Teardrop orbifold
= Teardrop orbifold
A teardrop orbifold is an <orbifold> whose underlying <topological space> is a two-sphere and which has exactly one <cone point of an orbifold> of order $m>1$. A neighborhood of that point is modeled on a disk modulo rotations by multiples of $2\pi/m$; every other point has trivial local group.
Topologically, a disk modulo this finite rotation group is still a disk. The local stabilizer is therefore extra orbifold information invisible in the underlying sphere. The singularity order distinguishes teardrops that have the same underlying <topological space>.
The teardrop is a standard example of a bad orbifold: it has no manifold orbifold covering. Cutting it into an ordinary disk and a disk with one order-$m$ cone point exposes incompatible nontrivial covering behavior along their common boundary. Thus being locally a finite quotient does not imply being globally covered by a manifold.<The Geometry and Topology of Three-Manifolds by William Thurston>[Chapter13, Definition13.2.3 and teardrop example, printed p304]{ref}