Corollary 2

Under the assumptions of proposition 1, suppose moreover that for every $i\in I$, the group $\pi_1(A_i, a(i))$ is reduced to the identity element and that, for every triple $(i_1, i_2, i_3)$ of pairwise distinct elements of I, the set $A_{i_1}\cap A_{i_2}\cap A_{i_3}$ is empty. The homomorphism $\mathsf{M}'':$ F(J-Fl(T)) $\rightarrow \pi_1(Y, a(i_0))$ induced by $\mathsf{M}$ by restriction is then an isomorphism.

Let $\pi ': F(J)\rightarrow$ F(J - Fl(T)) be the homomorphism which maps $[j]$ to $[j]$ if $j\in$ J-Fl(T) and which maps $[j]$ to the identity element if $j\in$ Fl(T); it is surjective and one has $\mathsf{M}''\circ \pi '=\mathsf{M}'$, where $\mathsf{M}'$ is the surjective homomorphism defined in corollary 1. It follows that $\mathsf{M}''$ is surjective and that its kernel is the smallest distinguished subgroup of F(J-Fl(T)) which contains the images under $\pi '$ of the elements described in loc. cit. But, by construction, $\pi '(j) =e$ if $j\in$ Fl(T) and the set K is empty, by hypothesis. The homomorphism $\mathsf{M}''$ is therefore an isomorphism.