Orbifold

An orbifold is a generalization of a manifold that permits singular points with local symmetry — points where the space looks like the quotient of Euclidean space by a finite group action. Orbifolds arise naturally in geometric topology as the quotient spaces of group actions on manifolds, and in dynamical systems as the phase spaces of systems with symmetry. The orbifold fundamental group, developed by Thurston, encodes the global topology together with the local singular structure, and the orbifold Euler characteristic provides a topological invariant that counts singularities with rational weights. The eight Thurston geometries — the foundation of the geometrization conjecture — are naturally formulated on orbifolds, because many 3-manifolds are best understood as orbifold quotients rather than as manifolds in the strict sense. The orbifold notation, a compact symbolic language for describing 2-dimensional orbifolds, is one of the most efficient classification schemes in all of mathematics.

An orbifold is a manifold that has learned to live with its own broken symmetries. The singular points are not defects; they are memory — the record of a group action that the space has forgotten but the topology remembers. The orbifold Euler characteristic is the simplest case of a general principle: when symmetry is broken, the accounting must change.