Let $c_1\in \Lambda_{x,y}(X),c_2\in \Lambda_{y,z}(X),c_3\in \Lambda_{z,u}(X)$ be representatives of $\gamma_1,\gamma_2$ and $\gamma_3$ respectively. Let $\varphi :\mathbf{I}\rightarrow \mathbf{I}$ be the function defined by

$t/2$ for $0\leqslant t\leqslant 1/2$,

(4) $\varphi (t) =t-1/4$ for $1/2\leqslant t\leqslant 3/4$,

$2t-1$ for $3/4\leqslant t\leqslant 1$.

The function $\varphi$ is affine on each of the three intervals $[0,1/2]$, $[1/2,3/4]$ and $[3/4,1]$; it is therefore continuous. One has $\varphi (0) = 0$ and $\varphi (1) = 1$. It follows from formula (1) of III, p. 256 defining the juxtaposition of paths that

By the example of III, p. 289, the paths $c_1*(c_2*c_3)$ and $(c_1*c_2)*c_3$ are strictly homotopic, whence equality (3).

Analogously, the function $\psi :\mathbf{I}\rightarrow \mathbf{I}$ defined by

$2t$ for $0\leqslant t\leqslant 1/2$,

1 for $1/2\leqslant t\leqslant 1$

is continuous and satisfies $\psi (0) = 0,\psi (1) = 1$. The equality $\gamma_1\varepsilon_y=\gamma_1$ then follows from the fact that

and the equality $\varepsilon_x\gamma_1=\gamma_1$ is proved analogously, whence (1).

The mapping $\sigma :\mathbf{I}\times \mathbf{I}\rightarrow X$ defined by

$c_1(2ts)$ for $0\leqslant t\leqslant 1/2$,

$c_1(2(1-t)s)$ for $1/2\leqslant t\leqslant 1$

is continuous; it is a strict homotopy joining the path $e_x$ to the path $c_1*\overline{c_1}$, whence the first equality in (2). The second results from the first and from the fact that, for every path $c$, one has $c=\overline{\overline{c}}$.

a) If the space X is arcwise connected, the groups $\pi_1(X, x)$ and $\pi_1(X, y)$ are isomorphic for all points $x$ and $y$ of X.

b) Let $x$ be a point of X; the following conditions are equivalent:

(i) The group $\pi_1(X, x)$ is trivial;

(ii) Two paths with origin $x$ in X which have the same end

are strictly homotopic;

(iii) Every loop with origin $x$ in X is strictly homotopic

to the constant loop with image $x$.

More precisely, let X be an arcwise connected topological space and let $x$ and $y$ be points of X. For every element $\delta$ of $\varpi_{x,y}(X)$, the mapping $u_{\delta}:\gamma \mapsto \delta \gamma \delta^{-1}$ of $\pi_1(X, y)$ into $\pi_1(X, x)$ is a group isomorphism whose inverse isomorphism is $u_{\delta^{-1}}$. For $\delta \in \varpi_{x,y}(X)$ and $\gamma \in \pi_1(X, y)$, we set $^{\delta}\gamma =u_{\delta}(\gamma )$. When $x=y$, one has $^{\delta}\gamma =$ Int($\delta$ )$(\gamma )$, i.e. $u_{\delta}=$ Int($\delta$ ).

Let $x,y,z$ be points of X. For $\delta \in \varpi_{x,y}(X)$ and $\eta \in \varpi_{y,z}(X)$, one has $u_{\delta \eta}=u_{\delta}\circ u_{\eta}$, which may also be written $^{\delta \eta}\gamma =^{\delta}(^{\eta}\gamma )$ for $\gamma \in \pi_1(X, z)$.

Let $x,y$ be points of X and let $\delta ,\delta '$ be elements of $\varpi_{x,y}(X)$. One has

(7) $u_{\delta'}=u_{\delta}\circ$ Int($\delta^{-1}\delta '$) $=$ Int($\delta '\delta^{-1}$)$\circ u_{\delta}$.

Let $c$ be a loop in X at $x$ and let $\gamma$ be its strict homotopy class. Put $\gamma_0= (f_0)_*(\gamma )$ and $\gamma_1= (f_1)_*(\gamma )$; these are the strict homotopy classes of $f_0\circ c$ and $f_1\circ c$. For $(t, s)\in \mathbf{I}\times \mathbf{I}$, put $\varphi (t, s) =\sigma (c(t), s)$. For every $t\in \mathbf{I}$, one has $\varphi (t,0) = (f_0\circ c)(t),\varphi (t,1) = (f_1\circ c)(t)$ and $\varphi (0, t) =\varphi (1, t) =d(t)$. The relation $\gamma_0\delta =\delta \gamma_1$ therefore follows from the following lemma.

