# 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 $\Phi$ between two objects $(G^{\prime},X^{\prime})$ and $(G,X)$ 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 $(g\phi)(X)\cap\phi(X)\neq\emptyset$, 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 $X$ 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. 1.

For $U\in\mathcal{U}$, $\mathcal{V}(U)=(G_{U},\tilde{U})\lx@stackrel{{\scriptstyle\tau}}{{\rightarrow}}U$ is a ramified cover $\tilde{U}\rightarrow U$ which identifies $\tilde{U}/G_{U}\cong U$

2. 2.

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

3. 3.

If $U\subset V\subset W\in\mathcal{U}$, $\phi_{WU}=\phi_{WV}\circ\phi_{VU}$

[1] Kawasaki T., The Signature theorem for V-manifolds. Topology 17 (1978), 75-83. MR0474432 (57:14072)

Title orbifold Orbifold 2013-03-22 15:40:06 2013-03-22 15:40:06 guffin (12505) guffin (12505) 8 guffin (12505) Definition msc 57M07 orbifold structure orbifold structure