##  [Orbifold](/orbifold-0) 

 Definition

An orbifold is a topological space (or more structured object) locally modeled on the quotient of Euclidean space R^n by a finite group action; charts record local isotropy groups so that singular points have well-defined finite stabilizers.

 

 

 

 

 

 





## Principle

Principle

The organizing idea is equivariant local modeling: spaces may be locally like R^n/G for finite G, allowing controlled singularities while retaining manifold-like atlas structure and notions of orbifold fundamental group and tangent orbibundles.

 

 

 

 

 





## Demonstration

Demonstration

Example: a 2-dimensional orbifold can be a quotient of the sphere by a finite rotation group producing cone points; in geometric group theory or stringy geometry, orbifold quotients of manifolds by finite group actions provide canonical examples.

 

 

 

 

## Misapplication

Misapplication

Calling any space with isolated singularities an orbifold without specifying compatible local finite group quotient charts or failing to track isotropy representations misapplies the concept.

 

 

 

 

 





## Consequence

Consequence

Correctly using orbifolds permits extension of manifold theories (cohomology, index theorems, curvature formulas) to settings with finite quotient singularities and yields modified invariants accounting for isotropy contributions.

 

 

 

 

## Reversal

Reversal

The reversal is a manifold: everywhere locally modeled on Euclidean space with trivial stabilizer; manifolds lack the prescribed finite-group local symmetries that generate orbifold singularities.

 

 

 

 

 





## Boundary

Boundary

Scope: spaces locally isomorphic to R^n/G for finite groups G with compatible chart transitions; excludes wild nonfinite quotients, general stratified spaces without finite isotropy, and spaces whose singularities lack local group quotient descriptions.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension exists between 'orbifold' and 'stack' or 'singular space': stacks record finer automorphism data and functoriality, while naive singular spaces may lack isotropy bookkeeping that orbifolds require.

 

 

 

 

 





## Synthesis

Synthesis

An orbifold is a manifold-like object with controlled finite-group quotient singularities: local charts are quotients R^n/G, isotropy groups are recorded in the atlas, and many differential-topological tools extend when adjusted for group actions and fixed-point contributions.