Let $\hat{D} := D \backslash \{0\}$ be a ball in $R^n$ with the origin $\{0\}$ removed. Assume that $\hat{D}$ has a structure as an orbifold (may be distinct from its standard manifold structure). Is it possible to extend the orbifold structure from $\hat{D}$ to $D$ ?



Extension of an orbifold structure from punctured balls to balls
No. This is not true in dimension three (nor in any higher dimension).


The three-dimensional orbifold structures allowed at a point are controlled by the list of finite subgroups of $\mathrm{SO}(3)$. In particular, there are at most three one-dimensional loci that converge at any point.


Let $S^2(2^m)$ be the two-orbifold which is topologically a two-sphere, decorated with $m$ orbifold points of order two. Suppose that $m > 3$. We form $\hat{D} = S^2(2^m) \times (0, 1]$. This is the desired example.




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Another way to think about this: the link of a point $x$ in an $n$-dimensional orbifold must be a spherical orbifold in dimension $n-1$. In the example above, (for $m > 4$) we found a particular hyperbolic orbifold structure on $S^2$. However *any* euclidean or hyperbolic orbifold structure on $S^2$ would have served as well.


