I am doing Exercise 1.3.19 from Hatcher's Algebraic Topology and I come to this part of the problem:
For $n = 3$ and $g\geq 3$, describe a normal covering space $\tilde{X}$ of $X=M_g$, the surface of genus $g$ explicitly with deck transformation group $G(\tilde{X})$ consisting of translations isomorphic to $\Bbb{Z}^3$.
Now in the case that $g = 3$ we can get such a covering space from the topological quotient map $p : \Bbb{R}^3 \to \Bbb{R}^3/\Bbb{Z}^3$, that is we quotient out $\Bbb{R}^3$ by the action of $\Bbb{Z}^3$ on $\Bbb{R}^3$ defined as follows. For $x \in \Bbb{R}^3$ and $(a,b,c) \in \Bbb{Z}^3$.
$(a,b,c) \cdot x = \text{translation by the vector $(a,b,c)$}.$
The orbit space is isomorphic to $M_3$ and so $\Bbb{R}^3$ is the desired covering space. However for $g > 3$ I am finding it hard to visualize such a covering space. How can I find one for $g > 3$?
Thanks.
Edit: I just realised the orbit space is $S^1 \times S^1 \times S^1$ and so the example above fails.