Here is the definition of an orbifold, from Fulton's The Geometry and Topology of 3-Manifolds:
An orbifold $O$ consists of a Hausdorff space $X_O$, with some additional structure. $X_O$ is to have a covering by a collection of open sets $\{U_i\}$ closed under finite intersections. To each $U_i$ is associated a finite group $\Gamma_i$, an action of $\Gamma_i$ on an open subset $\tilde U_i$ of $\mathbb{R}^n$ and a homeomorphism $\varphi_i\colon U_i \cong \tilde U_i / \Gamma_i$. Whenever $U_i \subset U_j$, there is to be an injective homomorphism $f_{ij}\colon\Gamma_i \hookrightarrow \Gamma_j$ and an embedding $\tilde\varphi_{ij}\colon \tilde U_i \hookrightarrow \tilde U_j$ equivariant with respect to $f_{ij}$ such that the two compositions $U_i \simeq \tilde U_i / \Gamma_i \to \tilde U_j/\Gamma_i \to \tilde U_j/\Gamma_j$ and $U_i \subset U_j \to \tilde U_j/\Gamma_j$ agree.
I want to construct a "spindle", whose underlying space is $S^2$, with an 2-fold orbifold point at the north pole and an 3-fold orbifold point at the south pole -- but I can't seem to make the construction work, can someone help me?
Here's what I've got: define $U_N$ resp. $U_S$ to be neighborhoods of the north resp. south poles, and $V := U_N \cap U_M$ to be the collar where they overlap. Define $\tilde U_N$ to be the unit disk, acted upon by $\Gamma_{U_N} := \mathbb{Z}/2\mathbb{Z}$ (which acts by rotations by $\pi$); define $\tilde U_S$ and $\Gamma_{U_S}$ similarly. What should $\tilde V$ and $\Gamma_V$ be? We have to have inclusions $V \hookrightarrow U_N$ and $V \hookrightarrow U_S$, so $\Gamma_V$ has to be the trivial group. So $\tilde V = V$. But now I don't see how the embeddings $\tilde \varphi_{VU_N}, \tilde \varphi_{VU_S}$ can be defined. (It seems like maybe this would all work if we didn't require $f_{ij}$ to be a monomorphism -- then maybe we could take $\tilde V$ to be a 6-fold cover of $V$.)
Help please!
You're right that this does not work with $V$. The problem is that $V$ is not simply connected, and its fundamental group is causing problems.
So... don't use $V$.
I would suggest instead covering $V$ by two simply connected neighborhoods $V_1,V_2$, and then using $\{U_N,U_S,V_1,V_2\}$ as the covering ... except I guess you'll have to throw in finite intersections of those to satisfy the definition, so choose $V_1,V_2$ carefully to make sure that all finite intersections are simply connected.