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Find the covering space of the torus $T=S^1\times S^1$ that is corresponding to the subgroup $2Z\oplus 3Z$ of $\pi_1(T,x_0)=Z^2$. What is the covering space corresponding to the trivial group?, do one of the covering spaces cover the another?

I've looked at Find all covering spaces of Torus $S^1 \times S^1$ up to isomorphism.

But did not realize what is covering space of $T$ here..

Can you please explain the way we find covering spaces (generally) and in this question

Lam18373
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    $T\cong\Bbb{R}^2/\Bbb{Z}^2$. Then $\Bbb{R}^2$ is the universal cover (corresponding to the trivial subgroup of $\pi_1(T)$). $\Bbb{R}^2/6 \Bbb{Z}^2\to \Bbb{R}^2/ \Bbb{Z}^2$ is a covering of degree $36$. The subgroup of $\pi_1(T)$ acting trivially on it is $6\Bbb{Z}^2$. – reuns Dec 02 '22 at 17:30

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The $e_n : S^1 \to S^1, e_n(z) = z^n$, are covering spaces corresponding to the subgroups $n\mathbb Z$. Moreover, $e_\infty : \mathbb R \to S^1, e_\infty(t) = e^{it}$ corresponds to the trivial subgroup of $\mathbb Z$. It is the universal covering. Therefore

  1. $e_2 \times e_3 : S^1 \times S^1 \to S^1 \times S^1$ corresponds to $2\mathbb Z \oplus 3\mathbb Z$.

  2. $e_\infty \times e_\infty : \mathbb R \times \mathbb R \to S^1 \times S^1$ corresponds to the trivial group.

Paul Frost
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