Let $C = \{(x, y, z)\in \mathbb{R}^3 : (x − 2)^2 + y^2 = 1, \, z = 0\}$, $X = \{(x, y, z)\in \mathbb{R}^3 : y = z = 0, x \in [−1, 1]\}$ and $T$ be the torus obtained by rotating $C$ around $y$ axis. Calculate the fundamental group of $Y=X \cup T$.
I've tried using the Seifert–Van Kampen theorem, taking the open and path connected subspaces such that $Y=U_1 \cup U_2$ as $Y = U_1-\text{the equatorial circumference of the torus}$ and $U_2=T\setminus \{(0,0,0)\}$. By appling some deformation retracts I've found out that $\pi_1(U_1)\simeq \mathbb{Z}*\mathbb{Z}$, $\pi_1(U_2)\simeq \mathbb{Z}\times \mathbb{Z}$ and $\pi_1(U_1\cap U_2)\simeq \mathbb{Z}$, where $\pi_1(A)$ denotes the fundamental group of $A$. Now I can't figure out what is $\pi_1(Y)$; in particular, using group presentations and relations, I can't understand what are the relations "induced" by the elements of $U_1\cap U_2$ in the presentation of $\pi_1(Y)$. Hopefully everything makes sense. Thanks everybody.