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This is from Munkres' Topology page 431:

Theorem 70.2 (Seifert-van Kampen theorem, classical version). Assume the hypotheses of the preceding theorem. Let

$$ j : \pi_1(U, x_0) * \pi_1(V, x_0) \longrightarrow \pi_1(X, x_0) $$

be the homomorphism of the free product that extends the homomorphisms $j_1$ and $j_2$ induced by inclusion. Then $j$ is surjective, and its kernel is the least normal subgroup $N$ of the free product that contains all elements represented by words of the form

$$ (i_{1}(g))^{-1} i_{2}(g)), $$

for $g \in \pi_{1}(U \cap V, x_{0}).$

and by hypotheses of the preceding theorem it means that $X$ is a topological space and $U$, $V$ open subsets of $X$ such that $X = U \cup V$ and $x_0 \in U \cap V$

$j_1:\pi_1(U,x_0)\to\pi_1(X,x_0)$ and $ j_2:\pi_1(V,x_0)\to\pi_1(X,x_0)$ are inclusion induced homomorphisms.

I read the proof and understood it but something is really bothering me and that is how do we know that the homomorphism $j$ mentioned in the theorem even exists?

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    $\begingroup$ This comes down to understanding the definition of the free product. $\endgroup$
    – Randall
    Mar 25 at 15:42
  • $\begingroup$ can you elaborate on it a bit? $\endgroup$ Mar 25 at 15:45
  • $\begingroup$ I'd start here: en.wikipedia.org/wiki/Free_product $\endgroup$
    – Randall
    Mar 25 at 15:46
  • $\begingroup$ I've read the definition before so if you could explain how this comes down to understanding the definition it would be appreciated $\endgroup$ Mar 25 at 15:49

1 Answer 1

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It tells you: $j_1:U\subset X$ and $j_2:V\subset X$ induce maps on $\pi_1$, $\pi_1(U)\to\pi_1(X)$ and $\pi_1(V)\to\pi_1(X)$, and these are genuine homomorphisms, and if you have two groups $A,B$ and a third group $C$ and homomorphisms $A\to C,B\to C$ then there is a unique associated homomorphism $A\ast B\to C$. In this case with $A=\pi_1(U),B=\pi_1(V),C=\pi_1(X)$, there is a unique homomorphism $j:\pi_1(U)\ast\pi_1(V)\to\pi_1(X)$ associated to $j_1,j_2$.

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  • $\begingroup$ is this a trivial theorem or has a name because it hasn't been mentioned in the book $\endgroup$ Mar 25 at 15:43
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    $\begingroup$ This is the defining property of the free product or sometimes also called the universal property of the free product, that any pair of morphisms into the same group extends uniquely to a morphism from the free product into the target group $\endgroup$ Mar 25 at 15:53
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    $\begingroup$ @Sven-Ole Behrend thank you, I got it. $\endgroup$ Mar 25 at 16:23
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    $\begingroup$ @DavoodKarimi check out tom leinster's basic category theory for funsies. the free product is the coproduct in groups. $\endgroup$ Mar 25 at 21:48

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