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I know the covering spaces of the of a torus $T^2$ are homeomorphic to $T^2,S^1\times\mathbb{R},\mathbb{R}^2$. I am interested in finding all of the covers with covering space $T^2$. The subgroups of $\pi_1(T^2)=\mathbb{Z}\times\mathbb{Z}$ with covering space $T^2$ are of the form $\langle (a,b),(c,d)\rangle\cong\mathbb{Z}\times\mathbb{Z}$.

The cover for $n\mathbb{Z}\times m\mathbb{Z}$ is $p:T^2\rightarrow T^2, p(x,y)=(x^n,x^m)$. How can I generalize this to arbitrary 2 dimensional subgroups.

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If the subgroup is spanned by two linearly independent $(a,b)$ and $(c,d)$, then the cover is $p(x,y) = (x^a y^b, x^c y^d)$. This map is surjective because of the independence of the spanning vectors.

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