Seifert-van Kampen Theorem
In particular: in particular, categories.
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The categorical formulation is nice and all, but this does not need category theory at all. Basic fundamental group topology suffices.
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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.