9163:

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 the

6307:, 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 in 7615:

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 group

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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 the triangle with the corresponding geodesic triangle in the

6607: 2653:. 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 3049:

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

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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 proper

3482:, 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 is 9035:

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 called

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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:

2706:. 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. 2625: 7787:

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 space

11254: 6243:, 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 in 8232:

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.

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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

5584:, 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. 3421:

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 and its isotropy groups.

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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 either

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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.

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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-zero

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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 of

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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.

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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.

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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 a

1866: 7796:. 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 stabilisers 1996: 9547: 1322:

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

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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

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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

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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.

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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 17

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General introductions to this material can be found in Peter Scott's 1983 notes and the expositions of Boileau, Maillot & Porti and Cooper, Hodgson & Kerckhoff.

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necessarily fixes all the vertices of σ. A straightforward inductive argument shows that such an action becomes regular on the barycentric subdivision; in particular

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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

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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.

5246:/ 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 groups 6529: 5382:Γ is naturally isomorphic to the edge-path group defined using edge-paths and vertex stabilisers for the barycentric subdivision of the fundamental domain in 9672:(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 (a 7885:. 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 Γ 6284:

The orbihedron fundamental group can be naturally identified with the orbispace fundamental group of the associated orbispace. This follows by applying the

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Mazzola and Tymoczko have participated in debate regarding their theories documented in a series of commentaries available at their respective web sites.

11456:. Group theory from a geometrical viewpoint, 26 March - 6 April 1990, ICPT, Trieste, Italy (proceedings). Singapore: World Scientific. pp. 491–503. 11016:. Group theory from a geometrical viewpoint, 26 March - 6 April 1990, ICPT, Trieste, Italy (proceedings). Singapore: World Scientific. pp. 504–540. 10910:. Group theory from a geometrical viewpoint, 26 March - 6 April 1990, ICPT, Trieste, Italy (proceedings). Singapore: World Scientific. pp. 193–253. 8916: 9082:, 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 the 10047: 7595:, must have girth at least 6. The original simplicial complex can be reconstructed using complexes of groups and the second barycentric subdivision. 7970: 6355:

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 the

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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 of

4167: 12909: 12456: 9006:. Orbifolding is therefore a general procedure of string theory to derive a new string theory from an old string theory in which the elements of 247: 9014:, but it also increases the number of states because of the extra twisted sectors. The result is usually a perfectly smooth, new string theory. 2535: 12511: 11647: 10444: 7188:. 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 on 11525: 9078:

are the twisted states, which are strings "stuck" at the fixed points. When the fields related with these twisted states acquire a non-zero

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has a geometric structure (in the sense of orbifolds). A complete proof of the theorem was published by Boileau, Leeb & Porti in 2005.

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This definition mimics the definition of a manifold in diffeology, which is a diffeological space locally diffeomorphic at each point to

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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 of

3759: 200:, post-Gromov, discrete groups have been studied in terms of the local curvature properties of orbihedra and their covering spaces. 11957: 9094:, the Fayet–Iliopoulos term is non-zero, and thereby deforms the theory (i.e. changes it) so that the singularity no longer exists 3459:

on the orbispace charts for which the gluing maps preserve distance. In this case each orbispace chart is usually required to be a

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Further examples of non-positively curved 2-dimensional complexes of groups have been constructed by Swiatkowski based on actions

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have been identified with the identity. Such a procedure reduces the number of states because the states must be invariant under

8920: 2686:, there is a canonical orbifold atlas over its orbit space, whose associated effective orbifold groupoid is Morita equivalent to 2710:

between groupoids, which carries more information than the underlying continuous map between the underlying topological spaces.

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Musical tones that differ by an octave (a doubling of frequency) are considered the same tone – this corresponds to taking the

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More generally, attached to an open covering of an orbifold by orbifold charts, there is the combinatorial data of a so-called

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to segments of an orbispace path lying in an orbispace chart: it is a straightforward variant of the classical proof that the

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Exactly as in the case of manifolds, differentiability conditions can be imposed on the gluing maps to give a definition of a

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Iglesias-Zemmour, Patrick; Laffineur, Jean-Pierre (2017). "Noncommutative Geometry & Diffeology: The Case Of Orbifolds".

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by a proper rigid action of a discrete group Γ, the orbifold fundamental group can be identified with Γ. In general it is an

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Musical tones depend on the frequency (pitch) of their fundamental, and thus are parametrized by the positive real numbers,

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When the orbifold group G is a discrete subgroup of spacetime isometries, and it has fixed points, then these usually have

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

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finite groups attached to vertices, inclusions to edges and transition elements, describing compatibility, to triangles.

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twist) – equivalently, as a solid torus in 3 dimensions with a cross-section an equilateral triangle and such a twist.

8948:, the word "orbifold" has a slightly new meaning. For mathematicians, an orbifold is a generalization of the notion of 7507:= 1 for each triangle. Generically this construction will not correspond to an action on a classical affine building. 1930: 11469: 11235: 11139: 11029: 10998: 10953: 10923: 10891: 10770: 10741: 10667: 10584: 10561: 10254: 10210: 9507: 9299:. Slices of cubes standing on end (with their long diagonals perpendicular to the plane of the image) form colored 2451:

of germs of its elements is an orbifold groupoid. Moreover, since by definition of orbifold atlas each finite group

12876: 12155: 11481: 9119: 8968:. In physics, the notion of an orbifold usually describes an object that can be globally written as an orbit space 6285: 137: 9504:), then quotienting by the integers (corresponding to differing by some number of octaves), yielding a circle (as 7168: 6278:

The orbihedron fundamental group is (tautologically) just the edge-path group of the associated complex of groups.

13106: 12841: 12138: 11747: 11159: 9220: 11549: 13699: 13694: 13468: 9083: 109: 9171: 9002:— namely the twisted sectors where the fields defined on the closed strings are periodic up to an action from 6347:. This is the simplest 2-dimensional example generalising the 1-dimensional "interval of groups" discussed in 13048: 12554: 12497: 12350: 11757: 10487: 10009: 9223:

in 1988. The role of orbifolds was first pointed out by Dixon, Harvey, Vafa and Witten around the same time.

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Like a manifold, an orbifold is specified by local conditions; however, instead of being locally modelled on

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and collaborators. One of the papers of Tymoczko was the first music theory paper published by the journal

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is a compact, connected, orientable, irreducible, atoroidal 3-orbifold with non-empty singular locus, then

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Beyond their manifold and various applications in mathematics and physics, orbifolds have been applied to

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There are a large number of possible Calabi–Yau manifolds (tens of thousands), hence the use of the term "

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There is a triangle Δ such that the stabiliser of its edges are the subgroups of order 3 generated by the

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the gluing maps are unique up to composition with group elements, i.e. any other possible gluing map from

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the gluing maps are unique up to composition with group elements, i.e. any other possible gluing map from

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have no twisted sector, and so the number of open string states is reduced by the orbifolding procedure.

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A complex of groups is developable if and only if for each simplex σ there is an injective homomorphism θ

3537: 13073: 7388:), a group of order 16·3·7. On the other hand, the stabiliser of a vertex is a subgroup of order 21 and 5554:

An edge-path group can then be defined. A similar structure is inherited by the barycentric subdivision

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generated by a 3-fold symmetry σ fixing a point and a cyclic permutation τ of all 7 points, satisfying

1219: 754: 3240: 3034:. The simplest approach (adopted by Haefliger and known also to Thurston) extends the usual notion of 1255: 579: 407: 378: 325: 253: 13288: 11922: 11787: 10168: 9739:), while there is a 1-dimensional singular set consisting of all tones being the same (the partition 9698: 9691:

The resulting orbifold is naturally stratified by repeated tones (properly, by integer partitions of

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This girth at each vertex is always even and, as observed by Stallings, can be described at a vertex

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and each of their finite intersections, if non-empty, is covered by an intersection of Γ-translates

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because the strings naturally propagate in a 10-dimensional space whilst the observed dimension of

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long before they were formally defined. One of the first classical examples arose in the theory of

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Boileau, Michel; Leeb, Bernhard; Porti, Joan (2005). "Geometrization of 3-dimensional orbifolds".

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is an orbifold (or orbispace), choose a covering by open subsets from amongst the orbifold charts

3570: 3434:, it is often convenient to have a slightly more general notion of orbifold, due to Haefliger. An 2400: 642: 282:

One topological space can carry different orbifold structures. For example, consider the orbifold

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in which the extra "hidden" variables live: when looking for realistic 4-dimensional models with

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and the link of the barycentre of σ in σ'. Restricting the complex of groups to the link of σ in

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Brin, Matthew (2007). "Seifert Fibered Spaces: Notes for a course given in the Spring of 1993".

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of the universe is 4. Formal constraints on the theories nevertheless place restrictions on the

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be a small 3-manifold. Let φ be a non-trivial periodic orientation-preserving diffeomorphism of

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The result using the Euclidean metric structure is not optimal. Angles α, β, γ at the vertices

6340: 5316:. Consistency can be checked using the fact that the restriction of the complex of groups to a 3460: 3431: 3417:" Indeed, there are smooth maps between orbifolds that do not lift locally as equivariant maps. 2966:: in this case the Kleinian group Γ is generated by hyperbolic reflections and the orbifold is 2786: 2737: 243: 223: 197: 10236: 9020:

propagating on the orbifolds are described, at low energies, by gauge theories defined by the

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of order 21 generated by the two order 3 elements stabilising the edges meeting at the vertex.

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and naturally a 2-dimensional orbifold. The corresponding group is an example of a hyperbolic

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Orbifold groupoids play the same role as orbifold atlases in the definition above. Indeed, an

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regions (colored by chord type) which represent the three-note chords at their centers, with

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unordered points (not necessarily distinct) in the circle, realized as the quotient of the

8988: 8200: 8173: 8146: 7947: 7941: 7291: 6319: 4447: 4420: 4351: 4071: 4051: 2508: 2481: 2427: 2342: 2144: 2007: 1879: 1721: 1694: 1609: 1582: 1191: 1163: 1073: 1045: 796: 696: 669: 552: 521: 494: 354: 189: 10796: 10639: 7881:). This link structure implies that the corresponding simplicial complex is necessarily a 6503:, say, as the length of the smallest word in the kernel of the natural homomorphism into Γ 289: 8: 13573: 13458: 13111: 13010: 12636: 12463: 12145: 12023: 12008: 11937: 11696: 11376: 11039:

Iglesias, Patrick; Karshon, Yael; Zadka, Moshe (June 2010). "Orbifolds as Diffeologies".

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The moduli space of chords: Dmitri Tymoczko on "Geometry and Music", Friday 7 Mar, 2:30pm

9866: 9742: 7393: 6626: 5363:, so that a subsimplices, given by subchains of simplices, is uniquely determined by the 2925: 1562: 227: 184:

holds after the quotient is compactified by the addition of two orbifold cusp points. In

12436: 11570: 11377:"Errata for "The geometries of 3-manifolds", Bull. London Math. Soc. 15 (1983), 401-487" 11267: 10817: 10466: 10291: 10246: 7178: 6602:{\displaystyle \Gamma _{AB}\star _{\,\Gamma _{ABC}}\Gamma _{AC}\rightarrow \Gamma _{A}.} 1315:

are equivalent if they can be consistently combined to give a larger orbifold atlas. An

13618: 13538: 13438: 13398: 13328: 13278: 13243: 13078: 12955: 12851: 12592: 12405: 12360: 12257: 12128: 11932: 11620: 11600: 11207: 11114: 11066: 11048: 10837: 10781: 10713: 10695: 10619: 10013: 9841: 9115: 7499:* and any two points uniquely determine the third. The groups produced have generators 6698: 5763: 5581: 5131: 5113: 3508:

Every orbifold has associated with it an additional combinatorial structure given by a

3309: 3213: 3193: 2955: 2689: 2380: 2356: 2202: 2182: 2050: 1906: 1297: 876: 470: 443: 13005: 11942: 11286: 11249: 10275: 6630: 2929: 207:, the word "orbifold" has a slightly different meaning, discussed in detail below. In 105: 13583: 13488: 13323: 13233: 13203: 12999: 12894: 12846: 12740: 12340: 12320: 12315: 12222: 12133: 11947: 11927: 11782: 11721: 11592: 11557: 11465: 11433: 11419: 11399: 11387: 11291: 11231: 11183: 11173: 11150: 11135: 11118: 11083: 11025: 10994: 10949: 10919: 10887: 10829: 10804: 10766: 10737: 10717: 10673: 10663: 10623: 10580: 10557: 10362: 10328: 10303: 10299: 10250: 10206: 10065: 10023: 9851: 9831: 9802: 9245: 7863:. The link of each vertex is given by the corresponding Cayley graph, so is just the 6382: 6348: 6289: 5405:

In two dimensions this is particularly simple to describe. The fundamental domain of

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the quotient simplicial structure on orbit-representatives of vertices is consistent;

3103: 3039: 3035: 3027: 2921: 1558: 434: 314: 235: 66: 11539: 11520: 11479:Świątkowski, Jacek (2001). "A class of automorphism groups of polygonal complexes". 11219: 11070: 10841: 2909: 2729:. Equivalently, it corresponds to the Morita equivalence class of the unit groupoid. 13593: 13528: 13498: 13378: 13318: 13283: 13228: 13218: 13131: 13083: 13041: 12946: 12939: 12932: 12925: 12918: 12836: 12626: 12534: 12478: 12272: 12227: 12150: 12121: 11979: 11912: 11907: 11902: 11892: 11684: 11667: 11604: 11584: 11534: 11516: 11502: 11490: 11457: 11411: 11360: 11352: 11337: 11319: 11281: 11271: 11199: 11168: 11106: 11058: 11017: 10984: 10970:. Astérisque. Vol. 116. Paris: Société Mathématique de France. pp. 70–97. 10939: 10911: 10868: 10821: 10756: 10705: 10609: 10549: 10295: 10057: 9074:

which are located at a locus point in spacetime. In the case of the orbifold these

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The local homomorphisms are all injective for a covering by contractible open sets.

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can be constructed for an orbifold by direct analogy with the construction of the

2789: 2719:

Any manifold without boundary is trivially an orbifold, where each of the groups Γ

81: 13673: 13628: 13578: 13563: 13553: 13448: 13413: 13238: 12808: 12582: 12421: 12330: 12160: 12116: 11882: 11443:. Astérisque. Vol. 46 (3rd ed.). Paris: Société Mathématique de France. 10989: 10450: 10356: 10322: 10008:. Orbifolds in mathematics and physics. Contemporary Mathematics. Vol. 310. 9806: 9407: 9304: 9300: 9240: 9236: 8237: 7447:. Each such action produces a bijection (or modified duality) between the points 7102: 7083: 5650: 5395: 3448: 3439: 3153:

An orbifold is a diffeological space locally diffeomorphic at each point to some

3070: 465: 222:

The main example of underlying space is a quotient space of a manifold under the

193: 74: 24:"manifolded". After two months of patiently saying "no, not a manifold, a manifol 13518: 11449: 11009: 10976: 10903: 10709: 10662:. Panoramas and Syntheses. Vol. 15. Paris: Société Mathématique de France. 10572: 9665: 5367:

of the simplices in the subchain. When an action satisfies this condition, then

3009:

is a finite cyclic group of rotations. This generalises Poincaré's construction.

13648: 13643: 13603: 13543: 13373: 13363: 13358: 13353: 13268: 13263: 13258: 13223: 13208: 13136: 12813: 12676: 12287: 12212: 12182: 12080: 12073: 12013: 11984: 11854: 11849: 11810: 11407: 11372: 11333: 11255:

Proceedings of the National Academy of Sciences of the United States of America

11227: 10733: 10579:. Progress in Mathematics. Vol. 83. Boston: Birkhäuser. pp. 189–201. 9021: 8953: 8894:

if it is closed, irreducible and does not contain any incompressible surfaces.

7905: 7812: 7409: 6647:

the homomorphisms of the vertex groups into the edge-path group are injections.

3055: 3023: 2963: 2959: 2913: 2025: 231: 212: 13533: 13453: 11494: 11415: 10944: 10761: 10324:

Gruppen und Kategorien in der Musik: Entwurf einer mathematischen Musiktheorie

9253: 8096:{\displaystyle \chi =\chi (X_{0})-\sum _{i}(1-1/n_{i})/2-\sum _{i}(1-1/m_{i})} 2657:) is a special kind of orbifold structure according to the second definition. 13688: 13638: 13623: 13598: 13588: 13558: 13503: 13478: 13408: 13403: 13368: 13343: 13303: 13093: 12745: 12661: 12539: 12520: 12473: 12297: 12292: 12277: 12267: 12217: 12194: 12068: 12028: 11969: 11917: 11716: 10553: 10307: 10061: 10043: 9999: 9669: 9308: 9135: 8999: 8996: 8945: 7897:

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

7864: 7603: 7401: 6942: 6374: 5577: 5109: 4249:{\displaystyle U_{i}\supset U_{i}\cap U_{j}\supset U_{i}\cap U_{j}\cap U_{k}} 2726: 2033: 307: 204: 132: 128: 13423: 11588: 11356: 11324: 11307: 10825: 10677: 9070:. In string theory, gravitational singularities are usually a sign of extra 7889:(modulo scalars) is only simply transitive on edges, not on oriented edges. 13668: 13508: 13388: 13338: 13308: 13293: 13101: 13068: 12960: 12886: 12866: 12803: 12666: 12400: 12395: 12237: 12204: 12177: 12085: 11726: 11596: 11506: 11295: 10833: 9661: 9232: 9165: 9060: 9050: 7625: 7120: 6843: 6508: 6390: 6352: 6323: 4629: 4621: 3483: 3456: 3452: 3110: 2037: 1687: 1685:

and other maps allowing arrows to be composed and inverted. It is called a

239: 216: 113: 70: 11364: 11276: 10851:"Groups acting simply transitively on the vertices of a building of type Ã 10849:

Cartwright, Donald; Mantero, Anna Maria; Steger, Tim; Zappa, Anna (1993).

10755:. Grundlehren der mathematischen Wissenschaften. Vol. 319. Springer. 10138: 10102: 9813:) C♯FA, DF♯A♯, D♯GB, and EG♯C (then they cycle: FAC♯ = C♯FA), with the 12 9213:

obtained by taking the quotient of the torus by the symmetry of inversion.

7598: 2620:{\displaystyle g\in (G_{X})_{x}\mapsto \mathrm {germ} _{x}(t\circ s^{-1})} 2036:, the discreteness condition implies that the isotropies must be actually 41: 13663: 13633: 13613: 13473: 13428: 13383: 13348: 13298: 13063: 13032: 12793: 12750: 12243: 12232: 12189: 12090: 11691: 10725: 10571:

Ballmann, Werner (1990). "Singular spaces of non-positive curvature". In

10433:

posted 28/Feb/08 – talk abstract and high-level mathematical description.

10261:

Orbifolds can be viewed as singular limits of smooth Calabi–Yau manifolds

9856: 9818: 9814: 9810: 4671:

of a complex of groups can be defined as a natural generalisation of the

2374: 1005: 36:, p. 300, section 13.2) explaining the origin of the word "orbifold" 10358:

The topos of music: geometric logic of concepts, theory, and performance

8859: 5258:

A complex of groups is developable if and only if the homomorphisms of Γ

3388:-map defined in a different choice of defining families is not always a 13608: 13548: 13483: 13146: 13126: 13025: 12984: 12798: 12468: 12426: 12252: 12165: 11797: 11701: 11612: 11211: 11110: 10983:. Progress in Mathematics. Vol. 83. Birkhäuser. pp. 203–213. 10873: 10850: 10614: 10595: 9154: 9127: 9095: 7124: 6660: 6204:

The group theoretic data of an orbihedron gives a complex of groups on

3361:-map thus defined is inconvenient in the point that a composite of two 3140: 3094:

if it arises as the quotient by a group action; otherwise it is called

1999: 185: 97: 11521:"Three-dimensional manifolds, Kleinian groups and hyperbolic geometry" 11082:. Mathematical Surveys and Monographs. American Mathematical Society. 9965: 6640:

the corresponding edge-path group, which can also be described as the

3943:

It is often more convenient and conceptually appealing to pass to the

2766:

fixing the common boundary; the quotient space can be identified with

246:

carries a natural orbifold structure, since it is the quotient of its

211:, it refers to the theory attached to the fixed point subalgebra of a 13523: 13313: 13253: 12899: 12773: 12694: 12689: 12684: 12282: 12247: 11952: 11839: 11053: 10700: 10150: 10018: 10003: 9451: 9436:(corresponding from moving from an ordered set to an unordered set). 9319:

notes, which are not necessarily distinct, as points in the orbifold

6231:. This key fact follows by noting that the star and link of a vertex 5394:

barycentric subdivision: as Haefliger observes using the language of

3127:

The orbifold structure on the universal covering orbifold is trivial.

