Let $A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ be a matrix with integer entries. Considering $S^{1}$ as the quotient space $\mathbb{R} / \mathbb{Z}$, show that the function

$$\begin{aligned} \varphi_{A}: S^{1} \times S^{1} & \longrightarrow S^{1} \times S^{1} \\ ([x],[y]) & \longmapsto([a x+b y],[c x+d y]) \end{aligned}$$

is well-defined and continuous. If in addition $\operatorname{det}(A)=\pm 1$, show that $\varphi_{A}$ is a homeomorphism.

State the Seifert-van Kampen theorem. Let $X_{A}$ be the space obtained by gluing together two copies of $S^{1} \times D^{2}$ along their boundaries using the homeomorphism $\varphi_{A}$. Show that the fundamental group of $X_{A}$ is cyclic and determine its order.