Proposition 5 (Van Kampen)

The homomorphism $\mathsf{Q}$ is surjective; its kernel is the smallest normal subgroup containing the elements

$\vartheta_{\ell_0}(v)\vartheta_{\ell}(v)^{-1}$ for $v\in \pi_1(B, b)$ and $\ell \in T$,

$\vartheta_{\ell_0}(v)\ell \vartheta_{\ell}(v)^{-1}\ell^{-1}$ for $v\in \pi_1(B, b)$ and $\ell \in L-T$.

The mapping $f: C\rightarrow$ A is proper, separated, with finite fibres (IV, p. 430, lemma 2); the spaces A and C are unlaceable. Moreover, $C\times_AC$ is the union of the diagonal $\Delta_C$ and of the disjoint subsets $B_{\ell}\times_AB_k=$ $(h_{\ell}, h_k)(B)$, for $(\ell , k)\in L^2$ with $\ell =\not k$. For such a pair $(\ell , k)$, we have $B_{\ell}\times_AB_k= N_{\ell}\times_AN_k$; hence this subset is open and closed in $C\times_AC$. Thus the diagonal $\Delta_C$ is open, and its complement in $C\times_AC$, a finite union of disjoint subsets homeomorphic to B, is locally connected. This proves that the mapping $f$ satisfies property (VK) (case (ii) of IV, p. 405). With a view to applying theorem 1 of IV, p. 408, we are going to define van Kampen data for $f$.

For each $i\in I$, let us choose as base point in $C_i$ the point $\mathsf{a}(i) =$ $b_{\sigma(i)}$.

Let J be the set of path-connected components of $C\times_AC$. The sets $\Delta_{C_i}$, for $i\in I$, are the connected components of the diagonal $\Delta_C$, which is open and closed in $C\times_AC$, therefore these sets belong to J. Analogously, the sets $[\ell_1, \ell_2] = B_{\ell_1}\times_A$ $B_{\ell_2}$ for $\ell_1$ and $\ell_2$ in L with $\ell_1=\not\ell_2$, belong to J. Since these sets form a partition of $C\times_AC$, they describe the set J.

Let $j$ be an element of J of the form $\Delta_{C_i}$, for $i\in I$. Let us choose as base point $\mathsf{b}(j)$ the point $(\mathsf{a}(i),\mathsf{a}(i)) = (b_{\sigma(i)}, b_{\sigma(i)})$ and let us take for path classes $\beta_1(j)$ and $\beta_2(j)$ the class of the constant path at $b_{\sigma(i)}$. Let $j$ be an element of J of the form $[\ell_1, \ell_2]$, where $\ell_1$ and $\ell_2$ are distinct elements of L. Let us then put $\mathsf{b}([\ell_1, \ell_2]) = (b_{\ell_1}, b_{\ell_2}),\beta_1([\ell_1, \ell_2]) =\beta_{\ell_1}$ and $\beta_2([\ell_1, \ell_2]) =\beta_{\ell_2}$.

Let $K =\pi_0(C\times_AC\times_AC)$.

Let $\Delta '_C$ denote the diagonal of the space $C\times_AC\times_AC$; this is a closed subset of $C\times_AC\times_AC$, because $f$ is separated, and it is homeomorphic to C. For every $i\in$ I, let $\Delta '_{C_i}$ also denote the image of $C_i$ under the diagonal mapping of C into $C\times_AC\times_AC$; these are open and closed subsets of $\Delta '_C$. For every triplet $(\ell_1, \ell_2, \ell_3)$ of elements of L, put also $[\ell_1, \ell_2, \ell_3] = B_{\ell_1}\times_AB_{\ell_2}\times_AB_{\ell_3}$; these are closed subsets of $C\times_A$ $C\times_AC$, homeomorphic to B. If the set $\{\ell_1, \ell_2, \ell_3\}$ has at least two elements, one has $[\ell_1, \ell_2, \ell_3] = N_{\ell_1}\times_AN_{\ell_2}\times_AN_{\ell_3}$, which implies that $[\ell_1, \ell_2, \ell_3]$ is also open in $C\times_AC\times_AC$. Moreover, $C\times_AC\times_AC$ is the union of the pairwise disjoint subsets $\Delta '_C$ and $[\ell_1, \ell_2, \ell_3]$, where $(\ell_1, \ell_2, \ell_3)$ runs through the set of triplets of elements of L which are not all equal to one and the same element. Thus, $\Delta_{C'}$, then the $\Delta '_{C_i}$, for $i\in I$, are open and closed in $C\times_AC\times_AC$. This implies that the set K of connected components of this space is the union of the following disjoint sets $K_0$ and $K_1$.

