A point in an orbifold whose neighborhood is modeled on the quotient of Euclidean space by a nontrivial finite group action, producing a localized singularity characterized by the isotropy (stabilizer) group and its linear representation.

Principle

Orbifold singularities arise when local symmetry groups identify directions; the neighborhood is locally R^n/G for a finite group G, and the conjugacy class of G (and its action) organizes the possible singular types and local invariants.

Demonstration

In a 2-dimensional cone orbifold, a cone point of order n is modeled on R2 modulo rotation by 2π/n; in 3D, a point with local stabilizer a finite subgroup of SO(3) yields quotient singularities classified by those finite rotation groups.

Misapplication

Treating orbifold singular points as ordinary manifold points or as purely topological singularities without recording the group action and isotropy information; or confusing orbifold points with branch points of different local models.

Consequence

Correct recognition allows computation of orbifold fundamental groups, orbifold Euler characteristics, and consistent application of covering space and index theorems adapted to the isotropy; it governs allowed local geometries and symmetry-preserving maps.

Reversal

The reversal is a regular point with trivial isotropy where local charts are genuine Euclidean neighborhoods; reversing removes group quotient data and restores manifold local structure.

Boundary

Applies to spaces with an orbifold atlas built from quotient charts R^n/G with finite G; excludes wild quotient constructions with non-finite stabilizers, non-effective actions, or accumulation of singular points that break orbifold axioms.

Semantic Tension

Tension exists between orbifold singular points and algebraic singularities with similar local topology but lacking a finite-group quotient description; distinguishing the group-action data is essential for orbifold-specific invariants.

Synthesis

An orbifold singular point is a local quotient singularity modeled on R^n by a finite group action; its isotropy group and representation determine the local geometry, topology, and permissible orbifold structures and maps.