Corollary 3

Under the hypotheses of Prop. 1, suppose in addition that the framework of the covering $(A_i)_{i\in I}$ is an oriented tree. The homomorphism $\mathsf{N}:_i*_{\in I}\pi_1(A_i, a(i))\rightarrow \pi_1(Y, a(i_0))$ deduced from $\mathsf{M}$ by restriction is then surjective; its kernel is the smallest normal subgroup of $_i*_{\in I}\pi_1(A_i, a(i))$ which contains the elements $\varphi_j(v)\psi_j(v)^{-1}$, for

every $j= (i_1, i_2,V)\in J$ and every $v\in \pi_1(V, b(j))$.

If the arcwise connected components of the intersections $A_i\cap A_{i'}$, for $i=\not i'$ are moreover simply connected, the homomorphism $\mathsf{N}$ is then an isomorphism.

Under the hypotheses of the corollary, the graph associated with the quiver Γ is a tree, whence T = Γ. It follows that the image of the group F(J) by the homomorphism $\mathsf{M}$ is reduced to the identity element.

Let

be the unique group homomorphism which induces the identity homomorphism on $\pi_1(A_i, a(i))$ and whose kernel contains F(J). We have $\mathsf{M}=\mathsf{N}\circ \rho$. Consequently, the homomorphism $\mathsf{N}$ is surjective and its kernel is the smallest normal subgroup of $_i*_{\in I}\pi_1(A_i, a(i))$ which

contains the images under $\rho$ of the elements defined by relations ($R_1$), ($R_2$) and ($R_3$) of Prop. 1. We have $\rho (j) = 1$ for every $j\in$ Fl(Γ). Since the framework of the covering $(A_i)_{i\in I}$ is a tree, one has $A_{i_1}\cap A_{i_2}\cap A_{i_3}=\emptyset$ for every triple $(i_1, i_2, i_3)$ of distinct elements of I. It follows that the kernel of $\mathsf{N}$ is the smallest normal subgroup containing the elements $\varphi_j(v)\psi_j(v)^{-1}$ for every $j= (i_1, i_2,V)\in J$ and every $v\in \pi_1(V, b(j))$.