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 be the space obtained by two tori by identifying the circle 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 equal the whole space minus a circle where . Only remove the circle from one torus. Then, let be the whole space minus the same circle from the other torus. Obviously, their union is the whole space, and both and 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 .
Now, if I calculate it a different way, I get confused about the quotient of the free product of and . Let be points on different tori, neither one is a point on the circles nor . Since is path connected, let be a path from to . Let be the space obtained by attaching to with the identification that follows the path , and the rest is unidentified. Obviously, deformation retracts to , so the two spaces are homotopic. Let and let (where is very small). While it is not obvious, I can show that deformation retracts to the wedge sum of three circles. It is obvious that is simply connected. However, the intersection of and 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 by is , which I know is not the correct fundamental group of . What am I missing?