In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold. It is a topological space (called the underlying space) with an orbifold structure.
Like a manifold, an orbifold is specified by local conditions; however, instead of being locally modelled on open subsets of $\mathbb{R}^n$, an orbifold is locally modelled on quotients of open subsets of $\mathbb{R}^n$ by finite group actions. The structure of an orbifold encodes not only that of the underlying quotient space, which need not be a manifold, but also that of the isotropy subgroups.
An $n$-dimensional orbifold is a Hausdorff topological space $X$, called the underlying space, with a covering by a collection of open sets $U_i$, closed under finite intersection. For each $U_i$, there is:
- An open subset $V_i$ of $\mathbb{R}^n$, invariant under a faithful linear action of a finite group $\Gamma_i$
- A continuous map $\phi_i$ of $V_i$ onto $U_i$ invariant under $\Gamma_i$, called an orbifold chart, which defines a homeomorphism between $V_i / \Gamma_i$ and $U_i$.
The collection of orbifold charts is called an orbifold atlas if the following properties are satisfied:
- For each inclusion $U_i \subset U_j$ there is an injective group homomorphism $f_{ij} : \Gamma_i \rightarrow \Gamma_j$.
- For each inclusion $U_i \subset U_j$ there is a $\Gamma_i$-equivariant homeomorphism $\psi_{ij}$, called a gluing map, of $V_i$ onto an open subset of $V_j$
- The gluing maps are compatible with the charts, i.e. $\phi_j·\psi_{ij} = \phi_i$
- The gluing maps are unique up to composition with group elements, i.e. any other possible gluing map from $V_i$ to $V_j$ has the form $g·\psi_{ij}$ for a unique $g$ in $\Gamma_j$.
Source: Wikipedia - Orbifold
