# 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**

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

(2)