(a) Let $V$ be the vector space of 3-dimensional upper-triangular matrices with real entries:

$$V=\left\{\left(\begin{array}{ccc} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{array}\right) \mid x, y, z \in \mathbb{R}\right\}$$

Let $\Gamma$ be the set of elements of $V$ for which $x, y, z$ are integers. Notice that $\Gamma$ is a subgroup of $G L_{3}(\mathbb{R})$; let $\Gamma$ act on $V$ by left-multiplication and let $N=\Gamma \backslash V$. Show that the quotient $\operatorname{map} V \rightarrow N$ is a covering map.

(b) Consider the unit circle $S^{1} \subseteq \mathbb{C}$, and let $T=S^{1} \times S^{1}$. Show that the map $f: T \rightarrow T$ defined by

$$f(z, w)=(z w, w)$$

is a homeomorphism.

(c) Let $M=[0,1] \times T / \sim$, where $\sim$ is the smallest equivalence relation satisfying

$$(1, x) \sim(0, f(x))$$

for all $x \in T$. Prove that $N$ and $M$ are homeomorphic by exhibiting a homeomorphism $M \rightarrow N$. [You may assume without proof that $N$ is Hausdorff.]

(d) Prove that $\pi_{1}(M) \cong \Gamma$.