Suppose I have two solid tori $U = S^1 \times B^2$ and $V = B^2 \times S^1$. Note that their boundary is the torus $\partial U = \partial V = S^1 \times S^1$. Let $X$ be the space that glues $U$ and $V$ along the boundary. I want to compute $\pi_1 (X)$.
I have convinced myself that $X = S^3$ so $\pi_1(X) =0$. However, I was wondering if I could possibly use Van Kampen to alternatively prove that $\pi_1(X) = 0$.
My attempt that led me to a different answer:
The choices of open sets I was thinking was $A=U \cup V' \subset X$ where $V' = \{ (x,y) \in V : ||x||> \frac{1}{2} \}$ and $B=U' \cup V$ defined similarly. Then, $A=U \cup V' \simeq U \simeq S^1 \simeq V \simeq U' \cup V=B$ with $A \cap B \simeq S^1 \times S^1 = \partial U$. Therefore, Van Kampen would've given me $\pi_1(X) \cong \pi_1(A) *_{\pi_1(A \cap B)} \pi_1(B) \cong \langle x,y | aba^{-1}b^{-1} = e \rangle \cong \mathbb{Z} \oplus \mathbb{Z}$. What did I do wrong here? $A \cap B$ seem path-connected since the boundary gives a way to connect any two points.