Example 1

This property is satisfied under one of the following assumptions:

(i) The spaces X and Y are delacable, the space $X\times_YX$ is locally arcwise connected, the mapping $f$ is surjective, proper, separated, with locally connected fibres;

(ii) The spaces X and Y are delacable, the mapping $f$ is surjective, proper, separated, with finite fibres, the diagonal $\Delta_X$ of $X\times_YX$ is open and its complement is locally connected;

(iii) The spaces X and Y are delacable, the mapping $f$ is surjective, open and possesses the path lifting property;

(iv) Every point of Y possesses a neighbourhood above which there exists a continuous section of the mapping $f$.

Indeed, under each of these assumptions, the mapping $f$ is surjective; it is also strict by virtue of I, p. 18, example 2. Finally, the morphism $\varpi '(f)$ is an isomorphism, both under the assumptions (i), (ii) or (iii) according to IV, p. 398, th. 1, and under assumption (iv) ( IV, p. 400, th. 2).

The prop. 5 of II, p. 208 describes the isotropy groups of the groupoid Coeg($f$). The purpose of this no$^o$ is to make explicit, when $f$ satisfies the property (VK), the description of the Poincaré groups of Y which is deduced from it by composition with the isomorphism of groupoids $\varpi '(f)$. The following nos$^{os}$ will be devoted to important particular cases.

Suppose that the mapping $f$ satisfies the property (VK).

Put $\mathsf{I}=\pi_0(X),\mathsf{J}=\pi_0(Z),\mathsf{K}=\pi_0(W)$; for $j\in \mathsf{J}$ and $s\in \{1,2\}$, put $i_s(j) =\pi_0(p_s)(j)$ ; for $k\in \mathsf{K}$ and $s,t\in \{1,2,3\}$, put $j_{st}(k) =\pi_0(q_{st})(k)$. Denote by $\mathsf{\Gamma}$ the skeleton of the pair $(\varpi (p_1), \varpi (p_2))$ of morphisms of groupoids from $\varpi (Z)$ into $\varpi (X)$ (II, p. 185, def. 3). It is the quiver $(\mathsf{I},\mathsf{J}, \pi_0(p_1), \pi_0(p_2))$, since the orbits of the Poincaré groupoid of a topological space are the arcwise connected components of this space.

Suppose moreover that Y is arcwise connected and nonempty. According to II, p. 200, remark 2, $\pi_0(\mathsf{\Gamma})$ is then in bijection with the set of orbits of the groupoid Coeg($f$); since the mapping $f$ satisfies the property (VK), the groupoid Coeg($f$) is isomorphic to the groupoid $\varpi (Y)$. The graph $\mathsf{\Gamma}$ is thus connected and nonempty.

Call van Kampen data of $f$ the data of the following elements:

(i) for every $i\in \mathsf{I}$, a point $\mathsf{a}(i)$ of the path-connected component $i$ of X ;

(ii) for every $j\in \mathsf{J}$, a point $\mathsf{b}(j) = (\mathsf{b}_1(j),\mathsf{b}_2(j))$ of the path-connected component $j$ of Z ;

(iii) for every $k\in \mathsf{K}$, a point $\mathsf{c}(k) = (\mathsf{c}_1(k),\mathsf{c}_2(k),\mathsf{c}_3(k))$ of the path-connected component $k$ of W;

(iv) for every $j\in \mathsf{J}$, the class $\beta_1(j)$ of a path in X joining $\mathsf{b}_1(j)$ to $\mathsf{a}(i_1(j))$ and the class $\beta_2(j)$ of a path in X joining $\mathsf{b}_2(j)$ to $\mathsf{a}(i_2(j))$;

(v) for every $k\in \mathsf{K}$ and for every pair $(s, t)$ equal to $(1,2)$, $(2,3)$ or $(1,3)$, the class $\gamma_{st}(k)$ of a path in Z joining $(\mathsf{c}_s(k),\mathsf{c}_t(k))$ to $\mathsf{b}(j_{st}(k))$;

(vi) a subquiver $\mathsf{T}$ of $\mathsf{\Gamma}$ whose associated graph is a maximal tree of the graph $\widetilde{\mathsf{\Gamma }}$;

(vii) an element $i_0$ of $\mathsf{I}$.

Let us choose a van Kampen datum of $f$. Then, $(\mathsf{a},\mathsf{b}, \beta_1, \beta_2,\mathsf{T}, i_0)$ is a basic equipment of the pair $(\varpi (p_1), \varpi (p_2))$ of groupoid morphisms from $\varpi (Z)$ into $\varpi (X)$ (II, p. 192, déf. 4). Moreover, the triples

and the classes of paths $(\gamma_{1,2}(k), \gamma_{2,3}(k), \gamma_{1,3}(k))$, where $k$ describes $\mathsf{K}$, define a complementary equipment of the pair $(\varpi (p_1), \varpi (p_2))$ (II, p. 208, déf. 3 ; II, p. 205, exemple ; II, p. 205, remarque). We shall say that the complete equipment of the coequalizer Coeg($f$) thus defined is deduced from the van Kampen datum of $f$ that we have chosen.

For every $j\in \mathsf{J}$, let $\varphi_j:\pi_1(Z,\mathsf{b}(j))\rightarrow \pi_1(X,\mathsf{a}(i_1(j)))$ and $\psi_j:\pi_1(Z,\mathsf{b}(j))\rightarrow \pi_1(X,\mathsf{a}(i_2(j)))$ denote the group homomorphisms defined by

$\varphi_j=$ Int($\beta_1(j)$)$^{-1}\circ (p_1)_*,v\mapsto \beta_1(j)^{-1}((p_1)_*(v))\beta_1(j)$

(1)

$\psi_j=$ Int($\beta_2(j)$)$^{-1}\circ (p_2)_*,v\mapsto \beta_2(j)^{-1}((p_2)_*(v))\beta_2(j)$,

for $v\in \pi_1(Z,\mathsf{b}(j))$. For every $k\in \mathsf{K}$ and every $s\in \{1,2,3\}$, let $\lambda_s(k)$ denote the class of a loop at the point $\mathsf{a}(i_s(k))$ in X defined by

(2)

$\lambda_2(k) =\beta_2(j_{12}(k))^{-1}\cdot ((p_2)_*(\gamma_{12}(k)))^{-1}\cdot ((p_1)_*(\gamma_{23}(k)))\cdot \beta_1(j_{23}(k))$,

Let $\tau$ denote the unique groupoid morphism from Grp($\mathsf{\Gamma}$) into $\varpi (Y)$ such that the mapping Som($\tau$ ) maps $i\in \mathsf{I}$ onto $f(\mathsf{a}(i))$ and Fl($\tau$ ) maps $j\in \mathsf{J}$ onto the class of paths $f_*(\beta_1(j))^{-1}f_*(\beta_2(j))$ joining $f(\mathsf{a}(i_1(j)))$

to $f(\mathsf{a}(i_2(j)))$ in Y. For $i\in \mathsf{I}$, let $d_i\in$ Grp($\mathsf{\Gamma}$) be the class of the unique path without backtracking joining $i_0$ to $i$ in the tree $\widetilde{\mathsf{T}}$ and put $\delta_i=$ $\tau (d_i)$.

If S is a set, recall that F(S) denotes the free group on S; the image in F(S) of an element $s\in S$ by the canonical mapping is denoted by $[s]$, or by $s$ if there is no possible confusion.