Is orbifolding of the torus a sophisticated spherification?

[arivero]

I was thinking... perhaps are all these orbifolds with conical singularities simply ironed versions of the typical manifolds of Kaluza Klein? I asked in math overflow about how to get spheres and CPn spaces from n-dimensional torus and the people who answered seem to look at it as usual business, even if difficult to prove in general cases. On other hand the underlying manifold of an orbifold is not mentioned very frequently in physics.

EDIT: It seems that string theoretists like to call to the S2 sphere a "pillow", as in this random quote: "Pictorially, the T 4/Z2 orbifold is a pillow with a torus over it everywhere except at the corners, where the fiber is a pillow"

 

[arivero]

Thanks for the bump. Actually the MO thread got some answers but the topic seems open.