The set $K_0$ is the set of components of the form $\Delta '_{C_i}$. Let $i\in I$ and let $k= \Delta '_{C_i}$. Put $\mathsf{c}(k) = (b_{\sigma(i)}, b_{\sigma(i)}, b_{\sigma(i)})$. For $(s, t)\in \{(1,2),(2,3),(1,3)\}$, choose for $\gamma_{st}(k)$ the class of the constant path at $(b_{\sigma(i)}, b_{\sigma(i)})$.

The set $K_1$ consists of the components of the form $k$ = $[\ell_1, \ell_2, \ell_3]$, where $\ell_1, \ell_2, \ell_3$ are three elements of B such that the set $\{\ell_1, \ell_2, \ell_3\}$ is of cardinal $\geqslant 2$. Put $\mathsf{c}(k) = (b_{\ell_1}, b_{\ell_2}, b_{\ell_3})$. Let $(s, t)\in \{(1,2),(1,3),(2,3)\}$. If $\ell_s=\ell_t$, take for $\gamma_{st}(k)$ the class $(\beta_{\ell_s}, \beta_{\ell_t})$; if $\ell_s=\not\ell_t$, take for $\gamma_{st}(k)$ the class of the constant path at $(b_{\ell_s}, b_{\ell_t})$.

It is then verified that, for every $k\in K$ and for every $s\in \{1,2,3\}$, the class of loops $\lambda_s(k)$ defined by relation (2) of IV, p. 407, is trivial.

The framework $\mathsf{\Gamma}$ of the pair of groupoid morphisms $(\varpi (p_1), \varpi (p_2))$ from $\varpi (C\times_AC)$ into $\varpi (C)$ has as vertices the set I and as oriented edges the set J. If $i\in I$, the arrow $j= \Delta_{C_i}$ has origin and end $i$; if $(\ell_1, \ell_2)$ is a pair of distinct elements of L, the arrow $j= [\ell_1, \ell_2]$ has origin $\eta (\ell_1)$ and end $\eta (\ell_2)$.

Let $\mathsf{T}$ be the subquiver of $\mathsf{\Gamma}$ whose set of vertices is I and whose arrows are those of the form $[\ell_0, \ell ]$, for $\ell \in T-\{\ell_0\}$. The associated graph of $\mathsf{T}$ is a maximal tree of $\widetilde{\mathsf{\Gamma }}$.

Put $i_0=\eta (\ell_0)$.

If $i\in I-\{i_0\}$, the unique path in $\mathsf{T}$ joining $i_0$ to $i$ is $(i_0,[\ell_0, \sigma (i)], i)$. The groupoid morphism from Grp($\mathsf{\Gamma}$) into $\varpi (Y)$ defined on p. 407 (and denoted by $\tau$ loc. cit.) maps $j$ to the identity element if $j= \Delta_{C_i}$, for $i\in I$, and maps $j= [\ell_1, \ell_2]$ to $f_*(\beta_{\ell_1})^{-1}f_*(\beta_{\ell_2})$, if $\ell_1$ and $\ell_2$ are distinct elements of L. For every $i\in I$, the path class $\delta_i$ defined loc. cit. and joining $b=f(b_{\sigma(i_0)})$ to $b=f(b_{\sigma(i)})$ is given by

because $\beta_{\ell}=e$ if $\ell \in T$.

Thus a van Kampen datum of the mapping $f$ has been defined. Consider then the unique group homomorphism

which coincides with $\pi_1(f, b_{\sigma(i)})$ on $\pi_1(C, b_{\sigma(i)})$ and such that

for $j\in J$. By IV, p. 408, Theorem 1, this homomorphism is surjective and its kernel is the smallest normal subgroup containing the relators $\mathsf{r}_1(j)$ (for $j\in$ Fl($\mathsf{T}$)), $\mathsf{r}_2(j, v)$ (for $j\in$ J and $v\in$ $\pi_1(C\times_AC,\mathsf{b}(j))$) and $\mathsf{r}_3(k)$ (for $k\in K$) defined, loc. cit., by equations ($R_1$)$, (R_2)$ and ($R_3$).