3000: 1869: 873: 286:

associated with a quotient space of the 2-sphere along a rotation by

11203: 9560:

ordered points on the circle, or equivalently a single point on the

9552:

Chords correspond to multiple tones without respect to order – thus

6402: 6339:

Historically one of the most important applications of orbifolds in

13169: 13058: 12783: 12778: 12715: 12631: 12441: 12431: 11822: 11643: 9025: 9017: 8949: 8143:

is the Euler characteristic of the underlying topological manifold

7693:

if the link is connected. The assumption on triangles implies that

7145:* with the Fano plane, σ can be taken to be the restriction of the 3926:

An easy inductive argument shows that every complex of groups on a

3031: 2751:

and its mirror image along their common boundary. There is natural

1575: 372: 62: 54: 50: 10886:. MSJ Memoirs. Vol. 5. Tokyo: Mathematical Society of Japan. 10795:

Callender, Clifton; Quinn, Ian; Tymoczko, Dmitri (18 April 2008).

10786: 10500:

Agustín-Aquino, Octavio Alberto; Mazzola, Guerino (14 June 2011).

10394: 9893: 6311:, with simplices identified isometrically along common boundaries. 5339:

always satisfies the following condition, weaker than regularity:

3951:. The vertices of this subdivision correspond to the simplices of 3701:

The group elements must in addition satisfy the cocycle condition

2977:

is a closed 2-manifold, new orbifold structures can be defined on

404:, an orbifold is locally modelled on quotients of open subsets of 12788: 12768: 12725: 12720: 11461: 11021: 10934:

Di Francesco, Philippe; Mathieu, Pierre; Sénéchal, David (1997).

10915: 10274:

Dixon, L.; Harvey, J. A.; Vafa, C.; Witten, E. (1 January 1985).

10192:, Vol. 1 and 2, Cambridge University Press, 1987, ISBN 0521357527 8991:

defined on an orbifold becomes singular near the fixed points of

6641: 6227:

is associated with an essentially unique orbihedron structure on

2341:). This definition shows that orbifolds are a particular kind of 12489: 10126: 10090: 7105:

of this vertex can be identified with the spherical building of

5347:·σ are subsimplices of some simplex τ, they are equal, i.e. σ = 4930:) is defined to be the subgroup of Γ generated by all products 2962:. Poincaré also gave a 3-dimensional version of this result for 2032:

Since the isotropy groups of proper groupoids are automatically

12038: 10241:, Theoretical and Mathematical Physics, Springer, p. 487, 3230:

is a finite linear group which may change from point to point.

1633:, and structural maps including the source and the target maps 8912:

admits a φ-invariant hyperbolic or Seifert fibered structure.

7807:

are involutions, none of the triangles need to be doubled. If

3790:{\displaystyle \pi \subset \rho \subset \sigma \subset \tau .} 3113:

open subset corresponds to a group Γ, then there is a natural

12760: 10225:, in *Trieste 1987, Proceedings, Superstrings '87* 1-88, 1987 9370: 8952:

that allows the presence of the points whose neighborhood is

7359:). This action leaves invariant a 2-dimensional subspace in 7206:

on the vertices of the building by passing to a subgroup of Γ

6656: 4625: 2792:

isometric action of a discrete group Γ, then the orbit space

10848: 10502:"On D. Tymoczko's critique of Mazzola's counterpoint theory" 10078: 8995:. However string theory requires us to add new parts of the 5625:' endowed with a rigid simplicial action of a finite group Γ 5596:

be a finite simplicial complex with barycentric subdivision

5123:

if it satisfies one of the following equivalent conditions:

10938:. Graduate Texts in Contemporary Physics. Springer-Verlag. 10933: 9971: 9953: 9695:) – the open set consists of distinct tones (the partition 9138:, the auxiliary compactified space must be a 6-dimensional 7628:

has girth at least 6. The associated group is generated by

7624:, not containing the identity, such that the corresponding 6267:' is canonically covered by a simplicial complex on which Γ 5398:, in this case the 3-skeleton of the fundamental domain of 4675:

of a simplicial complex. In the barycentric subdivision of

3495:

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

10660:

Three-dimensional orbifolds and their geometric structures

10235:

Blumenhagen, Ralph; Lüst, Dieter; Theisen, Stefan (2012),

6967:

on all vertices, edges and triangles in the building. Let

80:

Definitions of orbifold have been given several times: by

11096: 10882:

Cooper, Daryl; Hodgson, Craig; Kerckhoff, Steven (2000).

10156: 9626:

and omitting order corresponds to taking the quotient by

7441:

Cartwright et al. consider actions on buildings that are

6417:

complex of groups consisting of a triangle with vertices

2017:

is given by one of the following equivalent definitions:

1150:{\displaystyle \varphi _{j}\circ \psi _{ij}=\varphi _{i}} 927:{\displaystyle f_{ij}:\Gamma _{i}\rightarrow \Gamma _{j}} 196:, can be phrased in terms of 2-dimensional orbifolds. In 11149:

Köhler, Peter; Meixner, Thomas; Wester, Michael (1985).

9805:– spacing of 4/4/4 between tones), corresponding to the 5712:' extends to a simplicial action on the simplicial cone 5409:" has the same structure as the barycentric subdivision 10658:

Boileau, Michel; Maillot, Sylvain; Porti, Joan (2003).

9619:{\displaystyle T^{t}:=S^{1}\times \cdots \times S^{1},} 6637:

the orbispace of the triangle of groups is developable;

3329:

This definition is equivalent with Haefliger orbifolds.

2981:

i by removing finitely many disjoint closed discs from

2348: 1319:

is therefore an equivalence class of orbifold atlases.

10881: 10794: 10540:

Adem, Alejandro; Leida, Johann; Ruan, Yongbin (2007).

10400: 10234: 9497:{\displaystyle \mathbf {R} =\log _{2}\mathbf {R} ^{+}} 7867:, i.e. exactly the same as in the affine building for 5945:

the gluing maps are compatible with the charts, i.e. φ

4101:. For each such simplex there is an associated group Γ 10499: 10273: 10114: 9771: 9745: 9701: 9632: 9570: 9510: 9460: 9454:

base 2 of frequencies (yielding the real numbers, as

9415: 9378: 9325: 9263: 9174: 8203: 8176: 8149: 8113: 7973: 7950: 7294: 7167:

and τ to be multiplication by any element not in the

6726: 6676: 6532: 6318:

if and only if the link in each orbihedron chart has

6039: 6017: 5891: 5837: 5518: 5492: 5466: 5437: 5083: 4896: 4874: 4739: 4705: 4558: 4522: 4486: 4450: 4423: 4396: 4354: 4327: 4282: 4170: 4147: 4127: 4074: 4054: 3990: 3762: 3640: 3620: 3597: 3573: 3394: 3367: 3340: 3312: 3275: 3243: 3216: 3196: 3159: 3139:

Orbifolds can be defined in the general framework of

2692: 2666: 2633: 2538: 2511: 2484: 2457: 2430: 2403: 2383: 2359: 2312: 2277: 2251: 2225: 2205: 2185: 2147: 2103: 2077: 2053: 1933: 1909: 1882: 1784: 1751: 1724: 1697: 1639: 1612: 1585: 1363: 1336: 1300: 1258: 1222: 1194: 1166: 1107: 1101:

the gluing maps are compatible with the charts, i.e.

1076: 1048: 1014: 983: 943: 884: 838: 799: 757: 726: 699: 672: 645: 615: 582: 555: 524: 497: 473: 446: 410: 381: 328: 292: 256: 140: 123:

Historically, orbifolds arose first as surfaces with

11038: 10981:

Sur les groupes hyperboliques d'après Mikhael Gromov

10596:"Polygonal complexes and combinatorial group theory" 10577:

Sur les groupes hyperboliques d'après Mikhael Gromov

10382: 10144: 10108: 10096: 9982: 9980: 9905: 9883: 9881: 8919:, announced without proof in 1981; it forms part of 2377:

