Concrete examples of covering from the 3-torus to the 3-sphere There is a two-fold branched covering from 2-torus to the 2-sphere, $T^2 \rightarrow S^2$, whose covering transformation group is  generated by the map $x \mapsto -x$ (Note that $T^2$ is an abelian group).
I heard that there is a three-fold branched covering from the 3-torus to the 3-sphere. Then what would be the covering transformation group of this case? 
Probably it is trivial for topologists but could anyone can help me out?
 A: There is an algorithm, due to Montesinos, for converting a surgery diagram of a 3-manifold $M$ into a description of $M$ as a 3-fold (irregular; as remarked above, there is no regular branched cyclic covering $T^3 \to S^3$) cover of $S^3$. It is described nicely in Rolfsen, Knots and Links, Chapter 10.G. You need to start with a surgery description where all of the framings are $\pm 1$. 
For the 3-torus, such a description is easily found. $T^3$ is surgery on the Borromean rings, with framings 0 on each component. Add a $+1$ framed meridianal circle to each component of the Borromean rings, changing the framing on that component to $1$. The picture below shows what I mean. If you follow the description in Rolfsen's book, you will have the branched covering. I haven't tried to draw it, though.
 
A: First I think, the covering $T^2\to S^2$ has deck group $Z/2Z\oplus Z/2Z$ generated by $(x,y)\to (-x,y)$ and $(x,y)\to (x,-y)$.
Then for the 3-dimensional case, you consider the action of $$Z/2Z\oplus Z/2Z\oplus Z/2Z$$ on $T^3$, where the 3 generators send $$(x,y,z)$$ respectively to $(-x,y,z)$ or $(x,-y,z)$ or $(x,y,-z)$. The quotient by this action should be the 3-sphere, in analogy to the 2-dimensional case.
