Compute the fundamental group of the space obtained from two tori $S^{1} \times S^{1}$ by identifying a circle $S^{1} \times\left\{x_{0}\right\}$ in one torus with the corresponding circle $S^{1} \times\left\{x_{0}\right\}$ in the other torus.
solution : Let $X$ be the surface, the identification of two tori $S^{1} \times S^{1}$ as described in the exercise. And we know the fundamental group of the torus is $\mathbb{Z} \times \mathbb{Z}$. Let's assume the two tori $T_{1}, T_{2}$ are identified by "stacking" one on the other. i.e. If $a, b$ and $c, d$ are the generators of the fundamental groups respectively. And $a$ and $c$ are their longitudes. The way we stack the two tori will make $a$ and $c$ identified. To use the Van Kampen's Theorem, let $A$ be the top torus $T_{1}$ together with a strip of open neighborhood of $a$ on itself and on the bottom torus $T_{2} .$ Similarly, let $B$ the bottom torus $T_{2}$ together with a strip of open neighborhood of $c$ on itself and the top one $T_{1}$ Then $A$ and $B$ are open subset of $X$ and $A \cap B$ is open and path connected. since $A$ and $B$ deformation retracts to $T_{1}$ and $T_{2}$ respectively, so $\pi_{1}(A)=\pi_{1}(B)=\mathbb{Z} \times \mathbb{Z}$. since $A \cap B$ deformation retracts to a circle, we have $\pi_{1}(A \cap B) \simeq \mathbb{Z}$, the generator has its image $a, c$ in $A, B$ respectively.
By Van Kampen, $\pi_{1}(X)$ is isomorphic to the quotient of $\pi_{1}(A) * \pi_{1}(B)$ by the normal subgroup generated by $\left\langle a c^{-1}\right\rangle$ , $ \pi_{1}(X) \cong \frac{(\mathbb{Z} \times \mathbb{Z}) *(\mathbb{Z} \times \mathbb{Z})}{\left\langle a c^{-1}\right\rangle} \cong(\mathbb{Z} * \mathbb{Z}) \times \mathbb{Z} $
i can't understand why $ \frac{(\mathbb{Z} \times \mathbb{Z}) *(\mathbb{Z} \times \mathbb{Z})}{\left\langle a c^{-1}\right\rangle} \cong(\mathbb{Z} * \mathbb{Z}) \times \mathbb{Z}$ ? is solution true ?
Hint: Forget van Kampen's theorem, just observe that your space is homeomorphic to the product of the figure 8 graph and the circle. Now, use the formula for the fundamental group of product of two spaces. Now, use van Kampen's theorem to compute $\pi_1$ of the figure 8 graph.