Corollary 4

Under the hypotheses of Proposition 1, suppose that there exists a subset A of Y, connected by arcs and nonempty, such that the intersection $A_i\cap A_{i'}$ is equal to A for every pair $(i, i')$ of distinct elements of I. Let $a$ be a point of A. There exists a unique homomorphism $\varphi$ of the sum of the family of groups $(\pi_1(A_i, a))_{i\in I}$ amalgamated by $\pi_1(A, a)$ into $\pi_1(Y, a)$ which coincides with the homomorphism induced by the canonical injection of $A_i$ into Y, for every $i\in I$. The homomorphism $\varphi$ is an isomorphism.

In particular, if the group $\pi_1(A, a)$ is reduced to the identity element, the canonical homomorphism of the free product of the family of groups $(\pi_1(A_i, a))_{i\in I}$ into $\pi_1(Y, a)$ is an isomorphism.

For $i\in$ I, let us denote by $g_i:\pi_1(A, a)\rightarrow \pi_1(A_i, a)$ and $f_i:\pi_1(A_i, a)\rightarrow$ $\pi_1(Y, a)$ the canonical homomorphisms induced by the inclusions of A into $A_i$ and of $A_i$ into Y. Let us denote by $*\pi_1(A_i, a)$ the sum of the groups $\pi_1(A_i, a)$ amalgamated by $\pi_1(A, a)$. Let also$^Ap$ be the unique homomorphism of $_i*_{\in I}\pi_1(A_i, a)$ into $*_A\pi_1(A_i, a)$ which induces the identity on $\pi_1(A_i, a)$ (A, I, p. 80, prop. 4). The homomorphism $p$ is surjective and it follows from the definition of the monoid $*\pi_1(A_i, a)$ that its kernel is the smallest distinguished subgroup of$^A_i*_{\in I}\pi_1(A_i, a)$ which contains the

elements $g_i(v)g_{i'}(v)^{-1}$, for $i, i'\in I$ and $v\in \pi_1(A, a)$.

The homomorphisms $f_i\circ g_i:\pi_1(A, a)\rightarrow \pi_1(Y, a)$, for $i\in I$, are equal. It therefore follows from the universal property of amalgamated sums of monoids (A, I, p. 80, Prop. 4) that there exists a unique group homomorphism $\varphi$ (resp. $f$) from $*_A\pi_1(A_i, a)$ (resp. from $_i*_{\in I}\pi_1(A_i, a)$) into $\pi_1(Y, a)$ which induces the homomorphism $f_i$ on $\pi_1(A_i, a)$. We have $f=$ $\varphi \circ p$.

For $i\in$ I, let $u_i$ be the canonical injection of A into $A_i$ and $v_i$ the canonical injection of $A_i$ into Y. Let also $w$ be the canonical injection of A into Y. For every $i\in$ I, one has $v_i\circ u_i$ = $w$, hence $\pi_1(v_i, a)\circ \pi_1(u_i, a) =\pi_1(w, a)$. It then follows from the universal property of amalgamated sums of monoids (A, I, p. 80, Prop. 4) that there exists a unique group homomorphism $\varphi$ from $*_i\pi_1(A_i, a)$ into $\pi_1(Y, a)$ which induces the homomorphism $\pi_1(v_i, a)$ on $\pi_1(A_i, a)$. It remains to prove that $\varphi$ is an isomorphism.

The set J is identified with the set of pairs $(i, i')$ of elements of I such that $i < i'$. The set K is identified with the set of triples $(i_1, i_2, i_3)$ of elements of I such that $i_1< i_2< i_3$. Let us choose all the base points $a(i),b(j)$ and $c(k)$ equal to $a$ and all the paths $B(j)$, $C_{st}(k)$ equal to the constant path with image $a$. Let us also fix a point $i_0\in I$.

For every pair $(i, i')$ of distinct points of I, the framework Γ of the covering $(A_i)$ has exactly one arrow with endpoints $i$ and $i'$. The arrows of Γ one of whose endpoints is equal to $i_0$ are the arrows of an oriented maximal tree T.

For every element $k\in K$, one has $r_3(k) =j_{12}(k)j_{23}(k)j_{13}(k)^{-1}$; let R be the normal subgroup of F(J) generated by the elements $j\in$ Fl(T) and the elements $r_3(k),k\in$ K. Let us prove that R = F(J). It is enough to show that every element $j= (i_1, i_2)$ of J belongs to R. This is true, by hypothesis, if $i_1=i_0$ or $i_2=i_0$. Suppose $i_0< i_1$ and put $k= (i_0, i_1, i_2)$. This is an element of K such that $j=j_{23}(k) =j$. Moreover, $j_{12}(k)$ and $j_{13}(k)$ belong to Fl(T). It follows that $j$ belongs to R. The cases where $i_1< i_0< i_2$ or $i_2< i_0$ are treated in an analogous manner.

It then follows from Prop. 1 (IV, p. 412) that the homomorphism $f$ is surjective and that its kernel is the smallest distinguished subgroup containing the relators $r_2(j, v) =g_i(v)g_{i'}(v)^{-1}$, for $j= (i, i',A)\in J$ and $v\in \pi_1(A, a)$. In other words, Ker($f$) $=$ Ker($p$). It follows that $\varphi$ is an isomorphism, as was to be proved.

If $\pi_1(A, a)$ is reduced to the identity element, $p$ is an isomorphism, whence the second assertion.