Orbifold

Definition

Let $n \ge 0$ be an integer and let $X$ be a Hausdorff and second-countable topological space. An orbifold chart of dimension $n$ on an open set $U \subseteq X$ is a triple $(\tilde U, G, \varphi)$ with these parts. The set $\tilde U$ is a connected open subset of $\R^n$. The group $G$ is finite and has a faithful action on $\tilde U$ by diffeomorphisms. The map $\varphi \colon \tilde U \to U$ is continuous, and $\varphi(g x) = \varphi(x)$ for all $g \in G$ and $x \in \tilde U$. The induced map from the quotient space $\tilde U / G$ to $U$ is a homeomorphism. An embedding of a chart $(\tilde U_i, G_i, \varphi_i)$ into a chart $(\tilde U_j, G_j, \varphi_j)$ is a smooth embedding $\psi \colon \tilde U_i \to \tilde U_j$ with $\varphi_j \circ \psi = \varphi_i$. An orbifold atlas of dimension $n$ on $X$ is a family of charts $(\tilde U_i, G_i, \varphi_i)$ with these properties:

1. The sets $U_i = \varphi_i(\tilde U_i)$ cover $X$.
2. For any two charts and any point $x \in U_i \cap U_j$, there is a chart $(\tilde U_k, G_k, \varphi_k)$ of the atlas with $x \in U_k \subseteq U_i \cap U_j$.
3. Whenever $U_i \subseteq U_j$, there is an embedding of the chart $(\tilde U_i, G_i, \varphi_i)$ into the chart $(\tilde U_j, G_j, \varphi_j)$.

An atlas $\mathcal V$ refines an atlas $\mathcal U$ if every chart of $\mathcal V$ has an embedding into some chart of $\mathcal U$. Two atlases are equivalent if they have a common refinement. An $n$-dimensional orbifold is a space $X$ as above together with an equivalence class of $n$-dimensional orbifold atlases.