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 $\left({X, \tau}\right)$ be a topological space.
Let $U_1, U_2 \in \tau$ such that:
- $U_1 \cup U_2 = X$
- $U_1 \cap U_2 \ne \varnothing$ 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:
- $\pi_1 \left({X}\right) = \pi_1 \left({X, \ast}\right)$
It is to be shown that $\pi_1 \left(X\right)$ is the amalgamated free product:
- $\pi_1 \left({U_1}\right) *_{\pi_1 \left({U_1 \cap U_2}\right)} \pi_1 \left({U_2}\right)$
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Source of Name
This entry was named for Karl Johannes Herbert Seifert and Egbert Rudolf van Kampen.