orbifold
Roughly, an orbifold is the quotient^{} of a manifold by a finite group^{}. For example, take a sheet of paper and add a small crease perpendicular to one side at the halfway point. Then, line up the two halves of the side. This may be thought of as the plane ${\mathbb{R}}^{2}$ modulo the group ${\mathbb{Z}}^{2}$. Now, let us give the definition.
Define a category $\mathcal{X}$: The objects are pairs $(G,X)$, where $G$ is a finite group acting effectively on a connected smooth manifold $X$. A morphism $\mathrm{\Phi}$ between two objects $({G}^{\prime},{X}^{\prime})$ and $(G,X)$ is a family of open embeddings^{} $\varphi :{X}^{\prime}\to X$ which satisfy

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for each embedding $\varphi \in \mathrm{\Phi}$, there is an injective homomorphism^{} ${\lambda}_{\varphi}:{G}^{\prime}\to G$ such that $\varphi $ is ${\lambda}_{\varphi}$ equivariant

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For $g\in G$, we have
$g\varphi $ $:{X}^{\prime}\to X$ $g\varphi $ $:x\mapsto g\varphi (x)$ and if $(g\varphi )(X)\cap \varphi (X)\ne \mathrm{\varnothing}$, then $g\in {\lambda}_{\varphi}({G}^{\prime})$.

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$\mathrm{\Phi}=\{g\varphi ,g\in G$}, for any $\varphi \in \mathrm{\Phi}$
Now, we define orbifolds. Given a paracompact Hausdorff space $X$ and a nice open covering $\mathcal{U}$ which forms a basis for the topology^{} on $X$, an orbifold structure $\mathcal{V}$ on $X$ consists of

1.
For $U\in \mathcal{U}$, $\mathcal{V}(U)=({G}_{U},\stackrel{~}{U})\stackrel{\tau}{\to}U$ is a ramified cover $\stackrel{~}{U}\to U$ which identifies $\stackrel{~}{U}/{G}_{U}\cong U$

2.
For $U\subset V\in \mathcal{U}$, there exists a morphism ${\varphi}_{VU}({G}_{U},\stackrel{~}{U})\to ({G}_{V},\stackrel{~}{V})$ covering the inclusion

3.
If $U\subset V\subset W\in \mathcal{U}$, ${\varphi}_{WU}={\varphi}_{WV}\circ {\varphi}_{VU}$
[1] Kawasaki T., The Signature^{} theorem^{} for Vmanifolds. Topology 17 (1978), 7583. MR0474432 (57:14072)
Title  orbifold 

Canonical name  Orbifold 
Date of creation  20130322 15:40:06 
Last modified on  20130322 15:40:06 
Owner  guffin (12505) 
Last modified by  guffin (12505) 
Numerical id  8 
Author  guffin (12505) 
Entry type  Definition 
Classification  msc 57M07 
Synonym  orbifold structure 
Defines  orbifold structure 