OrbifoldEdit

An orbifold is a space that generalizes the familiar notion of a manifold by allowing controlled singularities that arise from local symmetries. Locally, an orbifold looks like the quotient of Euclidean space by a finite group action, so charts come equipped with a finite group that acts on a coordinate neighborhood. The global structure is described by an atlas of these local models together with compatibility data on overlaps. This framework captures spaces that are smooth apart from a prescribed, finite-type singularity pattern and is a standard setting in modern geometry and topology.

Orbifolds were introduced by Satake in the 1950s under the term V-manifolds and were later developed and popularized in geometric topology and differential geometry, notably by Thurston and collaborators. The language of orbifolds provides a natural way to study quotients by symmetry, moduli spaces, and geometric structures that are locally homogeneous but may fail to be globally so because of singular points. Invariants such as the orbifold fundamental group and the orbifold Euler characteristic encode both the global topology and the local symmetry data, and the theory has deep connections to algebraic geometry through stacks, for which the orbifold viewpoint offers a concrete geometric model in many settings. For broader context, see Satake and William Thurston; for a bridge to algebraic geometry, see Deligne–Mumford stack.

Definition and local structure

• Local models and charts

  • An orbifold is covered by charts (U, 𝐺, φ) where U is an open subset of R^n, 𝐺 is a finite group acting smoothly on U, and φ maps U onto an open subset of the topological space X such that φ factors through the quotient U/𝐺. The point x in X has a local neighborhood modeled on R^n/𝐺x with 𝐺x the stabilizer of a preimage point under φ.
  • The collection of compatible charts forms an atlas that specifies how different local models fit together along overlaps, including how the local symmetry groups relate when charts intersect. See group action and quotient space for background.

• Morphisms and maps

  • A map between orbifolds respects the local uniformizing data, in that it can be described by compatible families of equivariant maps between the local uniformizing charts. This leads to a notion of orbifold maps and suborbifolds, which generalize ordinary continuous maps and submanifolds.

• Manifolds as a special case

  • If every local group 𝐺x is trivial, an orbifold reduces to a manifold. Conversely, an orbifold with nontrivial local groups encodes singularities that reflect local symmetries rather than pathologies.

Global quotients and developable orbifolds

• Global quotients

  • A primary way to obtain orbifolds is by taking a manifold M and forming the quotient M/Γ under a properly discontinuous action of a finite or discrete group Γ. Such orbifolds are called developable or global quotient orbifolds, and their structure is tightly linked to the acting group and the fixed-point set of the action. This viewpoint connects to group action and quotient space.

• Developable versus non-developable

  • Not all orbifolds arise as global quotients of a manifold by a group action; some are non-developable, meaning they cannot be realized as such a quotient. This distinction motivates the broader perspective that orbifolds generalize quotients while retaining a concrete local model. See also Deligne–Mumford stack for an alternative viewpoint in algebraic geometry.

• Invariants and structure

  • The orbifold fundamental group captures paths and loops while recording how local symmetry data affects their liftings. The orbifold Euler characteristic extends the classical Euler characteristic by weighting contributions from singular points according to their local group orders. These invariants interact with notions of coverings, homotopy, and (co)homology theories adapted to orbifolds.

Examples

• Global quotients

  • Take R^n with a finite group G acting linearly; the quotient R^n/ G is a basic local model for an orbifold chart. Global constructions include quotients of spheres or tori by finite group actions, yielding a wide range of orbifold topologies.

• 2-dimensional orbifolds

  • The 2-sphere with a finite set of cone points, denoted S^2(p1, p2, ..., pk), is a classical family of 2D orbifolds. Each cone point represents a local cyclic symmetry of order pi. Triangular and football (or lens-type) orbifolds arise from combining such local data in various ways.

• Special terminology

  • A teardrop orbifold is a 2-sphere with a single cone point, while a football orbifold has two cone points of specified orders. These examples illuminate how local group data controls the global geometry.

Invariants and structures

• Orbifold fundamental group and coverings

  • The fundamental group of an orbifold generalizes the ordinary fundamental group by incorporating the local group actions, leading to orbifold coverings and a richer theory of loops with symmetry data.

• Orbifold Euler characteristic

  • This invariant adjusts the classical Euler characteristic by accounting for the order of the local symmetry groups at singular points, providing a useful scalar for comparing orbifolds and for understanding their geometric structure.

• Developable versus non-developable and stacks

  • The developable/non-developable distinction mirrors similar ideas in other areas of geometry. In algebraic geometry, stacks (such as Deligne–Mumford stacks) offer a parallel framework for handling objects with automorphisms, and the orbifold viewpoint often aligns with these concepts in differential geometry and topology.

In physics and applied contexts

• String theory and beyond

  • Orbifolds serve as a controlled method to construct new spaces by modding out by discrete symmetries, enabling tractable models in string theory and related areas. Orbifold conformal field theories and related techniques exploit the finite-group quotient structure to yield consistent quantum models and to study phenomena such as twisted sectors and enhanced symmetries.

• Geometric applications

  • In topology and geometry, orbifolds provide a natural setting for studying spaces that are locally homogeneous yet possess singularities, facilitating the study of moduli spaces, geometric structures on manifolds, and questions about curvature, tessellations, and symmetry breaking.

See also

• Satake
• William Thurston
• Deligne–Mumford stack
• Manifold
• Group action
• Quotient space
• V-manifold