Seifert–Van Kampen theorem

The seifert-van kampen theorem described the fundamental group of 2 surfaces glued together, as a function of the 2 original fundamental groups and the fundamental group of their intersection.

Example

Application to the torus:

• ${\pi }_1(A)=\mathbb{Z}\times \mathbb{Z}$
• ${\pi }_1(B)=\mathbb{Z}$
• ${\pi }_1(A\cap B)=\{e\}$ Hence ${\pi }_1(A\cup B)={\pi }_1(A){\star }_{{\pi }_1(A\cap B)}{\pi }_1(B)=(\mathbb{Z}\times \mathbb{Z}){\star }_{\{e\}}(\mathbb{Z})={\mathbb{Z}}^3$.