made up by all diffeomorphisms between open sets of

1861:{\displaystyle (G_{1})_{x}:=s^{-1}(x)\cap t^{-1}(x)} 11148: 10657: 9917: 6326:, depends only on the underlying complex of groups. 5558:' and its edge-path group is isomorphic to that of 10544:. Cambridge Tracts in Mathematics. Vol. 171. 9789: 9757: 9731: 9648: 9618: 9541: 9496: 9428: 9391: 9353: 9307:at the very center, surrounded by major and minor 9291: 9257:Animated slices of the three-dimensional orbifold 9205: 8216: 8189: 8162: 8135: 8095: 7956: 7463:and a collection of oriented triangles of points ( 7300: 6732: 6689: 6601: 6045: 6023: 5897: 5843: 5524: 5498: 5472: 5443: 5089: 4902: 4880: 4745: 4711: 4564: 4528: 4492: 4456: 4429: 4402: 4360: 4333: 4288: 4248: 4153: 4133: 4080: 4060: 3996: 3789: 3646: 3626: 3603: 3579: 3407: 3380: 3353: 3318: 3298: 3258: 3222: 3202: 3182: 2698: 2678: 2645: 2619: 2524: 2497: 2470: 2443: 2416: 2389: 2365: 2333: 2298: 2263: 2237: 2211: 2191: 2171: 2133: 2089: 2059: 1990: 1915: 1895: 1860: 1770: 1737: 1710: 1677: 1625: 1598: 1369: 1342: 1306: 1277: 1244: 1207: 1179: 1149: 1089: 1061: 1030: 996: 969: 926: 864: 812: 785: 739: 712: 685: 658: 628: 597: 568: 537: 510: 479: 452: 425: 396: 366: 343: 298: 271: 168: 11041:Transactions of the American Mathematical Society 10348: 9977: 9878: 7632:and an involution τ subject to (τg) = 1 for each 7475:), invariant under cyclic permutation, such that 5375:the action on the second barycentric subdivision 4628:, the complex of groups in this case arises as a 4020:: its vertices are the sets of the cover and its 3120:In fact the following conditions are equivalent: 1568: 13686: 11077: 10750: 10685: 10633:"Geometrizations of 3-manifolds with symmetries" 10418:– links to papers and to visualization software. 10084: 9959: 9941: 9929: 7643:In fact, if Γ acts in this way, fixing an edge ( 7119:) and the stabiliser can be identified with the 6475:into all the other groups and of an edge group Γ 6303:The orbispace associated to an orbihedron has a 4016:be the abstract simplicial complex given by the 1991:{\displaystyle (s,t):G_{1}\to G_{0}\times G_{0}} 238:. In particular this applies to any action of a 61:(for "orbit-manifold") is a generalization of a 11250:"On a generalization of the notion of manifold" 9542:{\displaystyle S^{1}=\mathbf {R} /\mathbf {Z} } 9168:of K3 surfaces and corresponds to the orbifold 5331:The action of Γ on the barycentric subdivision 3455:structures on an orbispace, given by invariant 1561:on the orbifold charts and the gluing maps are 824:The collection of orbifold charts is called an 491:, with a covering by a collection of open sets 11184:"An algebraic surface with K ample, (K) = 9, p 10884:Three-dimensional orbifolds and cone-manifolds 6271:acts, this defines an orbihedron structure on 2897:. Orbifolds that arise in this way are called 12505: 11628: 11526:Bulletin of the American Mathematical Society 11501: 10730:Introduction to Compact Transformation Groups 10539: 10354: 10052:. Cambridge Studies in Advanced Mathematics. 10042: 9899: 9315:Tymoczko models musical chords consisting of 9059:has such a singularity at the fixed point of 8939: 8921:his geometrization conjecture for 3-manifolds 8915:This theorem is a special case of Thurston's 8197:are the orders of the corner reflectors, and 7931:quotiented out by a dihedral group of order 2 6314:The orbispace associated to an orbihedron is 3022:. More sophisticated approaches use orbifold 3013: 518:, closed under finite intersection. For each 169:{\displaystyle \mathrm {SL} (2,\mathbb {Z} )} 33: 11508:The Geometry and Topology of Three-Manifolds 11312:Journal of the Mathematical Society of Japan 10975:Haefliger, André (1990). "Orbi-espaces". In 10593: 10205:, Vol. 2, Cambridge University Press, 1999, 10049:Introduction to Foliations and Lie Groupoids 9822:effected by movement between nearby points. 7892: 3134: 3061:If the orbifold arises as the quotient of a 2800:/Γ has natural orbifold structure: for each 11478: 11345:Bulletin of the London Mathematical Society 10901: 10314: 9031:More specifically, when the orbifold group 7392:is injective on this subgroup. Thus if the 2920:generated by reflections in the edges of a 2747:can be formed by gluing together a copy of 2024:a proper Lie groupoid whose isotropies are 2006:if both the source and the target maps are 828:if the following properties are satisfied: 12512: 12498: 11635: 11621: 11308:"The Gauss-Bonnet theorem for V-manifolds" 10751:Bridson, Martin; Haefliger, André (1999). 8850: 7916:quotiented out by a cyclic group of order 7651:), there is an involution τ interchanging 7519:containing inverses, but not the identity; 6381:= 2 (see below) as a step in producing an 5421:a finite 2-dimensional simplicial complex 5320:is equivalent to one with trivial cocycle 3930:is equivalent to a complex of groups with 3117:of Γ into the orbifold fundamental group. 11578: 11538: 11447: 11323: 11285: 11275: 11172: 11052: 11007: 10988: 10974: 10965: 10943: 10872: 10785: 10760: 10699: 10613: 10355:Mazzola, Guerino; Müller, Stefan (2002). 10120: 10017: 9972:Di Francesco, Mathieu & Sénéchal 1997 9911: 9439:Musically, this is explained as follows: 9235:at least as early as 1985 in the work of 9202: 9192: 7556:; and the relations correspond to edges ( 7101:generate the stabiliser of a vertex. The 7082:The stabiliser of a vertices of Δ is the 6854:, an element of order 3 given by σ(ζ) = ζ 6551: 5273:into a fixed discrete group Γ such that θ 3278: 3246: 3162: 3143:and have been proved to be equivalent to 2532:is automatically effective, i.e. the map 585: 413: 384: 331: 259: 159: 11642: 11547: 11515: 11218: 11125: 10594:Ballmann, Werner; Brin, Michael (1994). 10570: 10483:Tony Phillips' Take on Math in the Media 10388: 9998: 9923: 9252: 9206:{\displaystyle T^{4}/\mathbb {Z} _{2}\,} 7939:A compact 2-dimensional orbifold has an 7821:of order 14, generated by an involution 7772:By simple transitivity on the triangle ( 7597: 7572:) in that link. The graph with vertices 7532:= 1, invariant under cyclic permutation. 7341:, there is a homomorphism of the group Γ 7048:Γ is the smallest subgroup generated by 6655: 6471:There is an injective homomorphisms of Γ 6255:come with injective homomorphisms into Γ 5570:If a countable discrete group acts by a 5103: 2397:which preserve the transition functions 751:, which defines a homeomorphism between 40: 11181: 10753:Metric Spaces of Non-Positive Curvature 10320: 10005:Orbifolds as Groupoids: an Introduction 9103: 8224:are the orders of the elliptic points. 7792:/~ obtained by identifying inverses in 7430:on vertices must be simply transitive. 7366:and hence gives rise to a homomorphism 6644:of the triangle of groups, is infinite; 6491:into a vertex group all agree. (Often Γ 5359:' correspond to chains of simplices in 5262:into the edge-path group are injective. 3038:used in the standard definition of the 2660:Conversely, given an orbifold groupoid 2179:is the orbit space of the Lie groupoid 104:, after a vote by his students; and by 28:," we held a vote, and "orbifold" won. 13687: 11302: 11244: 10968:Groupoides d'holonomie et classifiants 10724: 10132: 9986: 9887: 6334: 5390:There is in fact no need to pass to a 5288:. In this case the simplicial complex 5145:/Γ has a natural simplicial structure; 4759:. Let Γ be the group generated by the 3503: 215:under the action of a finite group of 209:two-dimensional conformal field theory 12493: 11616: 11511:. Princeton University lecture notes. 11432: 11398: 11386: 11332: 11151:"The 2-adic affine building of type Ã 10188:M. Green, J. Schwartz and E. Witten, 10157:Iglesias-Zemmour & Laffineur 2017 9947: 9935: 6860:is the operator of multiplication by 6815:), a 3-dimensional vector space over 4583:There is a unique transition element 3233:This definition calls a few remarks: 3018:There are several ways to define the 2857:; finitely many neighbourhoods cover 361: 100:in the 1970s when he coined the name 65:. Roughly speaking, an orbifold is a 12910:Bogomol'nyi–Prasad–Sommerfield bound 10779: 10517: 10401:Callender, Quinn & Tymoczko 2008 9664:(two tones), this yields the closed 8854: 7503:, labelled by points, and relations 6651: 5238:The fundamental domain and quotient 3109:Note that if an orbifold chart on a 2679:{\displaystyle G\rightrightarrows M} 2349:Relation between the two definitions 2090:{\displaystyle G\rightrightarrows M} 11128:The Theory of Transformation Groups 10630: 9837:Euler characteristic of an orbifold 8923:. In particular it implies that if 7620:with a symmetric set of generators 7612:simply transitive on oriented edges 7177:, i.e. an order 7 generator of the 6239:', corresponding to a simplex σ of 5119:with finite quotient is said to be 4659:gives a 2-coboundary perturbation. 2835:, identified equivariantly with a Γ 2353:Given an orbifold atlas on a space 1557:if in addition there are invariant 49:In the mathematical disciplines of 13: 11099:Journal of Noncommutative Geometry 11078:Iglesias-Zemmour, Patrick (2013). 10797:"Generalized Voice-Leading Spaces" 10097:Iglesias, Karshon & Zadka 2010 7851:is generated by the 3 involutions 7433: 6587: 6571: 6553: 6534: 6401: 6198: 6105:These transition elements satisfy 4662: 4648:is a 2-cocycle in non-commutative 3488:Birkhoff curve shortening argument 3400: 3373: 3346: 3299:{\displaystyle \mathbb {R} ^{n}/G} 3183:{\displaystyle \mathbb {R} ^{n}/G} 2582: 2579: 2576: 2573: 2459: 1678:{\displaystyle s,t:G_{1}\to G_{0}} 1443:These transition elements satisfy 1266: 985: 970:{\displaystyle U_{i}\subset U_{j}} 915: 902: 865:{\displaystyle U_{i}\subset U_{j}} 774: 728: 617: 145: 142: 96:in the context of the geometry of 14: 13716: 12519: 11394:. Presse Universitaire de France. 11371: 11014:Complexes of groups and orbihedra 10169:Theorem of the hyperbolic medians 9556:notes (with order) correspond to 9024:. Open strings attached to these 8980:is a manifold (or a theory), and 7067:on the triangles in the building. 7056:, invariant under conjugation by 6910:is scalar multiplication by 6377:discovered the first example for 6216:' correspond to the simplices in 4722:, so that there is an injection ψ 4161:τ corresponding to intersections 3451:.) It is also useful to consider 2774:has a natural orbifold structure. 2047:on a Hausdorff topological space 1245:{\displaystyle g\circ \psi _{ij}} 786:{\displaystyle V_{i}/\Gamma _{i}} 11550:"The Geometry of Musical Chords" 11548:Tymoczko, Dmitri (7 July 2006). 11482:Quarterly Journal of Mathematics 9535: 9525: 9484: 9462: 9118:, the construction of realistic 8858: 6930:generate a discrete subgroup of 6286:simplicial approximation theorem 6148:as well as the cocycle relation 3259:{\displaystyle \mathbb {R} ^{n}} 2985:and gluing back copies of discs 2829:invariant under the stabiliser Γ 2219:by the equivalent relation when 1278:{\displaystyle g\in \Gamma _{j}} 609:linear action of a finite group 598:{\displaystyle \mathbb {R} ^{n}} 426:{\displaystyle \mathbb {R} ^{n}} 397:{\displaystyle \mathbb {R} ^{n}} 344:{\displaystyle \mathbb {Z} _{2}} 272:{\displaystyle \mathbb {Z} _{2}} 13107:Eleven-dimensional supergravity 11540:10.1090/S0273-0979-1982-15003-0 11338:"The geometries of 3-manifolds" 11192:American Journal of Mathematics 11160:Journal of Combinatorial Theory 10520:"Mazzola's Counterpoint Theory" 10511: 10493: 10474: 10458: 10436: 10421: 10406: 10267: 10238:Basic Concepts of String Theory 10228: 10215: 10195: 10182: 10173: 10162: 10036: 9992: 9847:Kawasaki's Riemann–Roch formula 9732:{\displaystyle t=1+1+\cdots +1} 9226: 8934: 7281:(ζ) and can be identified with 7202:Mumford also obtains an action 7089:The stabiliser of Δ is trivial. 6212:of the barycentric subdivision 5753:extends to a simplicial map of 5587: 5292:is canonically defined: it has 367:Definition using orbifold atlas 108:in the 1980s in the context of 11675:Differentiable/Smooth manifold 10979:; de la Harpe, Pierre (eds.). 10908:An invitation to Coxeter group 10575:; de La Harpe, Pierre (eds.). 10542:Orbifolds and Stringy Topology 10471:Harvard Magazine, Jan/Feb 2007 8130: 8117: 8090: 8063: 8039: 8012: 7996: 7983: 7552:in the link of a fixed vertex 6583: 6040: 6018: 5892: 5838: 5799:, the quotient of the star of 5519: 5493: 5467: 5438: 5130:admits a finite subcomplex as 5084: 4897: 4875: 4740: 4706: 4559: 4523: 4487: 4397: 4328: 4283: 3991: 3909:A complex of groups is called 3823:complex of groups by defining 3797:(This condition is vacuous if 3574: 2670: 2614: 2592: 2568: 2559: 2545: 2322: 2316: 2287: 2281: 2165: 2149: 2121: 2105: 2097:together with a homeomorphism 2081: 2071:class of an orbifold groupoid 1962: 1946: 1934: 1855: 1849: 1830: 1824: 1799: 1785: 1662: 1569:Definition using Lie groupoids 911: 226:action of a possibly infinite 163: 149: 1: 12555:Second superstring revolution 11063:10.1090/S0002-9947-10-05006-3 10532: 10488:American Mathematical Society 10223:Lectures On Complex Manifolds 10010:American Mathematical Society 9406:points on the circle) by the 7444:simply transitive on vertices 6487:. The three ways of mapping Γ 5858:', an injective homomorphism 5428:an orientation for all edges 3756:for every chain of simplices 3515: 3425: 2932:. If the triangle has angles 2853:under the exponential map at 2134:{\displaystyle |M/G|\simeq X} 1579:consists of a set of objects 1490:(guaranteeing associativity) 13049:Generalized complex manifold 12550:First superstring revolution 11224:Papers on Fuchsian functions 11174:10.1016/0097-3165(85)90070-6 10990:10.1007/978-1-4684-9167-8_11 10902:de la Harpe, Pierre (1991). 10300:10.1016/0550-3213(85)90593-0 9960:Bridson & Haefliger 1999 8258:Orders of corner reflectors 8136:{\displaystyle \chi (X_{0})} 7923:A corner reflector of order 7192:thus "lift" the formulas on 6690:{\displaystyle {\sqrt {-7}}} 6351:'s lectures on trees, where 6046:{\displaystyle \rightarrow } 6024:{\displaystyle \rightarrow } 5898:{\displaystyle \rightarrow } 5844:{\displaystyle \rightarrow } 5669:', identifying the quotient 5565: 5525:{\displaystyle \rightarrow } 5499:{\displaystyle \rightarrow } 5473:{\displaystyle \rightarrow } 5444:{\displaystyle \rightarrow } 5090:{\displaystyle \rightarrow } 4903:{\displaystyle \rightarrow } 4881:{\displaystyle \rightarrow } 4746:{\displaystyle \rightarrow } 4712:{\displaystyle \rightarrow } 4688:corresponding to edges from 4565:{\displaystyle \rightarrow } 4529:{\displaystyle \rightarrow } 4493:{\displaystyle \rightarrow } 4403:{\displaystyle \rightarrow } 4334:{\displaystyle \rightarrow } 4289:{\displaystyle \rightarrow } 3997:{\displaystyle \rightarrow } 3580:{\displaystyle \rightarrow } 3480:universal covering orbispace 3124:The orbifold is developable. 2417:{\displaystyle \varphi _{i}} 2021:a proper étale Lie groupoid; 659:{\displaystyle \varphi _{i}} 88:in the 1950s under the name 7: 13705:Group actions (mathematics) 12381:Classification of manifolds 11155:and its finite projections" 10710:10.4007/annals.2005.162.195 9825: 9354:{\displaystyle T^{n}/S_{n}} 9292:{\displaystyle T^{3}/S_{3}} 8890:A 3-manifold is said to be 7179:cyclic multiplicative group 6947:affine Bruhat–Tits building 6357:affine Bruhat–Tits building 6296:can be identified with its 6223:Every complex of groups on 5774:, carrying the centre onto 5112:of a discrete group Γ on a 4632:associated to the covering 3538:abstract simplicial complex 3476:orbispace fundamental group 3467:connecting any two points. 3408:{\displaystyle C^{\infty }} 3381:{\displaystyle C^{\infty }} 3354:{\displaystyle C^{\infty }} 3100:universal covering orbifold 3086:The orbifold is said to be 2713: 2471:{\displaystyle \Gamma _{i}} 1923:. A Lie groupoid is called 997:{\displaystyle \Gamma _{i}} 740:{\displaystyle \Gamma _{i}} 629:{\displaystyle \Gamma _{i}} 466:Hausdorff topological space 10: 13721: 12647:Non-critical string theory 10546:Cambridge University Press 10054:Cambridge University Press 9107: 8940:Orbifolds in string theory 8625: 8357: 8255:Orders of elliptic points 7784:), it follows that σ = 1. 