# Math4202 Topology II (Lecture 30)

## Algebraic Topology

We skipped a few chapters about Jordan curve theorem, which will be your final project soon. LOL, I will embedded the link once I'm done.

### Seifert-Van Kampen Theorem

#### The Seifert-Van Kampen Theorem

Let $X=U\cup V$ be a union of two open subspaces. Suppose that $U\cap V$, $U,V$ are path connected. Fix $x_0\in U\cap V$.

Let $H$ be a group (arbitrary). And now we assume $\phi_1,\phi_2$ be a group homomorphism, and $\phi_1:\pi_1(U,x_0)\to H$, and $\phi_2:\pi_1(V,x_0)\to H$.

![Seifert-Van Kampen Theorem](https://notenextra.trance-0.com/Math4202/Math4202_L30/Seifert-Van-Kampen-Theorem.png)

Let $i_1,i_2,j_1,j_2,i_{12}$ be group homomorphism induced by the inclusion maps.

Assume this diagram commutes.

$$
\phi_1\circ i_1=\phi_2\circ i_2
$$

There is a group homomorphism $\Phi:\pi_1(X,x_0)\to H$ making the diagram commute. $\Phi\circ j_1=\phi_1$ and $\Phi\circ j_2=\phi_2$.

We may change the base point using conjugations.