orbifolds

An orbifold is a space that is locally modeled on the quotient of Euclidean space by a finite group action. More precisely, an n-dimensional orbifold consists of an underlying topological space X equipped with an atlas of 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 X such that U/G is homeomorphic to that subset. Compatibility on overlaps is provided by equivariant diffeomorphisms between charts. The action is assumed to be effective, meaning that nontrivial elements of G act nontrivially on U. The local model U/G encodes the possible singularities: a point x in X has a nontrivial local group Gx exactly when x is a singular point of the orbifold.

Intuitively, orbifold singularities arise from points fixed by nontrivial elements of the local group. If all local groups are trivial, the orbifold is a manifold. A fundamental class of examples are global quotients M/G, where a finite group G acts on a manifold M and the quotient inherits an orbifold structure. Other examples include teardrop orbifolds (a 2-sphere with a single cone point) and football orbifolds S^2(p,q) with two cone points of orders p and q.

Historically, orbifolds were introduced by Satake (as V-manifolds) and further developed by Thurston, with important roles in geometric topology and mathematical physics. Key notions include the orbifold fundamental group, Euler characteristic, and orbifold cohomology. Maps between orbifolds are required to lift to equivariant maps between charts, and one can define orbibundles and Riemannian metrics compatible with the local group actions.