6305:canonical metric structure 5730:' (the simplicial join of 5254:if it arises in this way. 3958: 3801:has dimension 2 or less.) 3557:an injective homomorphism 3020:orbifold fundamental group 3014:Orbifold fundamental group 2816:and an open neighbourhood 1771:{\displaystyle x\in G_{0}} 1293:, two orbifold atlases of 1031:{\displaystyle \psi _{ij}} 13191: 13168: 13145: 13092: 12977: 12885: 12827: 12759: 12708: 12675: 12570: 12527: 12457:over commutative algebras 12414: 12373: 12306: 12203: 12099: 12046: 12037: 11873: 11796: 11735: 11655: 11416:10.1007/978-3-642-61856-7 11126:Kawakubo, Katsuo (1991). 11008:Haefliger, André (1991). 10966:Haefliger, André (1984). 10945:10.1007/978-1-4612-2256-9 10762:10.1007/978-3-662-12494-9 10321:Mazzola, Guerino (1985). 8262: 7893:Two-dimensional orbifolds 7803:When all the elements of 7034:generate a subgroup Γ of 6874:is the operator given by 6353:amalgamated free products 6056:, then there is a unique 5413:' of a complex of groups 4472:and gluing maps ψ : 4114:become the homomorphisms 4024:-simplices correspond to 3135:Orbifolds as diffeologies 2879:with corresponding group 1385:, then there is a unique 13183:Introduction to M-theory 12877:Wess–Zumino–Witten model 12819:Hanany–Witten transition 12545:History of string theory 12173:Riemann curvature tensor 11448:Stallings, John (1991). 11431:English translation of: 10554:10.1017/CBO9780511543081 10062:10.1017/cbo9780511615450 9872: 9092:vacuum expectation value 9080:vacuum expectation value 8960:by a finite group, i.e. 7912:, such as the origin of 7238:)-valued Hermitian form 6842:is the generator of the 6733:{\displaystyle \subset } 6701:of (1 − 8) in 6509:amalgamated free product 6467:for the triangle itself. 5616:', a simplicial complex 4923:, the edge-path group Γ( 4154:{\displaystyle \subset } 4134:{\displaystyle \subset } 3647:{\displaystyle \subset } 3627:{\displaystyle \subset } 3604:{\displaystyle \subset } 3147:'s original definition: 3104:universal covering space 1370:{\displaystyle \subset } 1343:{\displaystyle \subset } 12862:Vertex operator algebra 12562:String theory landscape 11589:10.1126/science.1126287 11495:10.1093/qjmath/52.2.231 11182:Mumford, David (1979). 11132:Oxford University Press 10826:10.1126/science.1153021 10046:; Mrcun, Janez (2003). 9120:phenomenological models 8851:3-dimensional orbifolds 7865:bipartite Heawood graph 7663:is made up of vertices 7604:bipartite Heawood graph 6868:, an element of order 7 6396:transitive on triangles 6208:, because the vertices 5914:equivariant simplicial 5882:for each directed edge 5828:for each directed edge 5604:structure consists of: 3945:barycentric subdivision 3804:Any choice of elements 2908:A classical theorem of 2627:is injective for every 2238:{\displaystyle x\sim y} 1551:differentiable orbifold 1069:onto an open subset of 131:with the action of the 45:23star Orbifold Example 13160:AdS/CFT correspondence 12915:Exceptional Lie groups 12857:Superconformal algebra 12829:Conformal field theory 12700:Montonen–Olive duality 12652:Non-linear sigma model 11965:Manifold with boundary 11680:Differential structure 10936:Conformal field theory 10276:"Strings on orbifolds" 9791: 9759: 9733: 9650: 9649:{\displaystyle S_{t},} 9620: 9543: 9498: 9430: 9393: 9355: 9312: 9293: 9207: 8252:Underlying 2-manifold 8218: 8191: 8164: 8137: 8097: 7958: 7675:in a symmetric subset 7606: 7302: 7147:Frobenius automorphism 6734: 6691: 6663: 6603: 6406: 6389:, but having the same 6341:geometric group theory 6047: 6025: 5899: 5845: 5747:of the cone. The map φ 5743:'), fixing the centre 5526: 5500: 5474: 5445: 5199:) are simplices, then 5091: 4904: 4882: 4747: 4713: 4566: 4530: 4494: 4458: 4431: 4404: 4362: 4335: 4290: 4250: 4155: 4135: 4105:and the homomorphisms 4082: 4062: 3998: 3791: 3648: 3628: 3614:for every inclusion ρ 3605: 3581: 3551:for each simplex σ of 3432:geometric group theory 3409: 3382: 3355: 3326:a finite linear group. 3320: 3300: 3260: 3224: 3204: 3184: 2945:for positive integers 2808:take a representative 2738:manifold with boundary 2700: 2680: 2647: 2646:{\displaystyle x\in X} 2621: 2526: 2499: 2472: 2445: 2418: 2391: 2367: 2335: 2334:{\displaystyle t(g)=y} 2300: 2299:{\displaystyle s(g)=x} 2265: 2264:{\displaystyle g\in G} 2239: 2213: 2199:(i.e. the quotient of 2193: 2173: 2135: 2091: 2061: 1992: 1917: 1897: 1862: 1772: 1739: 1712: 1679: 1627: 1600: 1371: 1344: 1308: 1279: 1246: 1209: 1181: 1151: 1091: 1063: 1032: 998: 971: 928: 866: 814: 787: 741: 714: 687: 660: 630: 599: 570: 539: 512: 481: 454: 427: 398: 345: 300: 273: 244:manifold with boundary 224:properly discontinuous 198:geometric group theory 188:theory, the theory of 170: 46: 30: 13700:Generalized manifolds 13695:Differential topology 13155:Holographic principle 13122:Type IIB supergravity 13117:Type IIA supergravity 12969:-form electrodynamics 12588:Bosonic string theory 11438:Arbres, amalgames, SL 11357:10.1112/blms/15.5.401 11325:10.2969/jmsj/00940464 11277:10.1073/pnas.42.6.359 10688:Annals of Mathematics 10445:The Geometry of Music 10442:Michael D. Lemonick, 10415:The Geometry of Music 10085:Iglesias-Zemmour 2013 9862:Ring of modular forms 9792: 9790:{\displaystyle 3=2+1} 9760: 9734: 9656:yielding an orbifold. 9651: 9621: 9544: 9499: 9431: 9429:{\displaystyle S_{n}} 9394: 9392:{\displaystyle T^{n}} 9356: 9294: 9256: 9208: 9124:dimensional reduction 9045:conical singularities 8249:Euler characteristic 8219: 8217:{\displaystyle m_{i}} 8192: 8190:{\displaystyle n_{i}} 8165: 8163:{\displaystyle X_{0}} 8138: 8098: 7959: 7957:{\displaystyle \chi } 7904:An elliptic point or 7829:of order 7 such that 7601: 7303: 7301:{\displaystyle \cap } 6831:-linear operators on 6735: 6692: 6659: 6604: 6405: 6316:non-positively curved 6048: 6026: 5900: 5846: 5527: 5501: 5475: 5446: 5355:Indeed, simplices in 5104:Developable complexes 5092: 4905: 4883: 4748: 4714: 4567: 4531: 4495: 4459: 4457:{\displaystyle \cap } 4432: 4430:{\displaystyle \cap } 4405: 4363: 4361:{\displaystyle \cap } 4336: 4291: 4251: 4156: 4136: 4121:. For every triple ρ 4083: 4081:{\displaystyle \cap } 4063: 4061:{\displaystyle \cap } 4018:nerve of the covering 3999: 3792: 3693:(here Ad denotes the 3649: 3629: 3606: 3582: 3484:non-positively curved 3410: 3383: 3356: 3321: 3301: 3261: 3225: 3205: 3185: 2701: 2681: 2648: 2622: 2527: 2525:{\displaystyle G_{X}} 2500: 2498:{\displaystyle V_{i}} 2473: 2446: 2444:{\displaystyle G_{X}} 2424:. In turn, the space 2419: 2392: 2368: 2336: 2301: 2266: 2240: 2214: 2194: 2174: 2172:{\displaystyle |M/G|} 2136: 2092: 2062: 2008:local diffeomorphisms 1993: 1918: 1898: 1896:{\displaystyle G_{1}} 1863: 1773: 1740: 1738:{\displaystyle G_{1}} 1713: 1711:{\displaystyle G_{0}} 1680: 1628: 1626:{\displaystyle G_{1}} 1601: 1599:{\displaystyle G_{0}} 1372: 1345: 1309: 1280: 1247: 1210: 1208:{\displaystyle V_{j}} 1182: 1180:{\displaystyle V_{i}} 1152: 1092: 1090:{\displaystyle V_{j}} 1064: 1062:{\displaystyle V_{i}} 1033: 999: 972: 929: 867: 815: 813:{\displaystyle U_{i}} 788: 742: 715: 713:{\displaystyle U_{i}} 688: 686:{\displaystyle V_{i}} 661: 631: 600: 571: 569:{\displaystyle V_{i}} 540: 538:{\displaystyle U_{i}} 513: 511:{\displaystyle U_{i}} 482: 455: 428: 399: 346: 301: 274: 171: 44: 21: 13074:Hořava–Witten theory 13021:Hyperkähler manifold 12709:Particles and fields 12657:Tachyon condensation 12642:Matrix string theory 12112:Covariant derivative 11663:Topological manifold 11410:. Berlin: Springer. 11392:Cours d'arithmétique 10645:on 30 September 2011 10145:Iglesias et al. 2010 10109:Iglesias et al. 2010 10056:. pp. 140–144. 10012:. pp. 205–222. 9769: 9743: 9699: 9630: 9568: 9508: 9458: 9413: 9376: 9323: 9261: 9172: 9104:Calabi–Yau manifolds 8989:quantum field theory 8201: 8174: 8147: 8111: 7971: 7948: 7942:Euler characteristic 7292: 6724: 6674: 6530: 6515:of the edge groups Γ 6259:. Since the link of 6037: 6015: 5889: 5835: 5516: 5490: 5464: 5435: 5081: 4894: 4872: 4737: 4703: 4556: 4520: 4484: 4448: 4421: 4394: 4352: 4325: 4280: 4168: 4145: 4125: 4072: 4052: 3988: 3760: 3638: 3618: 3595: 3571: 3430:For applications in 3392: 3365: 3338: 3310: 3273: 3241: 3214: 3194: 3157: 2954:, the triangle is a 2690: 2664: 2631: 2536: 2509: 2482: 2455: 2428: 2401: 2381: 2357: 2343:differentiable stack 2310: 2275: 2249: 2223: 2203: 2183: 2145: 2101: 2075: 2051: 1931: 1907: 1880: 1782: 1749: 1722: 1695: 1637: 1610: 1583: 1361: 1334: 1298: 1291:atlases on manifolds 1256: 1220: 1192: 1164: 1105: 1074: 1046: 1012: 981: 941: 882: 836: 797: 755: 724: 697: 670: 643: 613: 605:, invariant under a 580: 553: 522: 495: 471: 444: 408: 379: 355:Euler characteristic 326: 299:{\displaystyle \pi } 290: 254: 190:Seifert fiber spaces 182:Riemann–Roch theorem 138: 16:Generalized manifold 13112:Type I supergravity 13016:Calabi–Yau manifold 13011:Ricci-flat manifold 12990:Kaluza–Klein theory 12731:Ramond–Ramond field 12637:String field theory 12146:Exterior derivative 11748:Atiyah–Singer index 11697:Riemannian manifold 11571:2006Sci...313...72T 11454:Triangles of groups 11268:1956PNAS...42..359S 10860:Geometriae Dedicata 10818:2008Sci...320..346C 10601:Geometriae Dedicata 10464:Elizabeth Gudrais, 10292:1985NuPhB.261..678D 10247:2013bcst.book.....B 9867:Stack (mathematics) 9758:{\displaystyle t=t} 9140:Calabi–Yau manifold 9110:Calabi–Yau manifold 7811:is taken to be the 7544:label the vertices 7522:a set of relations 7394:congruence subgroup 7065:simply transitively 6963:). This group acts 6429:. There are groups 6345:triangles of groups 6335:Triangles of groups 4916:For a fixed vertex 3654:τ, a group element 3504:Complexes of groups 3058:of orbifold loops. 2478:acts faithfully on 1555:Riemannian orbifold 937:for each inclusion 832:for each inclusion 180:: a version of the 69:which is locally a 34:Thurston (1978–1981 13079:K-theory (physics) 12956:ADE classification 12593:Superstring theory 12452:Secondary calculus 12406:Singularity theory 12361:Parallel transport 12129:De Rham cohomology 11768:Generalized Stokes 11434:Serre, Jean-Pierre 11400:Serre, Jean-Pierre 11388:Serre, Jean-Pierre 10874:10.1007/BF01266617 10615:10.1007/BF01265309 10518:Tymoczko, Dmitri. 10190:Superstring theory 9900:Thurston 1978–1981 9842:Geometric quotient 9787: 9755: 9729: 9646: 9616: 9539: 9494: 9426: 9389: 9351: 9313: 9289: 9203: 9132:compactified space 9116:superstring theory 9076:degrees of freedom 9072:degrees of freedom 8870:. You can help by 8214: 8187: 8160: 8133: 8093: 8062: 8011: 7954: 7883:Euclidean building 7607: 7426:), the action of Γ 7400:is defined as the 7298: 7121:collineation group 6730: 6699:binomial expansion 6687: 6664: 6599: 6411:triangle of groups 6407: 6385:not isomorphic to 6251:, all the groups Γ 6058:transition element 6043: 6021: 5895: 5841: 5634:a simplicial map φ 5582:simplicial complex 5522: 5496: 5470: 5441: 5132:fundamental domain 5114:simplicial complex 5087: 4900: 4878: 4743: 4709: 4679:, take generators 4562: 4526: 4490: 4454: 4427: 4400: 4358: 4331: 4286: 4246: 4151: 4131: 4078: 4058: 3994: 3787: 3644: 3624: 3601: 3577: 3405: 3378: 3351: 3316: 3296: 3256: 3220: 3210:is an integer and 3200: 3180: 3115:local homomorphism 3028:classifying spaces 2956:fundamental domain 2696: 2676: 2655:classical orbifold 2643: 2617: 2522: 2495: 2468: 2441: 2414: 2387: 2373:, one can build a 2363: 2331: 2296: 2261: 2235: 2209: 2189: 2169: 2131: 2087: 2069:Morita equivalence 2067:is defined as the 2057: 2045:orbifold structure 1988: 1913: 1893: 1858: 1768: 1735: 1708: 1675: 1623: 1606:, a set of arrows 1596: 1559:Riemannian metrics 1387:transition element 1367: 1340: 1317:orbifold structure 1304: 1275: 1242: 1205: 1177: 1147: 1087: 1059: 1028: 994: 967: 924: 877:group homomorphism 862: 810: 783: 737: 710: 683: 656: 626: 595: 566: 535: 508: 477: 450: 435:isotropy subgroups 423: 394: 362:Formal definitions 341: 296: 269: 236:isotropy subgroups 166: 84:in the context of 47: 13682: 13681: 13464:van Nieuwenhuizen 13000:Why 10 dimensions 12905:Chern–Simons form 12872:Kac–Moody algebra 12852:Conformal algebra 12847:Conformal anomaly 12741:Magnetic monopole 12736:Kalb–Ramond field 12578:Nambu–Goto action 12487: 12486: 12369: 12368: 12134:Differential form 11788:Whitney embedding 11722:Differential form 11517:Thurston, William 11503:Thurston, William 11425:978-3-642-61858-1 11089:978-0-8218-9131-5 10812:(5874): 346–348. 10631:Boileau, Michel. 10412:Dmitri Tymoczko, 10368:978-3-7643-5731-3 10334:978-3-88538-210-2 10280:Nuclear Physics B 10135:, Footnote p.469. 10071:978-0-521-83197-0 10029:978-0-8218-2990-5 9852:Orbifold notation 9832:Branched covering 9803:equal temperament 8956:to a quotient of 8898:Orbifold Theorem. 8888: 8887: 8848: 8847: 8791:Projective plane 8614:Projective plane 8599:Projective plane 8053: 8002: 7515:a generating set 7226:>. The group Γ 7204:simply transitive 6949:corresponding to 6685: 6652:Mumford's example 6383:algebraic surface 6290:fundamental group 5296:-simplices (σ, xΓ 5036:is an edge-path, 4259:there are charts 3522:complex of groups 3510:complex of groups 3319:{\displaystyle G} 3223:{\displaystyle G} 3203:{\displaystyle n} 3040:fundamental group 2922:geodesic triangle 2918:reflection groups 2785:-manifold with a 2699:{\displaystyle G} 2390:{\displaystyle X} 2366:{\displaystyle X} 2212:{\displaystyle M} 2192:{\displaystyle G} 2060:{\displaystyle X} 2015:orbifold groupoid 1916:{\displaystyle x} 1544:complex of groups 1307:{\displaystyle X} 639:a continuous map 480:{\displaystyle X} 453:{\displaystyle n} 315:fundamental group 86:automorphic forms 67:topological space 13712: 13192:String theorists 13132:Lie superalgebra 13084:Twisted K-theory 13042:Spin(7)-manifold 12995:Compactification 12837:Virasoro algebra 12620:Heterotic string 12514: 12507: 12500: 12491: 12490: 12479:Stratified space 12437:Fréchet manifold 12151:Interior product 12044: 12043: 11741: 11637: 11630: 11623: 11614: 11613: 11608: 11582: 11554: 11544: 11542: 11512: 11498: 11475: 11444: 11429: 11406:. Translated by 11395: 11383: 11381: 11368: 11342: 11329: 11327: 11299: 11289: 11279: 11241: 11226:. Translated by 11215: 11178: 11176: 11145: 11122: 11111:10.4171/JNCG/319 11105:(4): 1551–1572. 11093: 11074: 11056: 11047:(6): 2811–2831. 11035: 11004: 10992: 10971: 10959: 10947: 10929: 10897: 10878: 10876: 10845: 10801: 10791: 10789: 10776: 10764: 10747: 10721: 10703: 10681: 10654: 10652: 10650: 10644: 10638:. Archived from 10637: 10627: 10617: 10590: 10567: 10527: 10526: 10524: 10515: 10509: 10508: 10506: 10497: 10491: 10478: 10472: 10462: 10456: 10440: 10434: 10425: 10419: 10410: 10404: 10398: 10392: 10386: 10380: 10379: 10377: 10375: 10352: 10346: 10345: 10343: 10341: 10318: 10312: 10311: 10271: 10265: 10263: 10232: 10226: 10219: 10213: 10199: 10193: 10186: 10180: 10177: 10171: 10166: 10160: 10154: 10148: 10142: 10136: 10130: 10124: 10118: 10112: 10106: 10100: 10094: 10088: 10082: 10076: 10075: 10040: 10034: 10033: 10021: 9996: 9990: 9984: 9975: 9969: 9963: 9957: 9951: 9945: 9939: 9933: 9927: 9921: 9915: 9909: 9903: 9897: 9891: 9885: 9807:augmented triads 9796: 9794: 9793: 9788: 9764: 9762: 9761: 9756: 9738: 9736: 9735: 9730: 9687: 9685: 9684: 9681: 9678: 9655: 9653: 9652: 9647: 9642: 9641: 9625: 9623: 9622: 9617: 9612: 9611: 9593: 9592: 9580: 9579: 9548: 9546: 9545: 9540: 9538: 9533: 9528: 9520: 9519: 9503: 9501: 9500: 9495: 9493: 9492: 9487: 9478: 9477: 9465: 9435: 9433: 9432: 9427: 9425: 9424: 9398: 9396: 9395: 9390: 9388: 9387: 9360: 9358: 9357: 9352: 9350: 9349: 9340: 9335: 9334: 9305:augmented triads 9298: 9296: 9295: 9290: 9288: 9287: 9278: 9273: 9272: 9212: 9210: 9209: 9204: 9201: 9200: 9195: 9189: 9184: 9183: 8917:orbifold theorem 8883: 8880: 8862: 8855: 8243: 8242: 8238:wallpaper groups 8223: 8221: 8220: 8215: 8213: 8212: 8196: 8194: 8193: 8188: 8186: 8185: 8169: 8167: 8166: 8161: 8159: 8158: 8142: 8140: 8139: 8134: 8129: 8128: 8102: 8100: 8099: 8094: 8089: 8088: 8079: 8061: 8046: 8038: 8037: 8028: 8010: 7995: 7994: 7963: 7961: 7960: 7955: 7927:: the origin of 7901:A boundary point 7724:. Thus, if σ = τ 7461:projective plane 7307: 7305: 7304: 7299: 6756: 6739: 6737: 6736: 6731: 6697:be given by the 6696: 6694: 6693: 6688: 6686: 6678: 6627:hyperbolic plane 6608: 6606: 6605: 6600: 6595: 6594: 6582: 6581: 6569: 6568: 6567: 6566: 6545: 6544: 6387:projective space 6309:hyperbolic space 6052: 6050: 6049: 6044: 6030: 6028: 6027: 6022: 5904: 5902: 5901: 5896: 5850: 5848: 5847: 5842: 5818:orbihedron chart 5697:This action of Γ 5608:for each vertex 5547:) is a triangle; 5535:is an edge and ( 5531: 5529: 5528: 5523: 5509:are edges, then 5505: 5503: 5502: 5497: 5479: 5477: 5476: 5471: 5450: 5448: 5447: 5442: 5096: 5094: 5093: 5088: 4909: 4907: 4906: 4901: 4887: 4885: 4884: 4879: 4752: 4750: 4749: 4744: 4718: 4716: 4715: 4710: 4650:sheaf cohomology 4618: 4571: 4569: 4568: 4563: 4535: 4533: 4532: 4527: 4499: 4497: 4496: 4491: 4463: 4461: 4460: 4455: 4436: 4434: 4433: 4428: 4409: 4407: 4406: 4401: 4367: 4365: 4364: 4359: 4340: 4338: 4337: 4332: 4295: 4293: 4292: 4287: 4255: 4253: 4252: 4247: 4245: 4244: 4232: 4231: 4219: 4218: 4206: 4205: 4193: 4192: 4180: 4179: 4160: 4158: 4157: 4152: 4140: 4138: 4137: 4132: 4087: 4085: 4084: 4079: 4067: 4065: 4064: 4059: 4003: 4001: 4000: 3995: 3922:= 1 everywhere. 