Greetings,

I am trying to figure out van Kampen's theorem. I understand it in some instances, but not in others. Here is an example:

Let $\displaystyle X$ be the space obtained by two tori $\displaystyle S^1\times S^1$ by identifying the circle $\displaystyle S^1 \times \{x_0\}$ of each torus. (The space looks like one torus stacked on top of a second torus). In order to compute the fundamental group, I want to apply van Kampen's theorem. So, let $\displaystyle A$ equal the whole space minus a circle $\displaystyle S^1 \times \{x_1\}$ where $\displaystyle x_1\neq x_0$. Only remove the circle from one torus. Then, let $\displaystyle B$ be the whole space minus the same circle from the other torus. Obviously, their union is the whole space, and both $\displaystyle A$ and $\displaystyle B$ deformation retract to a torus. Their intersection deformation retracts to a circle. So, the fundamental group is isomorphic to the quotient of the free product of the fundamental groups of two tori and the fundamental group of a circle. Hence, it is isomorphic to $\displaystyle \mathbb{Z}^3$.

Now, if I calculate it a different way, I get confused about the quotient of the free product of $\displaystyle A$ and $\displaystyle B$. Let $\displaystyle x_2,x_3$ be points on different tori, neither one is a point on the circles $\displaystyle S^1\times \{x_0\}$ nor $\displaystyle \{x_0\}\times S^1$. Since $\displaystyle X$ is path connected, let $\displaystyle p$ be a path from $\displaystyle x_2$ to $\displaystyle x_3$. Let $\displaystyle Y$ be the space obtained by attaching $\displaystyle I\times I$ to $\displaystyle X$ with the identification that $\displaystyle I\times \{0\}$ follows the path $\displaystyle p$, and the rest is unidentified. Obviously, $\displaystyle Y$ deformation retracts to $\displaystyle X$, so the two spaces are homotopic. Let $\displaystyle A=Y\setminus\{x_2,x_3\}$ and let $\displaystyle B=Y\setminus X \cup N_\epsilon(x_2) \cup N_\epsilon(x_3)$ (where $\displaystyle \epsilon$ is very small). While it is not obvious, I can show that $\displaystyle A$ deformation retracts to the wedge sum of three circles. It is obvious that $\displaystyle B$ is simply connected. However, the intersection of $\displaystyle A$ and $\displaystyle B$ is homotopic to the wedge sum of two circles. This is where I run into problems with van Kampen's theorem, since the quotient of $\displaystyle \mathbb{Z}^3$ by $\displaystyle \mathbb{Z}^2$ is $\displaystyle \mathbb{Z}$, which I know is not the correct fundamental group of $\displaystyle X$. What am I missing?