Let $q'$ be the unique homomorphism of F(L) into F(L - T) such that $q'(\ell ) =\ell$ if $\ell \in L-T$ and $q'(\ell ) =e$ otherwise. Let

be the unique group homomorphism which coincides with the identity on $\pi_1(C_i, b_{\sigma(i)})$, for $i\in$ I and such that one has $q(j) =e$ if $j= \Delta_{C_i}$, $q([\ell , \ell ']) =q'(\ell )^{-1}q'(\ell ')$ if $\ell$ and $\ell '$ are distinct elements of L. The homomorphism $q$ is surjective.

If $i\in I$ and $j= \Delta_{C_i}$, one has $\mathsf{Q}'(j) =e_b=\mathsf{Q}'(q(j))$. If $j= [\ell , \ell ']$, for $\ell$ and $\ell '$ in L, distinct, one has

Therefore, $\mathsf{Q}'$ = $\mathsf{Q}\circ q$. It follows that the homomorphism $\mathsf{Q}$ is surjective and that its kernel is the smallest normal subgroup of $(_i*_{\in I}\pi_1(C_i, b_{\sigma(i)}))*F(L-T)$ containing the images under $q$ of the relators $\mathsf{r}_1(j)$ (for $j\in$ Fl($\mathsf{T}$))$,\mathsf{r}_2(j, v)$ (for $j\in J$ and $v\in \pi_1(C\times_AC,\mathsf{b}(j))$) and $\mathsf{r}_3(k)$ (for $k\in K$).

If $\ell \in T-\{\ell_0\}$ and $j= [\ell_0, \ell ]$, one has $\mathsf{r}_1(j) =j$ and $q(\mathsf{r}_1(j)) =e$.

Let $k\in K$. One has $\mathsf{r}_3(k) =j_{12}(k)j_{23}(k)j_{13}(k)^{-1}$. If $k= \Delta '_{C_i}$, for $i\in I$, let us put $j= \Delta_{C_i}$; then one has

Let $\ell$ and $\ell '$ be distinct elements of L. If $k= [\ell , \ell , \ell ']$, one has therefore

If $k= [\ell , \ell ', \ell ]$, one gets

Moreover, if $k= [\ell ', \ell , \ell ]$, one has

Finally, if $k= [\ell_1, \ell_2, \ell_3]$, where $\ell_1, \ell_2, \ell_3$ are elements of L, pairwise distinct, one has

Let $i\in I$ and put $j= \Delta_{C_i}$. The group homomorphisms $\varphi_j$ and $\psi_j$, from $\pi_1(C\times_AC,\mathsf{b}(j)) =\pi_1(C_i,\mathsf{a}(i))$ into $\pi_1(C,\mathsf{a}(i))$ defined by relations (1) of IV, p. 407, are equal respectively to $(p_1)_*$ and to $(p_2)_*$. We have then, for $v\in \pi_1(C_i,\mathsf{a}(i)),\mathsf{r}_2(j, v) =vjv^{-1}j^{-1}$, whence $q(\mathsf{r}_2(j, v)) =e$.

Finally, let $j= [\ell , \ell ']$, where $\ell$ and $\ell '$ are distinct elements of L. The group homomorphism $\varphi_j:\pi_1(B_{\ell}\times_AB_{\ell'},\mathsf{b}(j))\rightarrow \pi_1(C_{\eta(\ell)}, b_{\tau(\ell)})$ defined loc. cit. is the homomorphism $\vartheta_{\ell}$, and the homomorphism $\psi_j$ is the homomorphism $\vartheta_{\ell'}$. We have then, for $v\in \pi_1(B_{\ell}\times_AB_{\ell'},\mathsf{b}(j))$

Let us distinguish four cases. If $\ell$ and $\ell '$ both belong to T, we have

When $\ell '=\ell_0$, we obtain the inverse of the element $\vartheta_{\ell_0}(v)\vartheta_{\ell}(v)^{-1}$. If $\ell \in T$ but $\ell '\notin T$, we have

Analogously, if $\ell \notin T$ and $\ell '\in T$, we have

Taking $\ell '=\ell_0$, we obtain an element conjugate to the inverse of $\vartheta_{\ell_0}(v)\ell \vartheta_{\ell}(v)^{-1}\ell^{-1}$. Finally, if neither $\ell$ nor $\ell '$ belongs to T, we have

These relations prove, on the one hand, that the elements announced in the proposition belong to the kernel of the homomorphism $\mathsf{Q}$, and, on the other hand, that these elements $q(\mathsf{r}_2(j, v))$ all belong to the smallest normal subgroup containing the elements announced in the statement. The proposition follows.