3796: 3794: 3793: 3788: 3748: 3737: 3711: 3653: 3651: 3650: 3645: 3633: 3631: 3630: 3625: 3610: 3608: 3607: 3602: 3586: 3584: 3583: 3578: 3547:a finite group Γ 3414: 3412: 3411: 3406: 3404: 3403: 3387: 3385: 3384: 3379: 3377: 3376: 3360: 3358: 3357: 3352: 3350: 3349: 3325: 3323: 3322: 3317: 3305: 3303: 3302: 3297: 3292: 3287: 3286: 3281: 3265: 3263: 3262: 3257: 3255: 3254: 3249: 3229: 3227: 3226: 3221: 3209: 3207: 3206: 3201: 3189: 3187: 3186: 3181: 3176: 3171: 3170: 3165: 3076: 3063:simply connected 3056:homotopy classes 2935: 2926:hyperbolic plane 2781:is a Riemannian 2762:on the manifold 2705: 2703: 2702: 2697: 2685: 2683: 2682: 2677: 2652: 2650: 2649: 2644: 2626: 2624: 2623: 2618: 2613: 2612: 2591: 2590: 2585: 2567: 2566: 2557: 2556: 2531: 2529: 2528: 2523: 2521: 2520: 2504: 2502: 2501: 2496: 2494: 2493: 2477: 2475: 2474: 2469: 2467: 2466: 2450: 2448: 2447: 2442: 2440: 2439: 2423: 2421: 2420: 2415: 2413: 2412: 2396: 2394: 2393: 2388: 2372: 2370: 2369: 2364: 2340: 2338: 2337: 2332: 2305: 2303: 2302: 2297: 2270: 2268: 2267: 2262: 2244: 2242: 2241: 2236: 2218: 2216: 2215: 2210: 2198: 2196: 2195: 2190: 2178: 2176: 2175: 2170: 2168: 2160: 2152: 2140: 2138: 2137: 2132: 2124: 2116: 2108: 2096: 2094: 2093: 2088: 2066: 2064: 2063: 2058: 1997: 1995: 1994: 1989: 1987: 1986: 1974: 1973: 1961: 1960: 1922: 1920: 1919: 1914: 1902: 1900: 1899: 1894: 1892: 1891: 1867: 1865: 1864: 1859: 1848: 1847: 1823: 1822: 1807: 1806: 1797: 1796: 1777: 1775: 1774: 1769: 1767: 1766: 1744: 1742: 1741: 1736: 1734: 1733: 1717: 1715: 1714: 1709: 1707: 1706: 1684: 1682: 1681: 1676: 1674: 1673: 1661: 1660: 1632: 1630: 1629: 1624: 1622: 1621: 1605: 1603: 1602: 1597: 1595: 1594: 1488:cocycle relation 1376: 1374: 1373: 1368: 1349: 1347: 1346: 1341: 1313: 1311: 1310: 1305: 1284: 1282: 1281: 1276: 1274: 1273: 1251: 1249: 1248: 1243: 1241: 1240: 1214: 1212: 1211: 1206: 1204: 1203: 1186: 1184: 1183: 1178: 1176: 1175: 1156: 1154: 1153: 1148: 1146: 1145: 1133: 1132: 1117: 1116: 1096: 1094: 1093: 1088: 1086: 1085: 1068: 1066: 1065: 1060: 1058: 1057: 1037: 1035: 1034: 1029: 1027: 1026: 1003: 1001: 1000: 995: 993: 992: 976: 974: 973: 968: 966: 965: 953: 952: 933: 931: 930: 925: 923: 922: 910: 909: 897: 896: 871: 869: 868: 863: 861: 860: 848: 847: 819: 817: 816: 811: 809: 808: 792: 790: 789: 784: 782: 781: 772: 767: 766: 746: 744: 743: 738: 736: 735: 720:invariant under 719: 717: 716: 711: 709: 708: 692: 690: 689: 684: 682: 681: 665: 663: 662: 657: 655: 654: 635: 633: 632: 627: 625: 624: 604: 602: 601: 596: 594: 593: 588: 575: 573: 572: 567: 565: 564: 544: 542: 541: 536: 534: 533: 517: 515: 514: 509: 507: 506: 489:underlying space 486: 484: 483: 478: 459: 457: 456: 451: 432: 430: 429: 424: 422: 421: 416: 403: 401: 400: 395: 393: 392: 387: 350: 348: 347: 342: 340: 339: 334: 305: 303: 302: 297: 278: 276: 275: 270: 268: 267: 262: 250:by an action of 178:upper half-plane 175: 173: 172: 167: 162: 148: 112:'s programme on 94:William Thurston 37: 13720: 13719: 13715: 13714: 13713: 13711: 13710: 13709: 13685: 13684: 13683: 13678: 13187: 13164: 13141: 13088: 13036: 13006:Kähler manifold 12973: 12950: 12943: 12936: 12929: 12922: 12881: 12842:Mirror symmetry 12823: 12809:Brane cosmology 12755: 12704: 12671: 12627:N=2 superstring 12613:Type IIB string 12608:Type IIA string 12583:Polyakov action 12566: 12523: 12518: 12488: 12483: 12422:Banach manifold 12415:Generalizations 12410: 12365: 12302: 12199: 12161:Ricci curvature 12117:Cotangent space 12095: 12033: 11875: 11869: 11828:Exponential map 11792: 11737: 11731: 11651: 11641: 11611: 11580:10.1.1.215.7449 11565:(5783): 72–74. 11552: 11472: 11441: 11430: 11426: 11408:Stillwell, John 11379: 11369: 11340: 11238: 11228:Stillwell, John 11220:Poincaré, Henri 11204:10.2307/2373947 11187: 11154: 11142: 11090: 11032: 11001: 10956: 10926: 10894: 10854: 10799: 10773: 10744: 10670: 10648: 10646: 10642: 10635: 10587: 10564: 10535: 10530: 10522: 10516: 10512: 10504: 10498: 10494: 10480:Tony Phillips, 10479: 10475: 10463: 10459: 10455:26 January 2007 10441: 10437: 10426: 10422: 10411: 10407: 10399: 10395: 10387: 10383: 10373: 10371: 10369: 10353: 10349: 10339: 10337: 10335: 10319: 10315: 10272: 10268: 10257: 10233: 10229: 10220: 10216: 10201:J. Polchinski, 10200: 10196: 10187: 10183: 10178: 10174: 10167: 10163: 10155: 10151: 10143: 10139: 10131: 10127: 10119: 10115: 10107: 10103: 10095: 10091: 10083: 10079: 10072: 10041: 10037: 10030: 9997: 9993: 9985: 9978: 9970: 9966: 9958: 9954: 9946: 9942: 9934: 9930: 9922: 9918: 9910: 9906: 9898: 9894: 9886: 9879: 9875: 9828: 9809:(thought of as 9770: 9767: 9766: 9744: 9741: 9740: 9700: 9697: 9696: 9682: 9679: 9676: 9675: 9673: 9637: 9633: 9631: 9628: 9627: 9607: 9603: 9588: 9584: 9575: 9571: 9569: 9566: 9565: 9534: 9529: 9524: 9515: 9511: 9509: 9506: 9505: 9488: 9483: 9482: 9473: 9469: 9461: 9459: 9456: 9455: 9420: 9416: 9414: 9411: 9410: 9408:symmetric group 9383: 9379: 9377: 9374: 9373: 9361:– the space of 9345: 9341: 9336: 9330: 9326: 9324: 9321: 9320: 9283: 9279: 9274: 9268: 9264: 9262: 9259: 9258: 9241:Dmitri Tymoczko 9237:Guerino Mazzola 9229: 9221:mirror symmetry 9196: 9191: 9190: 9185: 9179: 9175: 9173: 9170: 9169: 9151:singular points 9112: 9106: 9068: 9056: 9022:quiver diagrams 8942: 8937: 8884: 8878: 8875: 8868:needs expansion 8853: 8208: 8204: 8202: 8199: 8198: 8181: 8177: 8175: 8172: 8171: 8154: 8150: 8148: 8145: 8144: 8124: 8120: 8112: 8109: 8108: 8084: 8080: 8075: 8057: 8042: 8033: 8029: 8024: 8006: 7990: 7986: 7972: 7969: 7968: 7949: 7946: 7945: 7895: 7888: 7880: 7873: 7825:and an element 7820: 7688: 7436: 7434:Generalizations 7429: 7425: 7418: 7399: 7387: 7380: 7373: 7365: 7358: 7351: 7344: 7340: 7313: 7293: 7290: 7289: 7287: 7229: 7209: 7198: 7187: 7176: 7166: 7144: 7118: 7111: 7084:Frobenius group 7078: 7040: 7033: 7026: 7019: 6998: 6987: 6976: 6962: 6955: 6936: 6754: 6745: 6725: 6722: 6721: 6707: 6677: 6675: 6672: 6671: 6654: 6631:Poincaré metric 6590: 6586: 6574: 6570: 6556: 6552: 6550: 6546: 6537: 6533: 6531: 6528: 6527: 6522: 6518: 6514: 6506: 6494: 6490: 6486: 6482: 6478: 6474: 6466: 6459: 6455: 6451: 6444: 6440: 6436: 6372: 6365: 6337: 6324:Hadamard spaces 6298:edge-path group 6270: 6258: 6254: 6201: 6199:Main properties 6193: 6184: 6175: 6166: 6157: 6144: 6135: 6126: 6117: 6101: 6095: 6089: 6083: 6071: 6065: 6038: 6035: 6034: 6016: 6013: 6012: 6003: 5993: 5983: 5974: 5962: 5956: 5950: 5941: 5932: 5923: 5913: 5890: 5887: 5886: 5878: 5872: 5866: 5836: 5833: 5832: 5816:) and gives an 5811: 5798: 5792: 5783: 5761: 5752: 5742: 5729: 5720: 5711: 5702: 5692: 5683: 5677: 5660: 5648: 5639: 5630: 5624: 5590: 5568: 5517: 5514: 5513: 5491: 5488: 5487: 5465: 5462: 5461: 5436: 5433: 5432: 5396:category theory 5343:whenever σ and 5326: 5315: 5312:runs over Γ / Γ 5300:) where σ is a 5299: 5287: 5283: 5276: 5272: 5268: 5261: 5229: 5220: 5211: 5198: 5189: 5180: 5173: 5166: 5157: 5106: 5082: 5079: 5078: 5073: 5064: 5055: 5054: 5044: 5035: 5028: 5019: 5012: 5002: 5001: 4995: 4985: 4976: 4975: 4969: 4961: 4954: 4953: 4947: 4939: 4929: 4922: 4895: 4892: 4891: 4873: 4870: 4869: 4861: 4852: 4843: 4834: 4822: 4804: 4798: 4785: 4774:with relations 4773: 4767: 4758: 4738: 4735: 4734: 4733: 4727: 4704: 4701: 4700: 4687: 4673:edge path group 4669:edge-path group 4665: 4663:Edge-path group 4658: 4647: 4640: 4630:sheaf of groups 4616: 4604: 4595: 4589: 4579: 4557: 4554: 4553: 4552: 4543: 4521: 4518: 4517: 4516: 4507: 4485: 4482: 4481: 4480: 4471: 4449: 4446: 4445: 4444: 4422: 4419: 4418: 4417: 4395: 4392: 4391: 4390: 4381: 4375: 4353: 4350: 4349: 4348: 4326: 4323: 4322: 4321: 4312: 4303: 4281: 4278: 4277: 4276: 4267: 4240: 4236: 4227: 4223: 4214: 4210: 4201: 4197: 4188: 4184: 4175: 4171: 4169: 4166: 4165: 4146: 4143: 4142: 4126: 4123: 4122: 4120: 4113: 4104: 4100: 4099: 4073: 4070: 4069: 4053: 4050: 4049: 4048: 4047: 4036: 4011: 3989: 3986: 3985: 3984: 3975: 3961: 3939:= 1 everywhere. 3938: 3921: 3905: 3896: 3887: 3878: 3869: 3862: 3852: 3843: 3834: 3818: 3812: 3761: 3758: 3757: 3752: 3746: 3741: 3735: 3729: 3722: 3715: 3709: 3697:by conjugation) 3692: 3685: 3678: 3671: 3664: 3660: 3639: 3636: 3635: 3619: 3616: 3615: 3596: 3593: 3592: 3590: 3572: 3569: 3568: 3567: 3563: 3550: 3518: 3506: 3449:linear subspace 3440:locally compact 3428: 3399: 3395: 3393: 3390: 3389: 3372: 3368: 3366: 3363: 3362: 3345: 3341: 3339: 3336: 3335: 3311: 3308: 3307: 3288: 3282: 3277: 3276: 3274: 3271: 3270: 3250: 3245: 3244: 3242: 3239: 3238: 3215: 3212: 3211: 3195: 3192: 3191: 3172: 3166: 3161: 3160: 3158: 3155: 3154: 3137: 3079: 3074: 3024:covering spaces 3016: 3008: 2994: 2964:Kleinian groups 2953: 2944: 2933: 2930:Poincaré metric 2914:Fuchsian groups 2896: 2887: 2878: 2869: 2849: 2840: 2834: 2824: 2761: 2724: 2716: 2691: 2688: 2687: 2665: 2662: 2661: 2632: 2629: 2628: 2605: 2601: 2586: 2572: 2571: 2562: 2558: 2552: 2548: 2537: 2534: 2533: 2516: 2512: 2510: 2507: 2506: 2505:, the groupoid 2489: 2485: 2483: 2480: 2479: 2462: 2458: 2456: 2453: 2452: 2435: 2431: 2429: 2426: 2425: 2408: 2404: 2402: 2399: 2398: 2382: 2379: 2378: 2358: 2355: 2354: 2351: 2311: 2308: 2307: 2276: 2273: 2272: 2250: 2247: 2246: 2224: 2221: 2220: 2204: 2201: 2200: 2184: 2181: 2180: 2164: 2156: 2148: 2146: 2143: 2142: 2120: 2112: 2104: 2102: 2099: 2098: 2076: 2073: 2072: 2052: 2049: 2048: 2026:discrete spaces 1982: 1978: 1969: 1965: 1956: 1952: 1932: 1929: 1928: 1908: 1905: 1904: 1887: 1883: 1881: 1878: 1877: 1840: 1836: 1815: 1811: 1802: 1798: 1792: 1788: 1783: 1780: 1779: 1778:, i.e. the set 1762: 1758: 1750: 1747: 1746: 1729: 1725: 1723: 1720: 1719: 1702: 1698: 1696: 1693: 1692: 1669: 1665: 1656: 1652: 1638: 1635: 1634: 1617: 1613: 1611: 1608: 1607: 1590: 1586: 1584: 1581: 1580: 1571: 1553:. It will be a 1537: 1528: 1519: 1510: 1501: 1486:as well as the 1482: 1473: 1464: 1455: 1439: 1430: 1421: 1412: 1400: 1394: 1384: 1362: 1359: 1358: 1357: 1335: 1332: 1331: 1330: 1299: 1296: 1295: 1269: 1265: 1257: 1254: 1253: 1233: 1229: 1221: 1218: 1217: 1199: 1195: 1193: 1190: 1189: 1171: 1167: 1165: 1162: 1161: 1141: 1137: 1125: 1121: 1112: 1108: 1106: 1103: 1102: 1081: 1077: 1075: 1072: 1071: 1053: 1049: 1047: 1044: 1043: 1019: 1015: 1013: 1010: 1009: 988: 984: 982: 979: 978: 961: 957: 948: 944: 942: 939: 938: 918: 914: 905: 901: 889: 885: 883: 880: 879: 856: 852: 843: 839: 837: 834: 833: 804: 800: 798: 795: 794: 777: 773: 768: 762: 758: 756: 753: 752: 731: 727: 725: 722: 721: 704: 700: 698: 695: 694: 677: 673: 671: 668: 667: 650: 646: 644: 641: 640: 620: 616: 614: 611: 610: 589: 584: 583: 581: 578: 577: 560: 556: 554: 551: 550: 549:an open subset 529: 525: 523: 520: 519: 502: 498: 496: 493: 492: 472: 469: 468: 445: 442: 441: 417: 412: 411: 409: 406: 405: 388: 383: 382: 380: 377: 376: 369: 364: 335: 330: 329: 327: 324: 323: 291: 288: 287: 263: 258: 257: 255: 252: 251: 232:diffeomorphisms 194:Herbert Seifert 192:, initiated by 158: 141: 139: 136: 135: 125:singular points 116:under the name 106:André Haefliger 75:Euclidean space 39: 32: 17: 12: 11: 5: 13718: 13708: 13707: 13702: 13697: 13680: 13679: 13677: 13676: 13671: 13666: 13661: 13656: 13651: 13646: 13641: 13636: 13631: 13626: 13621: 13616: 13611: 13606: 13601: 13596: 13591: 13586: 13581: 13576: 13571: 13566: 13561: 13556: 13551: 13546: 13541: 13536: 13531: 13526: 13521: 13516: 13514:Randjbar-Daemi 13511: 13506: 13501: 13496: 13491: 13486: 13481: 13476: 13471: 13466: 13461: 13456: 13451: 13446: 13441: 13436: 13431: 13426: 13421: 13416: 13411: 13406: 13401: 13396: 13391: 13386: 13381: 13376: 13371: 13366: 13361: 13356: 13351: 13346: 13341: 13336: 13331: 13326: 13321: 13316: 13311: 13306: 13301: 13296: 13291: 13286: 13281: 13276: 13271: 13266: 13261: 13256: 13251: 13246: 13241: 13236: 13231: 13226: 13221: 13216: 13211: 13206: 13201: 13195: 13193: 13189: 13188: 13186: 13185: 13180: 13174: 13172: 13166: 13165: 13163: 13162: 13157: 13151: 13149: 13143: 13142: 13140: 13139: 13137:Lie supergroup 13134: 13129: 13124: 13119: 13114: 13109: 13104: 13098: 13096: 13090: 13089: 13087: 13086: 13081: 13076: 13071: 13066: 13061: 13056: 13051: 13046: 13045: 13044: 13039: 13034: 13030: 13029: 13028: 13018: 13008: 13003: 12997: 12992: 12987: 12981: 12979: 12975: 12974: 12972: 12971: 12963: 12958: 12953: 12948: 12941: 12934: 12927: 12920: 12912: 12907: 12902: 12897: 12891: 12889: 12883: 12882: 12880: 12879: 12874: 12869: 12864: 12859: 12854: 12849: 12844: 12839: 12833: 12831: 12825: 12824: 12822: 12821: 12816: 12814:Quiver diagram 12811: 12806: 12801: 12796: 12791: 12786: 12781: 12776: 12771: 12765: 12763: 12757: 12756: 12754: 12753: 12748: 12743: 12738: 12733: 12728: 12723: 12718: 12712: 12710: 12706: 12705: 12703: 12702: 12697: 12692: 12687: 12681: 12679: 12677:String duality 12673: 12672: 12670: 12669: 12664: 12659: 12654: 12649: 12644: 12639: 12634: 12629: 12624: 12623: 12622: 12617: 12616: 12615: 12610: 12603:Type II string 12600: 12590: 12585: 12580: 12574: 12572: 12568: 12567: 12565: 12564: 12559: 12558: 12557: 12552: 12542: 12540:Cosmic strings 12537: 12531: 12529: 12525: 12524: 12517: 12516: 12509: 12502: 12494: 12485: 12484: 12482: 12481: 12476: 12471: 12466: 12461: 12460: 12459: 12449: 12444: 12439: 12434: 12429: 12424: 12418: 12416: 12412: 12411: 12409: 12408: 12403: 12398: 12393: 12388: 12383: 12377: 12375: 12371: 12370: 12367: 12366: 12364: 12363: 12358: 12353: 12348: 12343: 12338: 12333: 12328: 12323: 12318: 12312: 12310: 12304: 12303: 12301: 12300: 12295: 12290: 12285: 12280: 12275: 12270: 12260: 12255: 12250: 12240: 12235: 12230: 12225: 12220: 12215: 12209: 12207: 12201: 12200: 12198: 12197: 12192: 12187: 12186: 12185: 12175: 12170: 12169: 12168: 12158: 12153: 12148: 12143: 12142: 12141: 12131: 12126: 12125: 12124: 12114: 12109: 12103: 12101: 12097: 12096: 12094: 12093: 12088: 12083: 12078: 12077: 12076: 12066: 12061: 12056: 12050: 12048: 12041: 12035: 12034: 12032: 12031: 12026: 12016: 12011: 11997: 11992: 11987: 11982: 11977: 11975:Parallelizable 11972: 11967: 11962: 11961: 11960: 11950: 11945: 11940: 11935: 11930: 11925: 11920: 11915: 11910: 11905: 11895: 11885: 11879: 11877: 11871: 11870: 11868: 11867: 11862: 11857: 11855:Lie derivative 11852: 11850:Integral curve 11847: 11842: 11837: 11836: 11835: 11825: 11820: 11819: 11818: 11811:Diffeomorphism 11808: 11802: 11800: 11794: 11793: 11791: 11790: 11785: 11780: 11775: 11770: 11765: 11760: 11755: 11750: 11744: 11742: 11733: 11732: 11730: 11729: 11724: 11719: 11714: 11709: 11704: 11699: 11694: 11689: 11688: 11687: 11682: 11672: 11671: 11670: 11659: 11657: 11656:Basic concepts 11653: 11652: 11640: 11639: 11632: 11625: 11617: 11610: 11609: 11545: 11533:(3): 357–381. 11513: 11499: 11489:(2): 231–247. 11476: 11470: 11445: 11439: 11424: 11396: 11384: 11365:2027.42/135276 11351:(5): 401–487. 11330: 11318:(4): 464–492. 11304:Satake, Ichirô 11300: 11262:(6): 359–363. 11246:Satake, Ichirô 11242: 11236: 11216: 11198:(1): 233–244. 11185: 11179: 11167:(2): 203–209. 11152: 11146: 11140: 11123: 11094: 11088: 11075: 11036: 11030: 11005: 10999: 10972: 10962: 10961: 10954: 10931: 10924: 10899: 10892: 10879: 10867:(2): 143–166. 10852: 10846: 10792: 10777: 10771: 10748: 10742: 10734:Academic Press 10722: 10683: 10668: 10655: 10628: 10608:(2): 165–191. 10591: 10585: 10568: 10562: 10536: 10534: 10531: 10529: 10528: 10510: 10492: 10490:, October 2006 10473: 10457: 10435: 10420: 10405: 10393: 10381: 10367: 10361:. Birkhäuser. 10347: 10333: 10327:. Heldermann. 