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

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.

Side notes about free product of two groups

Consider arbitrary group $G_1,G_2$ , then $G_1\times G_2$ is a group.

Note that the inclusion map $i_1:G_1\to G_1\times G_2$ is a group homomorphism and the inclusion map $i_2:G_2\to G_1\times G_2$ is a group homomorphism. The image of them commutes since $(e,g_2)(g_1,e)=(g_1,g_2)=(g_1,e)(e,g_2)$ .

The universal property

Then we want to have a group $G$ such that for all group homomorphism $\phi:G_1\to H$ and $G_2\to H$ , such that there always exists a map $\Phi: G\to H$ such that:

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

How to construct the free group?

We consider

G_1*G_2=S=\{g_1h_1g_2h_2:g_1,g_2\in G_1,h_1,h_2\in G_2\}/\sim

And we set $g_ie_{G_2}g_{i+1}\sim g_ig_{i+1}$ for $g_i\in G_1$ and $g_{i+1}\in G_2$ .

And $h_je_{G_1}h_{j+1}\sim h_jh_{j+1}$ for $h_j\in G_2$ and $h_{j+1}\in G_1$ .

And we define the group operation

(g_1 h_1\cdots g_k h_k)*(h_1' g_1'\cdots h_l' g_l')=g_1 h_1\cdots g_k h_k g_1' h_2'\cdots h_l' g_l'

And the inverse is defined

(g_1 h_1\cdots g_k h_k)^{-1}=h_k^{-1} g_k^{-1}\cdots h_1^{-1} g_1^{-1}

And $G=S$ is a well-defined group.

The homeomorphism $G\to H$ is defined as

\Phi((g_1 h_1\cdots g_k h_k))=\phi_1(g_1)\circ \phi_2(h_1)\circ \cdots \circ \phi_1(g_k)\circ \phi_2(h_k)

Note $\circ$ is the group operation in $H$ .

Group with such universal property is unique, so we don’t need to worry for that too much.

Back to the Seifert-Van Kampen Theorem:

Let $H=\pi_1(U,x_0)* \pi_1(V,x_0)$ .

Let $N$ be the least normal subgroup in the free product $H$ , containing $i_1(g)i_2(g)^{-1}$ , $\forall g\in \pi_1(U\cap V,x_0)$ .

Note $i_1(g)\in \pi_1(U,x_0)$ and $i_2(g)\in \pi_1(V,x_0)$ . You may think of them as $G_1,G_2$ in the free group descriptions.

Seifert-Van Kampen Theorem (classical version)

There is an isomorphism between $\pi_1(U,x_0)* \pi_1(V,x_0)/N$ and $\pi_1(U\cup V,x_0)$ .