Given a $n$-dimensional torus, is it always possible to find a discrete action to produce an orbifold such that its underlying space is the $n$-dimensional sphere? Or does it only happens for specific dimensions? Is there some particular property of the action $\Gamma$ -which I expect to be some realization of $\Bbb Z_n$- that I should check?

Bonus question: And to be a projective plane? I am thinking that according Arnold-Kuiper-Massey theorem one can see $\Bbb CP^2$ as a branched covering of $S^4$, so if $T^4$ could be a branched covering of $\Bbb CP^2$ we would have an interesting way to cover $S^4$

The personal motivation for this is, as usual, more fuzzy that the question itself but it could add some heat to the comments: I am interested on how to sequence manifolds in a way that increases some continuous symmetry. For manifolds with a metric the two extreme cases in a given dimension would be the n-Torus, where you can act with group $U(1)^n$ and the n-sphere, where you act with $SO(n+1)$. Of course the sphere that comes out from orbifolding is not exactly the sphere, one must consider the conical singularities and all the metric properties go amok, but still it is, to me, a hint of how do symmetries appear and disappear. For physics this is specially funny in dimension seven because you have also the the manifold $CP^2\times S^3$ where the isometry group has 14 generators and then you could try to produce a sequence 7 -> 14 -> 28

Given a $n$-dimensional torus, is it always possible to find a discrete action to produce an orbifold such that its underlying space is the $n$-dimensional sphere? Or does it only happens for specific dimensions? Is there some particular property of the action $\Gamma$ -which I expect to be some realization of $\Bbb Z_n$- that I should check?

Bonus question: And to be a projective plane? I am thinking that according Arnold-Kuiper-Massey theorem one can see $\Bbb CP^2$ as a branched covering of $S^4$, so if $T^4$ could be a branched covering of $\Bbb CP^2$ we would have an interesting way to cover $S^4$

The personal motivation for this is, as usual, more fuzzy that the question itself but it could add some heat to the comments: I am interested on how to sequence manifolds in a way that increases some continuous symmetry. For manifolds with a metric the two extreme cases in a given dimension would be the n-Torus, where you can act with group $U(1)^n$ and the n-sphere, where you act with $SO(n+1)$. Of course the sphere that comes out from orbifolding is not exactly the sphere, one must consider the conical singularities and all the metric properties go amok, but still it is, to me, a hint of how do symmetries appear and disappear. For physics this is specially funny in dimension seven because you have also the the manifold $CP^2\times S^3$ where the isometry group has 14 generators and then you could try to produce a sequence 7 -> 14 -> 28

EDIT: It seems than for n=2 the physicists call "pillows" to the spheres, in order to stress the existence of singular points. So the question is, when is the orbifold a n-pillow?