10313: 10266: 10255: 10227: 10214: 10194: 10181: 10172: 10161: 10149: 10137: 10125: 10121:Haefliger 1984 10113: 10101: 10089: 10077: 10070: 10044:Moerdijk, Ieke 10035: 10028: 10000:Moerdijk, Ieke 9991: 9976: 9964: 9952: 9940: 9928: 9916: 9912:Haefliger 1990 9904: 9892: 9876: 9874: 9871: 9870: 9869: 9864: 9859: 9854: 9849: 9844: 9839: 9834: 9827: 9824: 9786: 9783: 9780: 9777: 9774: 9754: 9751: 9748: 9728: 9725: 9722: 9719: 9716: 9713: 9710: 9707: 9704: 9658: 9657: 9645: 9640: 9636: 9615: 9610: 9606: 9602: 9599: 9596: 9591: 9587: 9583: 9578: 9574: 9550: 9537: 9532: 9527: 9523: 9518: 9514: 9491: 9486: 9481: 9476: 9472: 9468: 9464: 9448: 9423: 9419: 9399:(the space of 9386: 9382: 9348: 9344: 9339: 9333: 9329: 9286: 9282: 9277: 9271: 9267: 9228: 9225: 9217: 9216: 9215: 9214: 9199: 9194: 9188: 9182: 9178: 9108:Main article: 9105: 9102: 9064: 9052: 9038:winding states 8941: 8938: 8936: 8933: 8886: 8885: 8865: 8863: 8852: 8849: 8846: 8845: 8843: 8841: 8838: 8834: 8833: 8831: 8829: 8826: 8822: 8821: 8819: 8817: 8814: 8810: 8809: 8807: 8805: 8802: 8798: 8797: 8795: 8792: 8789: 8785: 8784: 8782: 8779: 8776: 8772: 8771: 8768: 8765: 8762: 8758: 8757: 8754: 8751: 8748: 8744: 8743: 8740: 8737: 8734: 8730: 8729: 8726: 8724: 8721: 8717: 8716: 8713: 8711: 8708: 8704: 8703: 8700: 8698: 8695: 8691: 8690: 8687: 8685: 8682: 8678: 8677: 8675: 8672: 8669: 8665: 8664: 8662: 8659: 8656: 8652: 8651: 8649: 8646: 8643: 8639: 8638: 8636: 8633: 8630: 8627: 8623: 8622: 8620: 8615: 8612: 8605: 8604: 8602: 8600: 8597: 8593: 8592: 8589: 8586: 8583: 8579: 8578: 8573: 8570: 8567: 8560: 8559: 8557: 8552: 8549: 8542: 8541: 8538: 8536: 8533: 8529: 8528: 8525: 8523: 8520: 8516: 8515: 8512: 8510: 8507: 8503: 8502: 8496: 8494: 8491: 8484: 8483: 8474: 8472: 8469: 8462: 8461: 8459: 8457: 8454: 8450: 8449: 8447: 8444: 8441: 8437: 8436: 8434: 8431: 8428: 8424: 8423: 8421: 8418: 8415: 8411: 8410: 8408: 8402: 8399: 8392: 8391: 8389: 8380: 8377: 8370: 8369: 8367: 8365: 8362: 8359: 8355: 8354: 8344: 8342: 8339: 8328: 8327: 8321: 8319: 8316: 8309: 8308: 8306: 8296: 8293: 8282: 8281: 8279: 8273: 8270: 8264: 8260: 8259: 8256: 8253: 8250: 8247: 8211: 8207: 8184: 8180: 8157: 8153: 8132: 8127: 8123: 8119: 8116: 8105: 8104: 8092: 8087: 8083: 8078: 8074: 8071: 8068: 8065: 8060: 8056: 8052: 8049: 8045: 8041: 8036: 8032: 8027: 8023: 8020: 8017: 8014: 8009: 8005: 8001: 7998: 7993: 7989: 7985: 7982: 7979: 7976: 7953: 7937: 7936: 7921: 7906:gyration point 7902: 7894: 7891: 7886: 7878: 7871: 7845: 7844: 7818: 7813:dihedral group 7770: 7769: 7714: 7713: 7684: 7659:. The link of 7534: 7533: 7520: 7435: 7432: 7427: 7423: 7416: 7410:Sylow subgroup 7397: 7385: 7378: 7371: 7363: 7356: 7349: 7342: 7338: 7311: 7297: 7285: 7275: 7274: 7230:preserves the 7227: 7207: 7196: 7185: 7174: 7164: 7142: 7138:. Identifying 7116: 7109: 7091: 7090: 7087: 7080: 7074: 7068: 7061: 7046: 7038: 7031: 7024: 7017: 7008: 7007: 6996: 6985: 6974: 6960: 6953: 6934: 6916: 6915: 6869: 6855: 6819:with basis 1, 6801: 6800: 6785: 6784: 6762: 6761: 6743: 6729: 6705: 6684: 6681: 6661:The Fano plane 6653: 6650: 6649: 6648: 6645: 6638: 6610: 6609: 6598: 6593: 6589: 6585: 6580: 6577: 6573: 6565: 6562: 6559: 6555: 6549: 6543: 6540: 6536: 6520: 6516: 6512: 6504: 6492: 6488: 6484: 6480: 6476: 6472: 6469: 6468: 6464: 6461: 6457: 6453: 6449: 6446: 6445:at each vertex 6442: 6438: 6434: 6370: 6363: 6336: 6333: 6332: 6331: 6327: 6312: 6301: 6282: 6279: 6276: 6268: 6256: 6252: 6221: 6200: 6197: 6196: 6195: 6189: 6180: 6171: 6162: 6153: 6146: 6145: 6140: 6131: 6122: 6113: 6103: 6102: 6097: 6091: 6085: 6079: 6067: 6063: 6042: 6020: 6006: 6005: 5999: 5989: 5979: 5970: 5964: 5958: 5952: 5946: 5943: 5937: 5928: 5919: 5909: 5894: 5880: 5874: 5868: 5862: 5840: 5807: 5794: 5788: 5779: 5757: 5748: 5738: 5725: 5716: 5707: 5698: 5695: 5694: 5688: 5679: 5673: 5656: 5644: 5635: 5632: 5626: 5620: 5589: 5586: 5567: 5564: 5552: 5551: 5548: 5521: 5495: 5469: 5455: 5440: 5426: 5388: 5387: 5380: 5353: 5352: 5329: 5328: 5324: 5313: 5297: 5285: 5281: 5274: 5270: 5266: 5263: 5259: 5250:is said to be 5236: 5235: 5225: 5216: 5207: 5194: 5185: 5178: 5171: 5162: 5155: 5149: 5146: 5135: 5105: 5102: 5086: 5069: 5060: 5050: 5046: 5040: 5033: 5024: 5017: 5010: 5004: 5003: 4999: 4991: 4987: 4981: 4973: 4967: 4963: 4959: 4951: 4945: 4941: 4937: 4927: 4920: 4899: 4877: 4863: 4862: 4857: 4848: 4839: 4830: 4818: 4811: 4810: 4800: 4794: 4781: 4769: 4763: 4754: 4742: 4729: 4723: 4708: 4683: 4664: 4661: 4656: 4645: 4636: 4600: 4591: 4587: 4575: 4561: 4548: 4544:and ψ" : 4539: 4525: 4512: 4503: 4489: 4476: 4467: 4453: 4440: 4426: 4413: 4399: 4386: 4377: 4371: 4357: 4344: 4330: 4317: 4308: 4299: 4285: 4272: 4263: 4257: 4256: 4243: 4239: 4235: 4230: 4226: 4222: 4217: 4213: 4209: 4204: 4200: 4196: 4191: 4187: 4183: 4178: 4174: 4150: 4130: 4118: 4109: 4102: 4095: 4091: 4077: 4057: 4045: 4041: 4032: 4028:intersections 4007: 3993: 3980: 3971: 3960: 3957: 3941: 3940: 3934: 3917: 3907: 3906: 3901: 3892: 3883: 3874: 3867: 3858: 3853: 3848: 3839: 3830: 3814: 3808: 3786: 3783: 3780: 3777: 3774: 3771: 3768: 3765: 3754: 3753: 3745: 3734: 3727: 3720: 3708: 3699: 3698: 3695:adjoint action 3690: 3683: 3676: 3669: 3665:such that (Ad 3662: 3658: 3643: 3623: 3612: 3600: 3588: 3576: 3565: 3561: 3555: 3548: 3517: 3514: 3505: 3502: 3501: 3500: 3427: 3424: 3419: 3418: 3402: 3398: 3375: 3371: 3348: 3344: 3334:The notion of 3330: 3327: 3315: 3295: 3291: 3285: 3280: 3267: 3253: 3248: 3219: 3199: 3179: 3175: 3169: 3164: 3136: 3133: 3132: 3131: 3128: 3125: 3077: 3015: 3012: 3011: 3010: 3004: 2999:is the closed 2990: 2971: 2960:triangle group 2949: 2940: 2916:as hyperbolic 2910:Henri Poincaré 2906: 2892: 2883: 2874: 2865: 2845: 2836: 2830: 2820: 2775: 2759: 2730: 2720: 2715: 2712: 2695: 2675: 2672: 2669: 2642: 2639: 2636: 2616: 2611: 2608: 2604: 2600: 2597: 2594: 2589: 2584: 2581: 2578: 2575: 2570: 2565: 2561: 2555: 2551: 2547: 2544: 2541: 2519: 2515: 2492: 2488: 2465: 2461: 2438: 2434: 2411: 2407: 2386: 2362: 2350: 2347: 2330: 2327: 2324: 2321: 2318: 2315: 2295: 2292: 2289: 2286: 2283: 2280: 2260: 2257: 2254: 2245:if there is a 2234: 2231: 2228: 2208: 2188: 2167: 2163: 2159: 2155: 2151: 2130: 2127: 2123: 2119: 2115: 2111: 2107: 2086: 2083: 2080: 2056: 2030: 2029: 2022: 1985: 1981: 1977: 1972: 1968: 1964: 1959: 1955: 1951: 1948: 1945: 1942: 1939: 1936: 1912: 1890: 1886: 1874:isotropy group 1857: 1854: 1851: 1846: 1843: 1839: 1835: 1832: 1829: 1826: 1821: 1818: 1814: 1810: 1805: 1801: 1795: 1791: 1787: 1765: 1761: 1757: 1754: 1732: 1728: 1705: 1701: 1672: 1668: 1664: 1659: 1655: 1651: 1648: 1645: 1642: 1620: 1616: 1593: 1589: 1573:Recall that a 1570: 1567: 1540: 1539: 1533: 1524: 1515: 1506: 1497: 1484: 1483: 1478: 1469: 1460: 1451: 1441: 1440: 1435: 1426: 1417: 1408: 1396: 1392: 1380: 1366: 1353: 1339: 1326: 1303: 1287: 1286: 1272: 1268: 1264: 1261: 1239: 1236: 1232: 1228: 1225: 1202: 1198: 1174: 1170: 1158: 1144: 1140: 1136: 1131: 1128: 1124: 1120: 1115: 1111: 1099: 1084: 1080: 1056: 1052: 1025: 1022: 1018: 1008:homeomorphism 991: 987: 964: 960: 956: 951: 947: 935: 921: 917: 913: 908: 904: 900: 895: 892: 888: 859: 855: 851: 846: 842: 826:orbifold atlas 822: 821: 807: 803: 780: 776: 771: 765: 761: 749:orbifold chart 734: 730: 707: 703: 680: 676: 653: 649: 637: 623: 619: 592: 587: 563: 559: 532: 528: 505: 501: 476: 449: 420: 415: 391: 386: 368: 365: 363: 360: 338: 333: 295: 266: 261: 213:vertex algebra 165: 161: 157: 154: 151: 147: 144: 110:Mikhail Gromov 73:quotient of a 20: 15: 9: 6: 4: 3: 2: 13717: 13706: 13703: 13701: 13698: 13696: 13693: 13692: 13690: 13675: 13672: 13670: 13667: 13665: 13662: 13660: 13659:Zamolodchikov 13657: 13655: 13654:Zamolodchikov 13652: 13650: 13647: 13645: 13642: 13640: 13637: 13635: 13632: 13630: 13627: 13625: 13622: 13620: 13617: 13615: 13612: 13610: 13607: 13605: 13602: 13600: 13597: 13595: 13592: 13590: 13587: 13585: 13582: 13580: 13577: 13575: 13572: 13570: 13567: 13565: 13562: 13560: 13557: 13555: 13552: 13550: 13547: 13545: 13542: 13540: 13537: 13535: 13532: 13530: 13527: 13525: 13522: 13520: 13517: 13515: 13512: 13510: 13507: 13505: 13502: 13500: 13497: 13495: 13492: 13490: 13487: 13485: 13482: 13480: 13477: 13475: 13472: 13470: 13467: 13465: 13462: 13460: 13457: 13455: 13452: 13450: 13447: 13445: 13442: 13440: 13437: 13435: 13432: 13430: 13427: 13425: 13422: 13420: 13417: 13415: 13412: 13410: 13407: 13405: 13402: 13400: 13397: 13395: 13392: 13390: 13387: 13385: 13382: 13380: 13377: 13375: 13372: 13370: 13367: 13365: 13362: 13360: 13357: 13355: 13352: 13350: 13347: 13345: 13342: 13340: 13337: 13335: 13332: 13330: 13327: 13325: 13322: 13320: 13317: 13315: 13312: 13310: 13307: 13305: 13302: 13300: 13297: 13295: 13292: 13290: 13287: 13285: 13282: 13280: 13277: 13275: 13272: 13270: 13267: 13265: 13262: 13260: 13257: 13255: 13252: 13250: 13247: 13245: 13242: 13240: 13237: 13235: 13232: 13230: 13227: 13225: 13222: 13220: 13217: 13215: 13212: 13210: 13207: 13205: 13202: 13200: 13197: 13196: 13194: 13190: 13184: 13181: 13179: 13178:Matrix theory 13176: 13175: 13173: 13171: 13167: 13161: 13158: 13156: 13153: 13152: 13150: 13148: 13144: 13138: 13135: 13133: 13130: 13128: 13125: 13123: 13120: 13118: 13115: 13113: 13110: 13108: 13105: 13103: 13100: 13099: 13097: 13095: 13094:Supersymmetry 13091: 13085: 13082: 13080: 13077: 13075: 13072: 13070: 13067: 13065: 13062: 13060: 13057: 13055: 13052: 13050: 13047: 13043: 13040: 13038: 13031: 13027: 13024: 13023: 13022: 13019: 13017: 13014: 13013: 13012: 13009: 13007: 13004: 13001: 12998: 12996: 12993: 12991: 12988: 12986: 12983: 12982: 12980: 12976: 12970: 12968: 12964: 12962: 12959: 12957: 12954: 12951: 12944: 12937: 12930: 12923: 12916: 12913: 12911: 12908: 12906: 12903: 12901: 12898: 12896: 12893: 12892: 12890: 12888: 12884: 12878: 12875: 12873: 12870: 12868: 12865: 12863: 12860: 12858: 12855: 12853: 12850: 12848: 12845: 12843: 12840: 12838: 12835: 12834: 12832: 12830: 12826: 12820: 12817: 12815: 12812: 12810: 12807: 12805: 12802: 12800: 12797: 12795: 12792: 12790: 12787: 12785: 12782: 12780: 12777: 12775: 12772: 12770: 12767: 12766: 12764: 12762: 12758: 12752: 12749: 12747: 12746:Dual graviton 12744: 12742: 12739: 12737: 12734: 12732: 12729: 12727: 12724: 12722: 12719: 12717: 12714: 12713: 12711: 12707: 12701: 12698: 12696: 12693: 12691: 12688: 12686: 12683: 12682: 12680: 12678: 12674: 12668: 12665: 12663: 12662:RNS formalism 12660: 12658: 12655: 12653: 12650: 12648: 12645: 12643: 12640: 12638: 12635: 12633: 12630: 12628: 12625: 12621: 12618: 12614: 12611: 12609: 12606: 12605: 12604: 12601: 12599: 12598:Type I string 12596: 12595: 12594: 12591: 12589: 12586: 12584: 12581: 12579: 12576: 12575: 12573: 12569: 12563: 12560: 12556: 12553: 12551: 12548: 12547: 12546: 12543: 12541: 12538: 12536: 12533: 12532: 12530: 12526: 12522: 12521:String theory 12515: 12510: 12508: 12503: 12501: 12496: 12495: 12492: 12480: 12477: 12475: 12474:Supermanifold 12472: 12470: 12467: 12465: 12462: 12458: 12455: 12454: 12453: 12450: 12448: 12445: 12443: 12440: 12438: 12435: 12433: 12430: 12428: 12425: 12423: 12420: 12419: 12417: 12413: 12407: 12404: 12402: 12399: 12397: 12394: 12392: 12389: 12387: 12384: 12382: 12379: 12378: 12376: 12372: 12362: 12359: 12357: 12354: 12352: 12349: 12347: 12344: 12342: 12339: 12337: 12334: 12332: 12329: 12327: 12324: 12322: 12319: 12317: 12314: 12313: 12311: 12309: 12305: 12299: 12296: 12294: 12291: 12289: 12286: 12284: 12281: 12279: 12276: 12274: 12271: 12269: 12265: 12261: 12259: 12256: 12254: 12251: 12249: 12245: 12241: 12239: 12236: 12234: 12231: 12229: 12226: 12224: 12221: 12219: 12216: 12214: 12211: 12210: 12208: 12206: 12202: 12196: 12195:Wedge product 12193: 12191: 12188: 12184: 12181: 12180: 12179: 12176: 12174: 12171: 12167: 12164: 12163: 12162: 12159: 12157: 12154: 12152: 12149: 12147: 12144: 12140: 12139:Vector-valued 12137: 12136: 12135: 12132: 12130: 12127: 12123: 12120: 12119: 12118: 12115: 12113: 12110: 12108: 12105: 12104: 12102: 12098: 12092: 12089: 12087: 12084: 12082: 12079: 12075: 12072: 12071: 12070: 12069:Tangent space 12067: 12065: 12062: 12060: 12057: 12055: 12052: 12051: 12049: 12045: 12042: 12040: 12036: 12030: 12027: 12025: 12021: 12017: 12015: 12012: 12010: 12006: 12002: 11998: 11996: 11993: 11991: 11988: 11986: 11983: 11981: 11978: 11976: 11973: 11971: 11968: 11966: 11963: 11959: 11956: 11955: 11954: 11951: 11949: 11946: 11944: 11941: 11939: 11936: 11934: 11931: 11929: 11926: 11924: 11921: 11919: 11916: 11914: 11911: 11909: 11906: 11904: 11900: 11896: 11894: 11890: 11886: 11884: 11881: 11880: 11878: 11872: 11866: 11863: 11861: 11858: 11856: 11853: 11851: 11848: 11846: 11843: 11841: 11838: 11834: 11833:in Lie theory 11831: 11830: 11829: 11826: 11824: 11821: 11817: 11814: 11813: 11812: 11809: 11807: 11804: 11803: 11801: 11799: 11795: 11789: 11786: 11784: 11781: 11779: 11776: 11774: 11771: 11769: 11766: 11764: 11761: 11759: 11756: 11754: 11751: 11749: 11746: 11745: 11743: 11740: 11736:Main results 11734: 11728: 11725: 11723: 11720: 11718: 11717:Tangent space 11715: 11713: 11710: 11708: 11705: 11703: 11700: 11698: 11695: 11693: 11690: 11686: 11683: 11681: 11678: 11677: 11676: 11673: 11669: 11666: 11665: 11664: 11661: 11660: 11658: 11654: 11649: 11645: 11638: 11633: 11631: 11626: 11624: 11619: 11618: 11615: 11606: 11602: 11598: 11594: 11590: 11586: 11581: 11576: 11572: 11568: 11564: 11560: 11559: 11551: 11546: 11541: 11536: 11532: 11528: 11527: 11522: 11518: 11514: 11510: 11509: 11505:(1978–1981). 11504: 11500: 11496: 11492: 11488: 11484: 11483: 11477: 11473: 11471:981-02-0442-6 11467: 11463: 11459: 11455: 11451: 11450:Ghys, Étienne 11446: 11442: 11435: 11427: 11421: 11417: 11413: 11409: 11405: 11401: 11397: 11393: 11389: 11385: 11378: 11374: 11366: 11362: 11358: 11354: 11350: 11346: 11339: 11335: 11331: 11326: 11321: 11317: 11313: 11309: 11305: 11301: 11297: 11293: 11288: 11283: 11278: 11273: 11269: 11265: 11261: 11257: 11256: 11251: 11247: 11243: 11239: 11237:3-540-96215-8 11233: 11229: 11225: 11221: 11217: 11213: 11209: 11205: 11201: 11197: 11193: 11189: 11180: 11175: 11170: 11166: 11162: 11161: 11156: 11147: 11143: 11141:0-19-853212-1 11137: 11133: 11129: 11124: 11120: 11116: 11112: 11108: 11104: 11100: 11095: 11091: 11085: 11081: 11076: 11072: 11068: 11064: 11060: 11055: 11050: 11046: 11042: 11037: 11033: 11031:981-02-0442-6 11027: 11023: 11019: 11015: 11011: 11010:Ghys, Étienne 11006: 11002: 11000:0-8176-3508-4 10996: 10991: 10986: 10982: 10978: 10977:Ghys, Étienne 10973: 10969: 10964: 10963: 10957: 10955:0-387-94785-X 10951: 10946: 10941: 10937: 10932: 10927: 10925:981-02-0442-6 10921: 10917: 10913: 10909: 10905: 10904:Ghys, Étienne 10900: 10895: 10893:4-931469-05-1 10889: 10885: 10880: 10875: 10870: 10866: 10862: 10861: 10856: 10847: 10843: 10839: 10835: 10831: 10827: 10823: 10819: 10815: 10811: 10807: 10806: 10798: 10793: 10788: 10783: 10778: 10774: 10772:3-540-64324-9 10768: 10763: 10758: 10754: 10749: 10745: 10743:0-12-128850-1 10739: 10735: 10731: 10727: 10723: 10719: 10715: 10711: 10707: 10702: 10697: 10693: 10689: 10684: 10679: 10675: 10671: 10669:2-85629-152-X 10665: 10661: 10656: 10641: 10634: 10629: 10625: 10621: 10616: 10611: 10607: 10603: 10602: 10597: 10592: 10588: 10586:0-8176-3508-4 10582: 10578: 10574: 10573:Ghys, Étienne 10569: 10565: 10563:9780521870047 10559: 10555: 10551: 10547: 10543: 10538: 10537: 10521: 10514: 10503: 10496: 10489: 10486: 10484: 10477: 10470: 10468: 10467:Mapping Music 10461: 10454: 10452: 10448: 10446: 10439: 10432: 10430: 10424: 10417: 10416: 10409: 10402: 10397: 10390: 10389:Tymoczko 2006 10385: 10370: 10364: 10360: 10359: 10351: 10336: 10330: 10326: 10325: 10317: 10309: 10305: 10301: 10297: 10293: 10289: 10285: 10281: 10277: 10270: 10262: 10258: 10256:9783642294969 10252: 10248: 10244: 10240: 10239: 10231: 10224: 10221:P. Candelas, 10218: 10212: 10211:0-521-63304-4 10208: 10204: 10203:String theory 10198: 10191: 10185: 10176: 10170: 10165: 10158: 10153: 10147:, Example 25. 10146: 10141: 10134: 10129: 10122: 10117: 10111:, Theorem 46. 10110: 10105: 10098: 10093: 10086: 10081: 10073: 10067: 10063: 10059: 10055: 10051: 10050: 10045: 10039: 10031: 10025: 10020: 10015: 10011: 10007: 10006: 10001: 9995: 9988: 9983: 9981: 9973: 9968: 9961: 9956: 9949: 9944: 9937: 9932: 9925: 9924:Poincaré 1985 9920: 9913: 9908: 9902:, Chapter 13. 