Examples of using van Kampen Theorem where the intersection is not a point

Question

The van Kampen theorem is:

If $X$ is the union of path-connected open sets $A_{\alpha}$ each containing the basepoint $x_0 \in X$, and if each intersection $A_{\alpha}\cap A_{\beta}$ is path-connected, then the homomorphism $\Phi : *_{\alpha}\pi_1(A_{\alpha}) \to \pi_1(X)$ is surjective.

In most examples I have seen, $A_{\alpha} \cap A_{\beta}$ is a point, (except some complicated examples). Can you find a simple example where $A_{\alpha}\cap A_{\beta}$ is not a point?

Answer 1

Every connected surface (i.e. topological two-manifold) can be obtained by identifying pairs of oriented edges of some $2n$-sided polygon. One way to calculate the fundamental group of a connected surface is to start with a corresponding polygon $P$ and let $D$ be a disc in the interior of $P$, then set $A_{\alpha} = D$ and $A_{\beta} = P\setminus\{x_0\}$ where $x_0 \in D$. Note that both $A_{\alpha}$ and $A_{\beta}$ are path-connected, and that $A_{\alpha}\cap A_{\beta}$ is a punctured disc, and is therefore path-connected. By applying the Seifert-van Kampen Theorem, you can then calculate the fundamental group of the surface.

An explicit example of this calculation for the torus can be found here.

Answer 2

For $n\ge2$ consider $X=\mathbb{S}^{n}$, $A_{1}=\mathbb{S}^{n}\setminus\{N\}$, and $A_{2}=\mathbb{S}^{n}\setminus\{S\}$ where $N$ and $S$ are the north and south pole respectively. Then $A_{1},A_{2},\text{and}\,A_{1}\cap A_{2}$ are all path connected. The intersection is homotopic to $\mathbb{S}^{n-1}$ and is not a point. The Seifert-van Kampen Theorem applies and I don't think this example is too difficult.