Help understanding van Kampen's theorem

**SlipEternal** · June 30th 2012, 01:21 PM

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 $X$ be the space obtained by two tori $S^1\times S^1$ by identifying the circle $S^1 \times \{x_0\}$ 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 $A$ equal the whole space minus a circle $S^1 \times \{x_1\}$ where $x_1\neq x_0$ . Only remove the circle from one torus. Then, let $B$ be the whole space minus the same circle from the other torus. Obviously, their union is the whole space, and both $A$ and $B$ 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 $\mathbb{Z}^3$ .

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

**SlipEternal** · July 4th 2012, 05:55 PM

I understand what I was doing wrong. Thank you anyway if you tried to offer advice.

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