9901: 9896: 9889: 9884: 9882: 9877: 9868: 9865: 9863: 9860: 9858: 9855: 9853: 9850: 9848: 9845: 9843: 9840: 9838: 9835: 9833: 9830: 9829: 9823: 9820: 9816: 9812: 9808: 9804: 9798: 9784: 9781: 9778: 9775: 9772: 9752: 9749: 9746: 9726: 9723: 9720: 9717: 9714: 9711: 9708: 9705: 9702: 9694: 9689: 9671: 9667: 9663: 9643: 9638: 9634: 9613: 9608: 9604: 9600: 9597: 9594: 9589: 9585: 9581: 9576: 9572: 9563: 9559: 9555: 9551: 9530: 9521: 9516: 9512: 9489: 9479: 9474: 9470: 9466: 9453: 9449: 9446: 9442: 9441: 9440: 9437: 9421: 9417: 9409: 9405: 9402: 9384: 9380: 9372: 9368: 9364: 9346: 9342: 9337: 9331: 9327: 9318: 9310: 9306: 9302: 9284: 9280: 9275: 9269: 9265: 9255: 9251: 9249: 9247: 9242: 9239:and later by 9238: 9234: 9224: 9222: 9197: 9186: 9180: 9176: 9167: 9162: 9161: 9160: 9159: 9158: 9156: 9152: 9148: 9143: 9141: 9137: 9136:supersymmetry 9133: 9129: 9125: 9121: 9117: 9111: 9101: 9099: 9096: 9093: 9087: 9085: 9084:Eguchi–Hanson 9081: 9077: 9073: 9069: 9067: 9063: 9058: 9057: 9055: 9046: 9041: 9039: 9034: 9029: 9027: 9023: 9019: 9015: 9013: 9009: 9005: 9001: 9000:Hilbert space 8998: 8997:closed string 8994: 8990: 8985: 8983: 8979: 8975: 8971: 8967: 8963: 8959: 8955: 8954:diffeomorphic 8951: 8947: 8946:string theory 8932: 8930: 8926: 8922: 8918: 8913: 8911: 8907: 8903: 8899: 8895: 8893: 8882: 8873: 8869: 8866:This section 8864: 8861: 8857: 8856: 8844: 8842: 8840:Moebius band 8839: 8836: 8835: 8832: 8830: 8827: 8824: 8823: 8820: 8818: 8816:Klein bottle 8815: 8812: 8811: 8808: 8806: 8803: 8800: 8799: 8796: 8793: 8790: 8787: 8786: 8783: 8780: 8777: 8774: 8773: 8769: 8766: 8763: 8760: 8759: 8755: 8752: 8749: 8746: 8745: 8741: 8738: 8735: 8732: 8731: 8727: 8725: 8722: 8719: 8718: 8714: 8712: 8709: 8706: 8705: 8701: 8699: 8696: 8693: 8692: 8688: 8686: 8683: 8680: 8679: 8676: 8673: 8670: 8667: 8666: 8663: 8660: 8657: 8654: 8653: 8650: 8647: 8644: 8641: 8640: 8637: 8634: 8631: 8628: 8624: 8621: 8619: 8616: 8613: 8611: 8607: 8606: 8603: 8601: 8598: 8595: 8594: 8590: 8587: 8584: 8581: 8580: 8577: 8574: 8571: 8568: 8566: 8562: 8561: 8558: 8556: 8553: 8550: 8548: 8544: 8543: 8539: 8537: 8534: 8531: 8530: 8526: 8524: 8521: 8518: 8517: 8513: 8511: 8508: 8505: 8504: 8501: 8497: 8495: 8492: 8490: 8486: 8485: 8482: 8478: 8475: 8473: 8470: 8468: 8464: 8463: 8460: 8458: 8455: 8452: 8451: 8448: 8445: 8442: 8439: 8438: 8435: 8432: 8429: 8426: 8425: 8422: 8419: 8416: 8413: 8412: 8409: 8407: 8403: 8400: 8398: 8394: 8393: 8390: 8388: 8384: 8381: 8378: 8376: 8372: 8371: 8368: 8366: 8363: 8360: 8356: 8352: 8348: 8345: 8343: 8340: 8338: 8334: 8330: 8329: 8325: 8322: 8320: 8317: 8315: 8311: 8310: 8307: 8304: 8300: 8297: 8294: 8292: 8288: 8284: 8283: 8280: 8277: 8274: 8271: 8269: 8265: 8261: 8257: 8254: 8251: 8248: 8245: 8244: 8241: 8239: 8234: 8231: 8225: 8209: 8205: 8182: 8178: 8155: 8151: 8125: 8121: 8114: 8085: 8081: 8076: 8072: 8069: 8066: 8058: 8054: 8050: 8047: 8043: 8034: 8030: 8025: 8021: 8018: 8015: 8007: 8003: 7999: 7991: 7987: 7980: 7977: 7974: 7967: 7966: 7965: 7951: 7944: 7943: 7934: 7930: 7926: 7922: 7920:of rotations. 7919: 7915: 7911: 7907: 7903: 7900: 7899: 7898: 7890: 7884: 7877: 7870: 7866: 7862: 7858: 7854: 7850: 7842: 7839: 7835: 7832: 7831: 7830: 7828: 7824: 7817: 7814: 7810: 7806: 7801: 7799: 7795: 7791: 7785: 7783: 7779: 7775: 7767: 7763: 7759: 7755: 7751: 7747: 7743: 7742: 7741: 7739: 7735: 7731: 7727: 7723: 7719: 7712: 7708: 7704: 7700: 7696: 7695: 7694: 7692: 7689:, generating 7687: 7682: 7678: 7674: 7670: 7666: 7662: 7658: 7654: 7650: 7646: 7641: 7639: 7635: 7631: 7627: 7623: 7619: 7614: 7613: 7605: 7600: 7596: 7594: 7590: 7587: 7583: 7579: 7575: 7571: 7567: 7563: 7559: 7555: 7551: 7547: 7543: 7539: 7536:The elements 7531: 7528: 7525: 7521: 7518: 7514: 7513: 7512: 7508: 7506: 7502: 7498: 7494: 7490: 7486: 7482: 7478: 7474: 7470: 7466: 7462: 7458: 7454: 7450: 7446: 7445: 7439: 7431: 7422: 7415: 7411: 7407: 7403: 7402:inverse image 7395: 7391: 7384: 7377: 7369: 7362: 7355: 7348: 7337: 7333: 7329: 7325: 7321: 7317: 7310: 7295: 7284: 7280: 7272: 7268: 7264: 7260: 7256: 7252: 7248: 7244: 7241: 7240: 7239: 7237: 7233: 7225: 7221: 7217: 7213: 7205: 7200: 7195: 7191: 7184: 7180: 7173: 7170: 7163: 7159: 7155: 7151: 7148: 7141: 7137: 7134: 7130: 7126: 7122: 7115: 7108: 7104: 7100: 7096: 7093:The elements 7088: 7085: 7081: 7077: 7073: 7069: 7066: 7062: 7059: 7055: 7051: 7047: 7044: 7037: 7030: 7023: 7016: 7013: 7012: 7011: 7005: 7002: 6995: 6991: 6984: 6980: 6973: 6970: 6969: 6968: 6966: 6959: 6952: 6948: 6944: 6941:) which acts 6940: 6933: 6929: 6925: 6921: 6918:The elements 6913: 6909: 6905: 6901: 6897: 6893: 6889: 6885: 6881: 6877: 6873: 6870: 6867: 6863: 6859: 6856: 6853: 6849: 6845: 6841: 6838: 6837: 6836: 6834: 6830: 6826: 6822: 6818: 6814: 6810: 6806: 6798: 6794: 6790: 6787: 6786: 6783: 6779: 6775: 6771: 6767: 6764: 6763: 6759: 6752: 6749: 6748: 6747: 6742: 6727: 6719: 6715: 6711: 6704: 6700: 6682: 6679: 6669: 6662: 6658: 6646: 6643: 6639: 6636: 6635: 6634: 6632: 6628: 6623: 6619: 6615: 6596: 6591: 6578: 6575: 6563: 6560: 6557: 6547: 6541: 6538: 6526: 6525: 6524: 6510: 6502: 6497: 6462: 6460:for each edge 6447: 6432: 6431: 6430: 6428: 6424: 6420: 6416: 6412: 6404: 6400: 6398: 6397: 6392: 6391:Betti numbers 6388: 6384: 6380: 6376: 6369: 6362: 6358: 6354: 6350: 6346: 6342: 6328: 6325: 6321: 6317: 6313: 6310: 6306: 6302: 6299: 6295: 6291: 6287: 6283: 6280: 6277: 6274: 6266: 6262: 6250: 6246: 6242: 6238: 6234: 6230: 6226: 6222: 6219: 6215: 6211: 6207: 6203: 6202: 6192: 6188: 6183: 6179: 6174: 6170: 6165: 6161: 6156: 6151: 6150: 6149: 6143: 6139: 6134: 6130: 6125: 6121: 6116: 6112: 6108: 6107: 6106: 6100: 6094: 6088: 6082: 6078: 6075: 6074: 6073: 6070: 6062: 6059: 6055: 6033: 6011: 6002: 5997: 5994:for a unique 5992: 5987: 5984:has the form 5982: 5978: 5973: 5969: 5965: 5961: 5955: 5949: 5944: 5940: 5936: 5931: 5927: 5922: 5917: 5912: 5907: 5885: 5881: 5877: 5871: 5865: 5861: 5857: 5853: 5831: 5827: 5826: 5825: 5823: 5819: 5815: 5810: 5806: 5802: 5797: 5791: 5787: 5782: 5777: 5773: 5769: 5765: 5760: 5756: 5751: 5746: 5741: 5737: 5733: 5728: 5724: 5719: 5715: 5710: 5706: 5701: 5691: 5687: 5682: 5676: 5672: 5668: 5664: 5659: 5655: 5652: 5647: 5643: 5638: 5633: 5629: 5623: 5619: 5615: 5611: 5607: 5606: 5605: 5603: 5599: 5595: 5585: 5583: 5579: 5578:proper action 5576: 5573: 5563: 5561: 5557: 5549: 5546: 5542: 5538: 5534: 5512: 5508: 5486: 5482: 5460: 5456: 5453: 5431: 5427: 5424: 5420: 5419: 5418: 5416: 5412: 5408: 5403: 5401: 5397: 5393: 5385: 5381: 5379:" is regular; 5378: 5374: 5373: 5372: 5370: 5366: 5362: 5358: 5350: 5346: 5342: 5341: 5340: 5338: 5334: 5323: 5319: 5311: 5307: 5303: 5295: 5291: 5280: 5264: 5257: 5256: 5255: 5253: 5249: 5245: 5241: 5233: 5228: 5224: 5219: 5215: 5210: 5206: 5202: 5197: 5193: 5188: 5184: 5177: 5170: 5165: 5161: 5154: 5150: 5147: 5144: 5140: 5137:the quotient 5136: 5133: 5129: 5126: 5125: 5124: 5122: 5118: 5115: 5111: 5110:proper action 5108:A simplicial 5101: 5099: 5077: 5072: 5068: 5063: 5059: 5053: 5049: 5043: 5039: 5032: 5027: 5023: 5016: 5009: 4998: 4994: 4990: 4984: 4980: 4972: 4966: 4958: 4950: 4944: 4936: 4933: 4932: 4931: 4926: 4919: 4914: 4912: 4890: 4868: 4860: 4856: 4851: 4847: 4842: 4838: 4833: 4829: 4826: 4825: 4824: 4821: 4816: 4808: 4803: 4797: 4793: 4789: 4784: 4780: 4777: 4776: 4775: 4772: 4766: 4762: 4757: 4732: 4726: 4721: 4699: 4695: 4691: 4686: 4682: 4678: 4674: 4670: 4660: 4655: 4652:and the data 4651: 4644: 4639: 4635: 4631: 4627: 4623: 4619: 4612: 4608: 4603: 4599: 4594: 4586: 4581: 4578: 4574: 4551: 4547: 4542: 4538: 4515: 4511: 4506: 4502: 4479: 4475: 4470: 4466: 4451: 4443: 4439: 4424: 4416: 4412: 4389: 4385: 4380: 4374: 4370: 4355: 4347: 4343: 4320: 4316: 4311: 4307: 4302: 4298: 4275: 4271: 4266: 4262: 4241: 4237: 4233: 4228: 4224: 4220: 4215: 4211: 4207: 4202: 4198: 4194: 4189: 4185: 4181: 4176: 4172: 4164: 4163: 4162: 4148: 4128: 4117: 4112: 4108: 4098: 4094: 4090: 4075: 4055: 4044: 4040: 4035: 4031: 4027: 4023: 4019: 4015: 4010: 4006: 3983: 3979: 3974: 3970: 3966: 3956: 3954: 3950: 3946: 3937: 3933: 3929: 3925: 3924: 3923: 3920: 3916: 3912: 3904: 3900: 3895: 3891: 3886: 3882: 3877: 3873: 3866: 3861: 3857: 3854: 3851: 3847: 3842: 3838: 3833: 3829: 3826: 3825: 3824: 3822: 3817: 3811: 3807: 3802: 3800: 3784: 3781: 3778: 3775: 3772: 3769: 3766: 3763: 3751: 3744: 3740: 3733: 3726: 3719: 3714: 3707: 3704: 3703: 3702: 3696: 3689: 3682: 3675: 3668: 3657: 3641: 3621: 3613: 3598: 3560: 3556: 3554: 3546: 3545: 3544: 3542: 3539: 3535: 3531: 3527: 3523: 3513: 3511: 3498: 3494: 3493: 3492: 3489: 3485: 3481: 3477: 3473: 3468: 3466: 3462: 3458: 3454: 3450: 3445: 3442:space with a 3441: 3437: 3433: 3423: 3416: 3396: 3369: 3342: 3331: 3328: 3313: 3293: 3289: 3283: 3268: 3251: 3236: 3235: 3234: 3231: 3217: 3197: 3177: 3173: 3167: 3152: 3148: 3146: 3145:Ichirô Satake 3142: 3129: 3126: 3123: 3122: 3121: 3118: 3116: 3112: 3107: 3105: 3101: 3097: 3093: 3089: 3084: 3082: 3072: 3068: 3064: 3059: 3057: 3053: 3052:orbifold loop 3048: 3047:orbifold path 3043: 3041: 3037: 3033: 3029: 3025: 3021: 3007: 3002: 2998: 2993: 2988: 2984: 2980: 2976: 2972: 2969: 2965: 2961: 2957: 2952: 2948: 2943: 2939: 2931: 2927: 2923: 2919: 2915: 2911: 2907: 2904: 2900: 2895: 2891: 2886: 2882: 2877: 2873: 2868: 2864: 2860: 2856: 2852: 2848: 2844: 2839: 2833: 2828: 2823: 2819: 2815: 2811: 2807: 2803: 2799: 2795: 2791: 2788: 2784: 2780: 2776: 2773: 2769: 2765: 2758: 2754: 2750: 2746: 2743: 2739: 2736:is a compact 2735: 2731: 2728: 2727:trivial group 2723: 2718: 2717: 2711: 2707: 2693: 2673: 2667: 2658: 2656: 2640: 2637: 2634: 2609: 2606: 2602: 2598: 2595: 2587: 2563: 2553: 2549: 2542: 2539: 2517: 2513: 2490: 2486: 2463: 2436: 2432: 2409: 2405: 2384: 2376: 2360: 2346: 2344: 2328: 2325: 2319: 2313: 2293: 2290: 2284: 2278: 2258: 2255: 2252: 2232: 2229: 2226: 2206: 2186: 2161: 2157: 2153: 2128: 2125: 2117: 2113: 2109: 2084: 2078: 2070: 2054: 2046: 2041: 2039: 2038:finite groups 2035: 2027: 2023: 2020: 2019: 2018: 2016: 2011: 2009: 2005: 2001: 1983: 1979: 1975: 1970: 1966: 1957: 1953: 1949: 1943: 1940: 1937: 1926: 1910: 1888: 1884: 1875: 1871: 1852: 1844: 1841: 1837: 1833: 1827: 1819: 1816: 1812: 1808: 1803: 1793: 1789: 1763: 1759: 1755: 1752: 1730: 1726: 1703: 1699: 1690: 1689: 1670: 1666: 1657: 1653: 1649: 1646: 1643: 1640: 1618: 1614: 1591: 1587: 1578: 1577: 1566: 1564: 1560: 1556: 1552: 1547: 1546:(see below). 1545: 1536: 1532: 1527: 1523: 1518: 1514: 1509: 1505: 1500: 1496: 1493: 1492: 1491: 1489: 1481: 1477: 1472: 1468: 1463: 1459: 1454: 1450: 1446: 1445: 1444: 1438: 1434: 1429: 1425: 1420: 1416: 1411: 1407: 1404: 1403: 1402: 1399: 1391: 1388: 1383: 1379: 1364: 1356: 1352: 1337: 1329: 1325: 1320: 1318: 1314: 1301: 1292: 1270: 1262: 1259: 1252:for a unique 1237: 1234: 1230: 1226: 1223: 1216:has the form 1215: 1200: 1196: 1172: 1168: 1159: 1142: 1138: 1134: 1129: 1126: 1122: 1118: 1113: 1109: 1100: 1097: 1082: 1078: 1054: 1050: 1041: 1023: 1020: 1016: 1007: 989: 962: 958: 954: 949: 945: 936: 919: 906: 898: 893: 890: 886: 878: 875: 857: 853: 849: 844: 840: 831: 830: 829: 827: 805: 801: 778: 769: 763: 759: 750: 732: 705: 701: 678: 674: 651: 647: 638: 621: 608: 590: 561: 557: 548: 547: 546: 530: 526: 503: 499: 490: 487:, called the 474: 467: 463: 460:-dimensional 447: 438: 436: 418: 389: 374: 359: 357: 356: 336: 321: 317: 316: 309: 293: 285: 280: 264: 249: 245: 241: 237: 233: 229: 225: 220: 218: 217:automorphisms 214: 210: 206: 205:string theory 201: 199: 195: 191: 187: 183: 179: 155: 152: 134: 133:modular group 130: 129:modular forms 126: 121: 119: 115: 114:CAT(k) spaces 111: 107: 103: 99: 95: 91: 87: 83: 82:Ichirō Satake 78: 76: 72: 68: 64: 60: 56: 52: 43: 38: 35: 29: 27: 19: 13204:Arkani-Hamed 13102:Supergravity 13069:Moduli space 13053: 12966: 12961:Dirac string 12887:Gauge theory 12867:Loop algebra 12804:Black string 12667:GS formalism 12446: 12401:Moving frame 12396:Morse theory 12386:Gauge theory 12178:Tensor field 12107:Closed/Exact 12086:Vector field 12054:Distribution 11995:Hypercomplex 11990:Quaternionic 11727:Vector field 11685:Smooth atlas 11562: 11556: 11530: 11524: 11507: 11486: 11480: 11462:10.1142/1235 11453: 11437: 11403: 11391: 11373:Scott, Peter 11348: 11344: 11334:Scott, Peter 11315: 11311: 11259: 11253: 11230:. Springer. 11223: 11195: 11191: 11164: 11163:. Series A. 11158: 11127: 11102: 11098: 11079: 11054:math/0501093 11044: 11040: 11022:10.1142/1235 11013: 10980: 10967: 10935: 10916:10.1142/1235 10907: 10883: 10864: 10858: 10809: 10803: 10752: 10729: 10726:Bredon, Glen 10701:math/0010185 10691: 10687: 10659: 10647:. Retrieved 10640:the original 10605: 10599: 10576: 10541: 10513: 10495: 10481: 10476: 10465: 10460: 10449: 10443: 10438: 10427: 10423: 10413: 10408: 10396: 10384: 10372:. Retrieved 10357: 10350: 10338:. Retrieved 10323: 10316: 10283: 10279: 10269: 10260: 10237: 10230: 10222: 10217: 10202: 10197: 10189: 10184: 10175: 10164: 10152: 10140: 10128: 10116: 10104: 10092: 10080: 10048: 10038: 10019:math/0203100 10004: 9994: 9967: 9955: 9943: 9931: 9919: 9907: 9895: 9819:minor chords 9815:major chords 9811:musical sets 9799: 9692: 9690: 9666:Möbius strip 9659: 9561: 9557: 9553: 9444: 9438: 9403: 9400: 9366: 9362: 9316: 9314: 9244: 9233:music theory 9230: 9227:Music theory 9218: 9166:moduli space 9144: 9113: 9088: 9065: 9061: 9053: 9048: 9042: 9037: 9032: 9030: 9016: 9011: 9007: 9003: 8992: 8986: 8981: 8977: 8973: 8969: 8965: 8961: 8957: 8943: 8935:Applications 8928: 8924: 8914: 8909: 8905: 8901: 8897: 8896: 8891: 8889: 8876: 8872:adding to it 8867: 8617: 8609: 8575: 8564: 8554: 8546: 8499: 8488: 8480: 8476: 8466: 8405: 8396: 8386: 8382: 8374: 8350: 8346: 8336: 8332: 8323: 8313: 8302: 8298: 8290: 8286: 8275: 8267: 8235: 8229: 8226: 8106: 7940: 7938: 7932: 7928: 7924: 7917: 7913: 7909: 7896: 7875: 7868: 7860: 7856: 7852: 7848: 7846: 7840: 7837: 7833: 7826: 7822: 7815: 7808: 7804: 7802: 7797: 7793: 7789: 7786: 7781: 7777: 7773: 7771: 7765: 7761: 7757: 7753: 7749: 7745: 7737: 7733: 7729: 7725: 7721: 7717: 7715: 7710: 7706: 7702: 7698: 7690: 7685: 7680: 7676: 7672: 7668: 7664: 7660: 7656: 7652: 7648: 7644: 7642: 7637: 7633: 7629: 7626:Cayley graph 7621: 7617: 7611: 7610: 7608: 7592: 7588: 7585: 7581: 7577: 7573: 7569: 7565: 7561: 7557: 7553: 7549: 7545: 7541: 7537: 7535: 7529: 7526: 7523: 7516: 7509: 7504: 7500: 7496: 7492: 7488: 7484: 7480: 7476: 7472: 7468: 7464: 7459:of a finite 7457:flag complex 7452: 7448: 7443: 7442: 7440: 7437: 7420: 7413: 7405: 7389: 7382: 7375: 7367: 7360: 7353: 7346: 7335: 7331: 7327: 7323: 7319: 7315: 7308: 7282: 7278: 7276: 7270: 7266: 7262: 7258: 7254: 7250: 7246: 7242: 7235: 7231: 7223: 7219: 7215: 7211: 7201: 7193: 7189: 7182: 7171: 7161: 7157: 7153: 7149: 7139: 7135: 7132: 7128: 7113: 7106: 7098: 7094: 7092: 7075: 7071: 7057: 7053: 7049: 7042: 7035: 7028: 7021: 7014: 7009: 7003: 7000: 6993: 6989: 6982: 6978: 6971: 6965:transitively 6964: 6957: 6950: 6938: 6931: 6927: 6923: 6919: 6917: 6911: 6907: 6903: 6899: 6895: 6891: 6887: 6883: 6879: 6875: 6871: 6865: 6861: 6857: 6851: 6847: 6844:Galois group 6839: 6835:as follows: 6832: 6828: 6824: 6820: 6816: 6812: 6808: 6804: 6802: 6796: 6792: 6788: 6781: 6777: 6773: 6769: 6765: 6757: 6750: 6740: 6717: 6713: 6709: 6702: 6667: 6665: 6621: 6617: 6613: 6611: 6500: 6498: 6470: 6426: 6422: 6418: 6414: 6410: 6408: 6395: 6394: 6378: 6367: 6360: 6344: 6343:has been to 6338: 6304: 6272: 6264: 6260: 6248: 6244: 6240: 6236: 6232: 6228: 6224: 6217: 6213: 6209: 6205: 6190: 6186: 6181: 6177: 6172: 6168: 6163: 6159: 6154: 6147: 6141: 6137: 6132: 6128: 6123: 6119: 6114: 6110: 6104: 6098: 6092: 6086: 6080: 6076: 6068: 6060: 6057: 6053: 6031: 6009: 6007: 6000: 5995: 5990: 5985: 5980: 5976: 5971: 5967: 5959: 5953: 5947: 5938: 5934: 5929: 5925: 5920: 5915: 5910: 5905: 5883: 5875: 5869: 5863: 5859: 5855: 5851: 5829: 5821: 5817: 5813: 5808: 5804: 5800: 5795: 5789: 5785: 5780: 5775: 5771: 5767: 5758: 5754: 5749: 5744: 5739: 5735: 5731: 5726: 5722: 5717: 5713: 5708: 5704: 5699: 5696: 5689: 5685: 5680: 5674: 5670: 5666: 5662: 5657: 5653: 5645: 5641: 5636: 5627: 5621: 5617: 5613: 5609: 5601: 5597: 5593: 5591: 5574: 5571: 5569: 5559: 5555: 5553: 5544: 5540: 5536: 5532: 5510: 5506: 5484: 5480: 5458: 5451: 5429: 5422: 5414: 5410: 5406: 5404: 5399: 5391: 5389: 5383: 5376: 5368: 5364: 5360: 5356: 5354: 5348: 5344: 5336: 5332: 5330: 5321: 5317: 5309: 5305: 5304:-simplex of 5301: 5293: 5289: 5278: 5251: 5247: 5243: 5239: 5237: 5231: 5226: 5222: 5217: 5213: 5208: 5204: 5200: 5195: 5191: 5186: 5182: 5175: 5168: 5163: 5159: 5152: 5142: 5138: 5127: 5120: 5116: 5107: 5097: 5075: 5070: 5066: 5061: 5057: 5051: 5047: 5041: 5037: 5030: 5025: 5021: 5014: 5007: 5005: 4996: 4992: 4988: 4982: 4978: 4970: 4964: 4956: 4948: 4942: 