A Structure Theorem for Bad 3-Orbifolds

We explicitly construct a collection of bad 3-orbifolds, $\mathcal{X}$, satisfying the following properties: \begin{enumerate} \item The underlying topological space of any $X\in \mathcal{X}$ is homeomorphic to $S^2\times I$ or $(S^2\times S^1)\mathrm{\backslash }{B}^3$. \item The boundary of any $X\in \mathcal{X}$ consists of one or two spherical 2-orbifolds. \item Any bad 3-orbifold is obtained from a good 3-orbifold by repeating, finitely many times, the following operation: remove one or two orbifold-balls, and glue in some $X\in \mathcal{X}$. \end{enumerate} Conversely, any bad 3-orbifold $\text{OO}$ contains some $X\in \mathcal{X}$ as a sub-orbifold; we call removing $X$ and capping the resulting boundary \em cut-and-cap.\em\ Then by cutting-and-capping finitely many times we obtain a good orbifold.