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 ${ℝ}^2$ modulo the group ${\mathbb{Z}}^2$. Now, let us give the definition.

Define a category $𝒳$:The objects are pairs $(G,X)$, where $G$ is a finite group actingeffectively on a connected smooth manifold $X$. A morphism $\mathrm{\Phi }$ betweentwo objects $(G^{\prime },X^{\prime })$ and $(G,X)$ is a family of openembeddings $\varphi :X^{\prime }\to X$ which satisfy

• 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

• 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 $g\in {\lambda }_{\varphi }(G^{\prime })$.

• $\mathrm{\Phi }=\{g\varphi ,g\in G$}, for any $\varphi \in \mathrm{\Phi }$

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

1. For $U\in 𝒰$, $𝒱(U)=(G_U,\tilde{U})\stackrel{\tau }{\to }U$ is a ramified cover $\tilde{U}\to U$ whichidentifies $\tilde{U}/G_U\cong U$

2. For $U\subset V\in 𝒰$, there exists a morphism${\varphi }_{VU}(G_U,\tilde{U})\to (G_V,\tilde{V})$ covering theinclusion

3. If $U\subset V\subset W\in 𝒰$, ${\varphi }_{WU}={\varphi }_{WV}\circ {\varphi }_{VU}$

References:

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