The Seifert-van Kampen Theorem Example 1

On The Seifert-van Kampen Theorem page we stated the very important Seifert-van Kampen theorem. We will now look at some examples of applying the theorem. More examples can be found on the following pages:

• The Seifert-van Kampen Theorem Example 1

• The Seifert-van Kampen Theorem Example 2

• The Seifert-van Kampen Theorem Example 3

Definition: The Bouquet of $n$ Circles is the topological space obtained by gluing together $n$ circles at a single common shared point.

For example, the bouquet of $2$ circles is depicted below:

Example 1

Let $X$ be the bouquet of two circles (as depicted above). Use the Seifert-van Kampen theorem to find the fundamental group of $X$.

Let $U_1$ and $U_2$ be the open sets as shown:

(Where the endpoints of the cut circles are not included in $U_1$ and $U_2$).

Then $U_1 \cap U_2$ is:

Observe that the circle is a deformation retract of $U_1$, and is also a deformation retract of $U_2$. Also, a single point is a deformation retract of $U_1 \cap U_2$. Therefore we have that:

(1)

\begin{align} \quad \pi_1(U_1, x) &\cong \mathbb{Z} = \langle \alpha : \emptyset \rangle \\ \quad \pi_1(U_2, x) &\cong \mathbb{Z} = \langle \beta : \emptyset \rangle \\ \quad \pi_1(U_1 \cap U_2, x) &\cong \{ 1 \} \end{align}

The generators of $\pi_1(U_1 \cap U_2, x)$ are simple the identity element. Therefore by the Seifert-van Kampen theorem:

(2)

\begin{align} \quad \pi_1(X, x) & \cong \langle \alpha, \beta : \emptyset \cup \{ i_{1*}(h) = i_{2^*}(h) : h \in \pi_1(U_1 \cap U_2, x) \} \rangle \\ & \cong \langle \alpha, \beta : 1 = 1 \rangle \\ & \cong \langle \alpha, \beta : \emptyset \rangle \end{align}