Seifert-van Kampen Theorem

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Although this article appears correct, it's inelegant. There has to be a better way of doing it. The categorical formulation is nice and all, but this does not need category theory at all. Basic fundamental group topology suffices. You can help Proof Wiki by redesigning it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Improve}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Theorem

The functor $\pi_1 : \mathbf{Top_\bullet} \to \mathbf{Grp}$ preserves pushouts of inclusions.

Proof

Let $\struct {X, \tau}$ be a topological space.

Let $U_1, U_2 \in \tau$ such that:

$U_1 \cup U_2 = X$

$U_1 \cap U_2 \ne \O$ is connected

Let $\ast \in U_1 \cap U_2$.

Let:

$i_k : U_1 \cap U_2 \hookrightarrow U_k$

$j_k : U_k \hookrightarrow U_1 \cup U_2$

be inclusions.

For the sake of simplicity let:

$\map {\pi_1} X = \map {\pi_1} {X, \ast}$

It is to be shown that $\map {\pi_1} X$ is the amalgamated free product:

$\map {\pi_1} {U_1} *_{\map {\pi_1} {U_1 \cap U_2} } \map {\pi_1} {U_2}$

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Source of Name

This entry was named for Karl Johannes Herbert Seifert and Egbert Rudolf van Kampen.