Lemma 2

The mapping $f: C\rightarrow A$ is proper and separated; its fibres are finite.

Let us prove that $f$ is closed. Let X be a closed subset of C. Let us prove that its image is closed in A. For each $\ell \in L$, $X\cap B_{\ell}$ is closed in $B_{\ell}$, hence the space $Y =\bigcup_{\ell\in L}h^{-1}_{\ell}(X\cap B_{\ell})$ is closed in B since L is finite. The saturation $X^*$ of X for the equivalence relation R is then equal to $X\cup \bigcup_{\ell\in L}h_k(Y)$, hence is closed in C. Consequently, $f(X) =f(X^*)$ is closed in A, which was to be proved.

The fibres of $f$ are the equivalence classes of the relation R; they are finite. Consequently, the mapping $f$ is proper (TG, I, p. 75, Theorem 1).

Lastly, let us show that $f$ is separated. Let $x$ and $y$ be distinct points of C which have the same image under $f$. There therefore exists a point $b\in B$ and distinct elements $\ell$ and $m\in L$ such that $x=h_{\ell}(b)$ and $y=h_m(b)$. Consequently, $N_{\ell}$ and $N_m$ are disjoint neighbourhoods of $x$ and $y$ in C, whence the required assertion by virtue of Prop. 1 of I, p. 25.

Assume in addition the following hypotheses:

– the space A is simply connected and connected;

– the space C is simply connected;

– the space B is connected and locally arcwise connected.

Let us put $I =\pi_0(C)$; if $i$ is an element of I, let $C_i$ denote the connected component $i$ of C. Let $\eta : L\rightarrow I$ denote the mapping which associates to an element $\ell \in L$ the connected component of C containing $B_{\ell}$. The mapping $\eta$ is surjective. For, arguing by contradiction, consider a connected component X of C which meets no set $B_{\ell}$. It is an open and closed subset of C, saturated with respect to the equivalence relation R. Hence its image $f(X)$ is an open and closed subset of A, disjoint from B. Since A is assumed connected and B nonempty, $f(X)$ is empty, whence a contradiction.

Let $\sigma : I\rightarrow L$ be a section of the mapping $\eta$; put $\tau =\sigma \circ \eta$ and $T =\sigma (I)$.

Choose a point $b$ of B. For every $\ell \in L$, put $b_{\ell}=h_{\ell}(b)$ and choose the class $\beta_{\ell}$ of a path joining $b_{\ell}$ to $b_{\tau(\ell)}$ in C; if $\ell =\tau (\ell )$, choose for $\beta_{\ell}$ the class of the constant path whose image is $b_{\ell}$. For every $\ell \in L$, let $\vartheta_{\ell}$ denote the homomorphism from $\pi_1(B, b)$ into $\pi_1(C, b_{\tau(\ell)})$ defined by

$\vartheta_{\ell}(v) =$ Int($\beta_{\ell}$)$^{-1}((h_{\ell})_*(v)) =\beta_{\ell}^{-1}(h_{\ell})_*(v)\beta_{\ell}$,

for every $v\in \pi_1(B, b)$. Finally, fix an element $\ell_0$ of L such that $\ell_0=$ $\tau (\ell_0)$.

Let $\mathsf{Q}$ be the unique group homomorphism

such that $\mathsf{Q}(\ell ) =f_*(\beta_{\ell})$ for every $\ell \in$ L - T and which coincides with $\pi_1(f, b_{\sigma(i)})$ on $\pi_1(C_i, b_{\sigma(i)})$ for every $i\in I$.