PlanetMath: orbifold

(more info)

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orbifold

(Definition)

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 , where is a finite group acting effectively on a connected smooth manifold . A morphism $\Phi$ between two objects $(G^\prime, X^\prime)$ and is a family of open embeddings $\phi:X^\prime\rightarrow X$ which satisfy

for each embedding $\phi\in\Phi$ , there is an injective homomorphism $\lambda_\phi:G^\prime\rightarrow G$ such that $\phi$ is $\lambda_\phi$ equivariant
For $g\in G$ , we have

$\displaystyle g\phi$ $\displaystyle :X^\prime\rightarrow X$

$\displaystyle g\phi$ $\displaystyle :x\mapsto g\phi(x)$

and if $(g\phi)(X) \cap \phi(X) \ne \emptyset$ , then $g\in \lambda_\phi(G^\prime)$ .
$\Phi = \{g\phi, g\in G$ }, for any $\phi \in \Phi$

Now, we define orbifolds. Given a paracompact Hausdorff space and a nice open covering $\mathcal U$ which forms a basis for the topology on , an orbifold structure $\mathcal V$ on consists of

For $U\in \mathcal U$ , $\mathcal V(U) = (G_U,\tilde U)\stackrel \tau \rightarrow U$ is a ramified cover $\tilde U \rightarrow U$ which identifies $\tilde U\slash G_U \cong U$
For $U\subset V \in \mathcal U$ , there exists a morphism $\phi_{VU}(G_U, \tilde U)\rightarrow (G_V, \tilde V)$ covering the inclusion
If $U\subset V\subset W \in \mathcal U$ , $\phi_{WU} = \phi_{WV}\circ\phi_{VU}$