4934: 4924: 4917: 4915: 4910: 4888: 4866: 4864: 4858: 4854: 4849: 4845: 4840: 4836: 4831: 4827: 4819: 4814: 4812: 4806: 4801: 4795: 4791: 4787: 4782: 4778: 4770: 4764: 4760: 4755: 4730: 4724: 4719: 4697: 4693: 4689: 4684: 4680: 4676: 4668: 4666: 4653: 4642: 4637: 4633: 4622:sheaf theory 4614: 4610: 4606: 4601: 4597: 4592: 4584: 4582: 4576: 4572: 4549: 4545: 4540: 4536: 4513: 4509: 4508:, ψ' : 4504: 4500: 4477: 4473: 4468: 4464: 4441: 4437: 4414: 4410: 4387: 4383: 4378: 4372: 4368: 4345: 4341: 4318: 4314: 4309: 4305: 4300: 4296: 4273: 4269: 4264: 4260: 4258: 4115: 4110: 4106: 4096: 4092: 4088: 4042: 4038: 4033: 4029: 4025: 4021: 4013: 4008: 4004: 3981: 3977: 3972: 3968: 3964: 3962: 3952: 3948: 3942: 3935: 3931: 3927: 3918: 3914: 3910: 3908: 3902: 3898: 3893: 3889: 3884: 3880: 3875: 3871: 3864: 3859: 3855: 3849: 3845: 3840: 3836: 3831: 3827: 3820: 3815: 3809: 3805: 3803: 3798: 3755: 3749: 3742: 3738: 3731: 3724: 3717: 3712: 3705: 3700: 3687: 3680: 3673: 3666: 3655: 3558: 3552: 3543:is given by 3540: 3533: 3529: 3525: 3521: 3519: 3509: 3507: 3496: 3479: 3475: 3471: 3469: 3463:with unique 3461:length space 3453:metric space 3443: 3435: 3429: 3420: 3333: 3232: 3150: 3149: 3138: 3119: 3114: 3111:contractible 3108: 3099: 3095: 3091: 3087: 3085: 3080: 3066: 3060: 3051: 3046: 3044: 3019: 3017: 3005: 2996: 2991: 2986: 2982: 2978: 2974: 2967: 2950: 2946: 2941: 2937: 2902: 2898: 2893: 2889: 2884: 2880: 2875: 2871: 2866: 2862: 2858: 2854: 2850: 2846: 2842: 2837: 2831: 2826: 2821: 2817: 2813: 2809: 2805: 2801: 2797: 2793: 2782: 2778: 2771: 2767: 2763: 2756: 2752: 2748: 2744: 2741: 2733: 2721: 2708: 2659: 2654: 2352: 2044: 2042: 2031: 2014: 2012: 2003: 1924: 1873: 1688:Lie groupoid 1686: 1574: 1572: 1554: 1550: 1548: 1543: 1541: 1534: 1530: 1525: 1521: 1516: 1512: 1507: 1503: 1498: 1494: 1487: 1485: 1479: 1475: 1470: 1466: 1461: 1457: 1452: 1448: 1442: 1436: 1432: 1427: 1423: 1418: 1414: 1409: 1405: 1397: 1389: 1386: 1381: 1377: 1354: 1350: 1327: 1323: 1321: 1316: 1294: 1288: 1188: 1070: 1039: 872:there is an 825: 823: 748: 747:, called an 488: 461: 439: 373:open subsets 370: 352: 319: 312: 308:homeomorphic 283: 281: 240:finite group 234:with finite 221: 202: 122: 117: 101: 89: 79: 71:finite group 58: 48: 31: 25: 22: 18: 13564:Silverstein 13064:Orientifold 12799:Black holes 12794:Black brane 12751:Dual photon 12346:Levi-Civita 12336:Generalized 12308:Connections 12258:Lie algebra 12190:Volume form 12091:Vector flow 12064:Pushforward 12059:Lie bracket 11958:Lie algebra 11923:G-structure 11712:Pushforward 11692:Submanifold 10694:: 195–290. 10374:26 February 10340:26 February 10286:: 678–686. 10133:Satake 1957 9987:Bredon 1972 9888:Satake 1956 9857:Orientifold 9155:K3 surfaces 9086:spacetime. 8728:2, 2, 2, 2 8674:2, 2, 2, 2 7576:and edges ( 7169:prime field 6373:); in 1979 6330:orbispaces. 5784:identifies 5649:' onto the 5252:developable 4641:; the data 3591:whenever σ 3486:, then the 3151:Definition: 3088:developable 2912:constructs 2899:developable 2841:-subset of 2375:pseudogroup 1927:if the map 1872:called the 1038:, called a 1006:equivariant 977:there is a 545:, there is 98:3-manifolds 13689:Categories 13584:Strominger 13579:Steinhardt 13574:Staudacher 13489:Polchinski 13439:Nanopoulos 13399:Mandelstam 13379:Kontsevich 13219:Berenstein 13147:Holography 13127:Superspace 13026:K3 surface 12985:Worldsheet 12900:Instantons 12528:Background 12469:Stratifold 12427:Diffeology 12223:Associated 12024:Symplectic 12009:Riemannian 11938:Hyperbolic 11865:Submersion 11773:Hopf–Rinow 11707:Submersion 11702:Smooth map 11080:Diffeology 10649:6 December 10533:References 9948:Scott 1983 9936:Serre 1970 9128:space-time 9047:, because 8626:Parabolic 7451:and lines 7125:Fano plane 6906:, so that 6294:polyhedron 6072:such that 5916:gluing map 5812:, with St( 5602:orbihedron 5588:Definition 5575:simplicial 5417:, namely: 4596:such that 3821:equivalent 3819:yields an 3516:Definition 3426:Orbispaces 3141:diffeology 2770:, so that 2755:action of 2753:reflection 2000:proper map 1563:isometries 1401:such that 1040:gluing map 186:3-manifold 118:orbihedron 90:V-manifold 13619:Veneziano 13499:Rajaraman 13394:Maldacena 13284:Gopakumar 13234:Dijkgraaf 13229:Curtright 12895:Anomalies 12774:NS5-brane 12695:U-duality 12690:S-duality 12685:T-duality 12351:Principal 12326:Ehresmann 12283:Subbundle 12273:Principal 12248:Fibration 12228:Cotangent 12100:Covectors 11953:Lie group 11933:Hermitian 11876:manifolds 11845:Immersion 11840:Foliation 11778:Noether's 11763:Frobenius 11758:De Rham's 11753:Darboux's 11644:Manifolds 11575:CiteSeerX 11119:126092375 10787:0711.1346 10718:119624092 10624:119617693 10308:0550-3213 9721:⋯ 9601:× 9598:⋯ 9595:× 9480:⁡ 9452:logarithm 9147:landscape 9122:requires 8879:July 2008 8358:Elliptic 8312:1/2 + 1/2 8115:χ 8070:− 8055:∑ 8051:− 8019:− 8004:∑ 8000:− 7981:χ 7975:χ 7964:given by 7952:χ 7908:of order 7455:* in the 7408:of the 2- 7296:∩ 6827:. Define 6772:− 1)/2 = 6728:⊂ 6680:− 6629:with the 6588:Γ 6584:→ 6572:Γ 6554:Γ 6548:⋆ 6535:Γ 6041:→ 6019:→ 5893:→ 5839:→ 5762:onto the 5566:Orbihedra 5520:→ 5494:→ 5468:→ 5439:→ 5230:for some 5085:→ 5045:lies in Γ 4898:→ 4876:→ 4741:→ 4728: : Γ 4707:→ 4560:→ 4524:→ 4488:→ 4452:∩ 4425:∩ 4398:→ 4356:∩ 4329:→ 4284:→ 4234:∩ 4221:∩ 4208:⊃ 4195:∩ 4182:⊃ 4149:⊂ 4129:⊂ 4076:∩ 4056:∩ 4026:non-empty 3992:→ 3913:whenever 3782:τ 3779:⊂ 3776:σ 3773:⊂ 3770:ρ 3767:⊂ 3764:π 3642:⊂ 3622:⊂ 3599:⊂ 3575:→ 3564: : Γ 3465:geodesics 3436:orbispace 3401:∞ 3374:∞ 3347:∞ 3071:extension 3065:manifold 3032:groupoids 3001:unit disc 2787:cocompact 2671:⇉ 2638:∈ 2607:− 2599:∘ 2569:↦ 2543:∈ 2460:Γ 2406:φ 2256:∈ 2230:∼ 2126:≃ 2082:⇉ 1976:× 1963:→ 1870:Lie group 1868:, is the 1842:− 1834:∩ 1817:− 1756:∈ 1663:→ 1365:⊂ 1338:⊂ 1267:Γ 1263:∈ 1231:ψ 1227:∘ 1139:φ 1123:ψ 1119:∘ 1110:φ 1017:ψ 986:Γ 955:⊂ 916:Γ 912:→ 903:Γ 874:injective 850:⊂ 775:Γ 729:Γ 648:φ 618:Γ 353:orbifold 313:orbifold 294:π 242:; thus a 13674:Zwiebach 13629:Verlinde 13624:Verlinde 13599:Townsend 13594:Susskind 13529:Sagnotti 13494:Polyakov 13449:Nekrasov 13414:Minwalla 13409:Martinec 13374:Knizhnik 13369:Klebanov 13364:Kapustin 13329:'t Hooft 13264:Fischler 13199:Aganagić 13170:M-theory 13059:Conifold 13054:Orbifold 13037:manifold 12978:Geometry 12784:M5-brane 12779:M2-brane 12716:Graviton 12632:F-theory 12447:Orbifold 12442:K-theory 12432:Diffiety 12156:Pullback 11970:Oriented 11948:Kenmotsu 11928:Hadamard 11874:Types of 11823:Geodesic 11648:Glossary 11597:16825563 11519:(1982). 11436:(1983). 11402:(2003). 11390:(1970). 11370:Errata: 11336:(1983). 11306:(1957). 11296:16578464 11248:(1956). 11222:(1985). 11188:= q = 0" 11071:15210173 10842:35229232 10834:18420928 10728:(1972). 10678:56349823 10002:(2002). 9826:See also 9026:D-branes 9018:D-branes 8950:manifold 8828:Annulus 8715:3, 3, 3 8702:2, 4, 4 8689:2, 3, 6 8661:3, 3, 3 8648:2, 4, 4 8635:2, 3, 6 8540:2, 3, 5 8527:2, 3, 4 8514:2, 3, 3 8446:2, 3, 5 8433:2, 3, 4 8420:2, 3, 3 7495:lies on 7487:lies on 7479:lies on 7326:. Since 7318:) where 6943:properly 6708:and set 5778:; thus φ 4977:· ··· · 4382: : 4313: : 4268: : 3536:) on an 3190:, where 3073:of Γ by 2928:for the 2714:Examples 2141:, where 1691:if both 1576:groupoid 607:faithful 462:orbifold 351:and its 306:; it is 102:orbifold 63:manifold 59:orbifold 55:geometry 51:topology 13604:Trivedi 13589:Sundrum 13554:Shenker 13544:Seiberg 13539:Schwarz 13509:Randall 13469:Novikov 13459:Nielsen 13444:Năstase 13354:Kallosh 13339:Gibbons 13279:Gliozzi 13269:Friedan 13259:Ferrara 13244:Douglas 13239:Distler 12789:S-brane 12769:D-brane 12726:Tachyon 12721:Dilaton 12535:Strings 12391:History 12374:Related 12288:Tangent 12266:) 12246:) 12213:Adjoint 12205:Bundles 12183:density 12081:Torsion 12047:Vectors 12039:Tensors 12022:) 12007:) 12003:, 12001:Pseudo− 11980:Poisson 11913:Finsler 11908:Fibered 11903:Contact 11901:) 11893:Complex 11891:) 11860:Section 11605:2877171 11567:Bibcode 11558:Science 11452:(ed.). 11264:Bibcode 11212:2373947 11012:(ed.). 10906:(ed.). 10814:Bibcode 10805:Science 10288:Bibcode 10243:Bibcode 9817:and 12 9686:⁠ 9674:⁠ 9564:-torus 9404:ordered 9301:Voronoi 9246:Science 8908:. Then 8671:Sphere 8658:Sphere 8645:Sphere 8632:Sphere 8443:Sphere 8430:Sphere 8417:Sphere 8401:Sphere 8379:Sphere 8364:Sphere 8353:> 1 8326:> 1 8305:> 1 8295:Sphere 8278:> 1 8272:Sphere 7740:, then 7584:), for 7123:of the 7063:Γ acts 6945:on the 6882:) = 1, 6753:= exp 2 6642:colimit 6507:of the 6375:Mumford 5572:regular 5318:simplex 5181:, ..., 5167:) and ( 5158:, ..., 5121:regular 5020:, ..., 3959:Example 3928:simplex 3457:metrics 2924:in the 2725:is the 2034:compact 1289:As for 176:on the 13669:Zumino 13664:Zaslow 13649:Yoneya 13639:Witten 13559:Siegel 13534:Scherk 13504:Ramond 13479:Ooguri 13404:Marolf 13359:Kaluza 13344:Kachru 13334:Hořava 13324:Harvey 13319:Hanson 13304:Gubser 13294:Greene 13224:Bousso 13209:Atiyah 12761:Branes 12571:Theory 12356:Vector 12341:Koszul 12321:Cartan 12316:Affine 12298:Vector 12293:Tensor 12278:Spinor 12268:Normal 12264:Stable 12218:Affine 12122:bundle 12074:bundle 12020:Almost 11943:Kähler 11899:Almost 11889:Almost 11883:Closed 11783:Sard's 11739:(list) 11603: 11595: 11577: 11468: 11422: 11294: 11287:528292 11284: 11234: 11210: 11138: 11117: 11086: 11069: 11028: 10997: 10952: 10922: 10890: 10840: 10832: 10769: 10740: 10716: 10676: 10666: 10622: 10583: 10560: 10365: 10331: 10306: 10253: 10209: 10068: 10026: 9670:triads 9668:; for 9309:triads 8976:where 8804:Torus 8498:2, 2, 8404:2, 2, 8266:1 + 1/ 8170:, and 8107:where 7491:* and 7404:under 7210:= < 6926:, and 6898:(1) = 6823:, and 6746:. Let 6511:over Γ 6479:into Γ 6415:simple 5873:into Γ 5600:'. An 5269:from Γ 5006:where 4696:where 4626:gerbes 4012:. Let 3911:simple 3835:= (Ad 2995:where 2790:proper 2742:double 2740:, its 2002:, and 1925:proper 358:is 1. 248:double 13609:Turok 13519:Roček 13484:Ovrut 13474:Olive 13454:Neveu 13434:Myers 13429:Mukhi 13419:Moore 13389:Linde 13384:Klein 13309:Gukov 13299:Gross 13289:Green 13274:Gates 13254:Dvali 13214:Banks 12464:Sheaf 12238:Fiber 12014:Rizza 11985:Prime 11816:Local 11806:Curve 11668:Atlas 11601:S2CID 11553:(PDF) 11404:Trees 11380:(PDF) 11341:(PDF) 11208:JSTOR 11115:S2CID 11067:S2CID 11049:arXiv 10838:S2CID 10800:(PDF) 10782:arXiv 10714:S2CID 10696:arXiv 10643:(PDF) 10636:(PDF) 10620:S2CID 10523:(PDF) 10505:(PDF) 10014:arXiv 9873:Notes 9662:dyads 9371:torus 8892:small 8794:2, 2 8781:2, 2 8778:Disk 8764:Disk 8750:Disk 8742:2, 2 8736:Disk 8723:Disk 8710:Disk 8697:Disk 8684:Disk 8585:Disc 8582:1/12 8569:Disc 8551:Disc 8535:Disc 8532:1/60 8522:Disc 8519:1/24 8509:Disc 8506:1/12 8493:Disc 8471:Disc 8456:Disc 8440:1/30 8427:1/12 8349:> 8341:Disk 8335:+ 1/2 8318:Disk 8301:> 8246:Type 7847:then 7374:into 7345:into 7265:*) + 7010:Then 6850:over 6519:and Γ 6483:and Γ 6413:is a 6349:Serre 6320:girth 6292:of a 5933:into 5908:, a Γ 5770:) of 5721:over 5684:with 5678:' / Γ 5580:on a 5392:third 5365:sizes 5335:' of 5234:in Γ. 4768:and Γ 4376:and φ 3863:= 3444:rigid 3415:-map. 3306:with 3003:and Γ 2271:with 2004:étale 1998:is a 1042:, of 693:onto 464:is a 228:group 92:; by 57:, an 13634:Wess 13614:Vafa 13524:Rohm 13424:Motl 13349:Kaku 13314:Guth 13249:Duff 12331:Form 12233:Dual 12166:flow 12029:Tame 12005:Sub− 11918:Flat 11798:Maps 11593:PMID 11466:ISBN 11420:ISBN 11292:PMID 11232:ISBN 11136:ISBN 11084:ISBN 11026:ISBN 10995:ISBN 10950:ISBN 10920:ISBN 10888:ISBN 10855:, I" 10830:PMID 10767:ISBN 10738:ISBN 10674:OCLC 10664:ISBN 10651:2007 10581:ISBN 10558:ISBN 10451:Time 10376:2012 10363:ISBN 10342:2012 10329:ISBN 10304:ISSN 10251:ISBN 10207:ISBN 10066:ISBN 10024:ISBN 9660:For 8900:Let 8414:1/6 8289:+ 1/ 8263:Bad 7859:and 7764:) = 7760:, σ( 7756:) = 7752:, σ( 7748:) = 7728:and 7716:for 7705:) = 7683:= Γ 7671:for 7655:and 7602:The 7370:of Γ 7334:) = 7288:(f) 7257:* + 7253:) = 7156:) = 7103:link 7097:and 7052:and 7027:and 6894:and 6890:) = 6803:Let 6666:Let 6620:and 6359:for 6109:(Ad 6066:in Γ 5998:in Γ 5867:of Γ 5764:star 5734:and 5651:link 5592:Let 5483:and 5308:and 5151:if ( 5056:and 4823:and 4817:in Γ 4813:for 4667:The 4624:and 4609:" = 4590:in Γ 4068:··· 3813:in Γ 3661:in Γ 3497:good 3478:and 3470:Let 3098:. A 3092:good 3036:loop 2970:/ Γ. 2903:good 2306:and 1718:and 1447:(Ad 1395:in Γ 793:and 53:and 26:dead 13644:Yau 13569:Sơn 13549:Sen 12253:Jet 11585:doi 11563:313 11535:doi 11491:doi 11458:doi 11412:doi 11361:hdl 11353:doi 11320:doi 11282:PMC 11272:doi 11200:doi 11196:101 11169:doi 11107:doi 11059:doi 11045:362 11018:doi 10985:doi 10940:doi 10912:doi 10869:doi 10822:doi 10810:320 10757:doi 10706:doi 10692:162 10610:doi 10550:doi 10296:doi 10284:261 10058:doi 9471:log 9114:In 8944:In 8874:. 8563:1/2 8487:1/2 8331:1/2 8230:bad 7720:in 7697:τ·( 7679:of 7636:in 7591:in 7540:in 7505:xyz 7483:*, 7412:of 7277:on 7222:, − 7181:of 7160:of 7079:'s. 6990:ρσρ 6864:on 6846:of 6768:= ( 6513:ABC 6493:ABC 6489:ABC 6473:ABC 6465:ABC 6456:, Γ 6452:, Γ 6441:, Γ 6437:, Γ 6263:in 6235:of 6191:jkm 6182:ijm 6173:ikm 6164:ijk 6115:ijk 6090:= ψ 6081:ijk 6064:ijk 6008:If 5975:to 5957:= φ 5924:of 5854:of 5820:at 5803:in 5793:/ Γ 5766:St( 5703:on 5665:in 5661:of 5640:of 5612:of 5457:if 5325:ρστ 5284:= θ 5074:if 4986:· e 4962:· e 4940:· e 4865:if 4859:ijk 4799:= ψ 4692:to 4646:ρστ 4602:ρστ 4588:ρστ 4550:ijk 4514:ijk 4388:ijk 4379:ijk 4141:σ 3963:If 3947:of 3936:ρστ 3919:ρστ 3894:ρστ 3860:ρστ 3728:πρτ 3721:ρστ 3670:ρστ 3659:ρστ 3096:bad 3090:or 3045:An 3030:of 3026:or 2989:/ Γ 2973:If 2901:or 2825:of 2812:in 2804:in 2777:If 2732:If 2013:An 1903:at 1876:of 1535:jkm 1526:ijm 1517:ikm 1508:ijk 1453:ijk 1410:ijk 1393:ijk 1187:to 666:of 576:of 440:An 375:of 322:is 318:of 230:of 203:In 13691:: 12945:, 12938:, 12931:, 12924:, 12244:Co 11599:. 11591:. 11583:. 11573:. 11561:. 11555:. 11529:. 11523:. 11487:52 11485:. 11464:. 11418:. 11375:. 11359:. 11349:15 11347:. 11343:. 11314:. 11310:. 11290:. 11280:. 11270:. 11260:42 11258:. 11252:. 11206:. 11194:. 11190:. 11165:38 11157:. 11134:. 11130:. 11113:. 11103:12 11101:. 11065:. 11057:. 11043:. 11024:. 10993:. 10948:. 10918:. 10865:47 10863:. 10857:. 10836:. 10828:. 10820:. 10808:. 10802:. 10765:. 10736:. 10732:. 10712:. 10704:. 10690:. 10672:. 10618:. 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6096:·ψ 6093:jk 6087:ik 6084:·ψ 5991:ij 5988:·ψ 5954:ij 5951:·ψ 5921:ij 5864:ij 5824:. 5562:. 5543:, 5539:, 5386:". 5351:·σ 5282:στ 5242:= 5212:= 5141:= 5100:. 5071:ij 5062:ji 5029:, 5013:, 4955:· 4913:. 4850:ij 4841:jk 4835:= 4832:ik 4802:ij 4796:ij 4790:· 4786:· 4783:ij 4765:ij 4725:ij 4685:ij 4657:στ 4580:. 4541:ij 4478:ij 4319:ij 4310:ij 4304:, 4119:στ 4111:ij 4037:= 3976:: 3903:ρτ 3888:)· 3885:στ 3876:ρσ 3868:ρσ 3856:g' 3850:στ 3844:)· 3841:στ 3832:στ 3828:f' 3810:στ 3750:ρσ 3739:στ 3730:= 3723:) 3691:στ 3684:ρσ 3679:= 3677:ρτ 3672:)· 3634:σ 3562:στ 3520:A 3512:. 3499:). 3083:. 3042:. 2888:Γ 2796:= 2345:. 2040:. 2010:. 1809::= 1565:. 1520:= 1511:)· 1499:km 1480:ij 1471:jk 1465:= 1462:ik 1456:)· 1437:ij 1428:jk 1422:= 1419:ik 437:. 279:. 219:. 120:. 77:. 13035:2 13033:G 13002:? 12967:p 12952:) 12949:8 12947:E 12942:7 12940:E 12935:6 12933:E 12928:4 12926:F 12921:2 12919:G 12917:( 12513:e 12506:t 12499:v 12262:( 12242:( 12018:( 11999:( 11897:( 11887:( 11650:) 11646:( 11636:e 11629:t 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2827:m 2822:m 2818:V 2814:M 2810:m 2806:X 2802:x 2798:M 2794:X 2783:n 2779:M 2772:N 2768:N 2764:M 2760:2 2757:Z 2749:N 2745:M 2734:N 2722:i 2694:G 2674:M 2668:G 2641:X 2635:x 2615:) 2610:1 2603:s 2596:t 2593:( 2588:x 2583:m 2580:r 2577:e 2574:g 2564:x 2560:) 2554:X 2550:G 2546:( 2540:g 2518:X 2514:G 2491:i 2487:V 2464:i 2437:X 2433:G 2410:i 2385:X 2361:X 2329:y 2326:= 2323:) 2320:g 2317:( 2314:t 2294:x 2291:= 2288:) 2285:g 2282:( 2279:s 2259:G 2253:g 2233:y 2227:x 2207:M 2187:G 2166:| 2162:G 2158:/ 2154:M 2150:| 2129:X 2122:| 2118:G 2114:/ 2110:M 2106:| 2085:M 2079:G 2055:X 2028:. 1984:0 1980:G 1971:0 1967:G 1958:1 1954:G 1950:: 1947:) 1944:t 1941:, 1938:s 1935:( 1911:x 1889:1 1885:G 1856:) 1853:x 1850:( 1845:1 1838:t 1831:) 1828:x 1825:( 1820:1 1813:s 1804:x 1800:) 1794:1 1790:G 1786:( 1764:0 1760:G 1753:x 1731:1 1727:G 1704:0 1700:G 1671:0 1667:G 1658:1 1654:G 1650:: 1647:t 1644:, 1641:s 1619:1 1615:G 1592:0 1588:G 1538:. 1531:g 1529:· 1522:g 1513:g 1504:g 1502:( 1495:f 1476:f 1474:· 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Index