Proposition 2

Tag 023D, permanent. Cite it as [Bourbaki, Tag 023D].

Topologie algébrique, chapter IV, § 5. Théorème de van Kampen

There exists a unique group homomorphism

\mathsf{M}:\pi_1(B, a)*\pi_1(C, a)*F(\pi_0(A)-\{j_0\})\rightarrow \pi_1(X, a)

which coincides with $\iota_B$ in $\pi_1(B, a)$ , $\iota_C$ in $\pi_1(C, a)$ and $\mu$ in $F(\pi_0(A)$ $\{j_0\})$ . This homomorphism is surjective and its kernel is the smallest normal subgroup containing the elements

\varphi_j(v)j\psi_j(v)^{-1}j^{-1}

for $j\in \pi_0(A)-\{j_0\}$ and $v\in \pi_1(A, a_j)$ , and the elements

\varphi_0(v)\psi_0(v)^{-1}

for $v\in \pi_1(A, a)$ .

The framework Γ of the covering of X defined by the family $(B,C)$ has two vertices $b$ and $c$ corresponding to the two sets B and C. The set of its arrows is equal to $\pi_0(A)$ ; they join the point $b$ to the point $c$ . The graph associated with the subquiver of Γ whose only arrow is $j_0$ is a maximal tree of $\widetilde{\Gamma}$ . The proposition then results from IV, p. 412, prop. 1.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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Cites

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