Visualizing a covering space of a torus Note that the fundamental group of the torus $T$ is given by $\pi_1(T)=\Bbb Z \times \Bbb Z =<a,b~|~aba^{-1}b^{-1}>$. By the Galois correspondence of covering spaces, I know that there is a unique (up to covering isomorphism) covering space of $T$ corresponding to (the conjugacy class of) the subgroup $<a^3, a^2b>$. Can I visualize this covering space using square diagrams? I want to describe the covering space explicitly, but I have no idea.
 A: Subdivide the universal covering space $\mathbb R^2$ in the usual manner as squares, with vertical lines $x=m$ and horizontal lines $y=n$ for integers $m,n \in \mathbb Z$.
To visualize the desired covering space, draw two vectors based at the origin: $v = \langle 3,0 \rangle$ corresponding to $a^3$; and $w = \langle 2,1 \rangle$ corresponding to $a^2 b$. Let $P$ be the parallelogram determined by the vectors $v,w$, which form two sides of $P$, the other two sides then being determined. Now glue opposite sides of $P$ to form a quotient space $S$. And as usual, gluing opposite sides of the unit square $Q = [0,1] \times [0,1]$ gives the covering space $T$. 
You can then use the pattern of intersections of the parallelograph $P$ with unit squares $[m,m+1] \times [n,n+1]$ to define the desired covering map $Q \mapsto T$.
It is theoretically more straightforward to view this construction using orbit spaces of deck transformations. If I have $(a,b) \in \mathbb R^2$ let me use $\tau_{(a,b)}$ to represent the translation $\tau_{a,b}(x,y) (x+a,y+b)$. Thus we can think of $T$ as the quotient of $\mathbb R^2$ by the action of the deck group $\langle \tau_{(1,0)},\tau_{(0,1)} \rangle$ (with fundamental domain $[0,1] \times [0,1]$), so
$$T = \mathbb R^2 / \langle \tau_{(1,0)},\tau_{(0,1)} \rangle
$$
and we can think of $S$ as the quotient of $\mathbb R^2$ by the action of the deck group $\langle \tau_{(3,0)}, \tau_{(2,1)} \rangle$ (with fundamental domain $P$), so 
$$S = \mathbb R^2 / \langle \tau_{(3,0)}, \tau_{(2,1)} \rangle
$$
This way, the desired quotient map $S \mapsto T$ can be precisely defined as the map
$$S = \mathbb R^2 / \langle \tau_{(3,0)}, \tau_{(2,1)} \rangle \mapsto  = \mathbb R^2 / \langle \tau_{(1,0)},\tau_{(0,1)} \rangle = T
$$
that is induced by the identity map $\mathbb R^2 \mapsto \mathbb R^2$.
A: All of the coverings of the torus are homeomorphic to $\mathbb R^2,\mathbb R\times S^1$, or $T=S^1\times S^1$. Any subgroup of $\mathbb Z^2$ spanned by two independent elements corresponds to the covering space $T$. What changes is the covering map, which in this case is given by $p(x,y)=(3x,2x+y)$.