Let $c$ denote the path in $\mathbf{I}\times \mathbf{I}$ obtained by juxtaposing the paths $t\mapsto (t,0)$ and $t\mapsto (1, t)$; let $d$ denote the path in $\mathbf{I}\times \mathbf{I}$ obtained by juxtaposing the paths $t\mapsto (0, t)$ and $t\mapsto (t,1)$. The paths $c$ and $d$ have the same origin $(0,0)$ and the same end $(1,1)$. The mapping $(t, s)\mapsto$ $(1-s)c(t)+sd(t)$ of $\mathbf{I}\times \mathbf{I}$ into $\mathbf{I}\times \mathbf{I}$ is a strict homotopy joining $c$ to $d$. One has $c_0*d_1=\varphi \circ c$ and $d_0*c_1=\varphi \circ d$; these two paths are therefore strictly homotopic.

Let $\sigma$ be a pointed homotopy at $x$ joining $f_0$ to $f_1$. With the notation of Prop. $3,\delta$ is the class of a constant path with image $f_0(x) =$ $f_1(x)$. The assertion follows.

Let $g$ be a continuous mapping of Y into X, inverse to $f$ up to homotopy. Let $x$ be a point of X. It follows from Prop. 3 applied to the homotopic mappings Id$_X$ and $g\circ f: X\rightarrow X$ that the mapping $\pi_1(g\circ f, x)$ is an isomorphism of the group $\pi_1(X, x)$ onto the group $\pi_1(X, g\circ f(x))$. Since

the homomorphism $\pi_1(f, x)$ is injective and the homomorphism $\pi_1(g, f(x))$ is surjective. Since the mapping $g$ is also a homotopy equivalence, the homomorphism $\pi_1(g, f(x))$ is injective; it is therefore an isomorphism. Hence $\pi_1(f, x)$ is an isomorphism.

Cor. 3 applies in particular when X is the numerical space of $n$ dimensions $\mathbf{R}^n$ and more generally when X is a subset of the space $\mathbf{R}^n$ which is star-shaped (III, p. 234) with respect to one of its points.

Let $\varphi$ denote this morphism of groupoids. The mapping deduced from it by passing to the vertices is the identity mapping $X\rightarrow \prod_jX_j$. Let $x= (x_j)$ and $y= (y_j)$ be two points of X. Let $\varphi_{x,y}$ denote the mapping of $\varpi_{x,y}(X)$ into $\prod_j\varpi_{x_j,y_j}(X_j)$ deduced from $\varphi$. If for every $j\in J,c_j:\mathbf{I}\rightarrow$ $X_j$ is a path joining $x_j$ to $y_j$, the mapping $t\mapsto (c_j(t))$ is a path in X joining $x$ to $y$ (TG, I, p. 25, prop. 1). This proves that $\varphi_{x,y}$ is surjective. Let $c$ and $d$ be two paths in X joining $x$ to $y$. Suppose that for every $j\in J$ there exists a strict homotopy $\sigma_j:\mathbf{I}\times \mathbf{I}\rightarrow X_j$ joining pr$_j\circ c$ to pr$_j\circ d$. Then the mapping $(t, s)\mapsto (\sigma_j(t, s))$ of $\mathbf{I}\times \mathbf{I}$ into X is a strict homotopy joining $c$ to $d($loc. cit.). This proves that $\varphi_{x,y}$ is injective.

This isomorphism is said to be canonical. In the sequel, we shall often identify $\pi_1(X, x)$ with $\prod_{j\in J}\pi_1(X_j, x_j)$ by means of this isomorphism.

$\rho :$ lim$\longrightarrow_{i\in I}\varpi (A_i)\rightarrow \varpi (X)$,

induced by the canonical injections of $A_i$ into X, is an isomorphism.

If $i\leqslant j$, let us denote by $\rho_{j,i}$ the groupoid morphism $\varpi (A_i)\rightarrow \varpi (A_j)$ induced by the injection of $A_i$ into $A_j$. Since the mapping induced by $\rho_{j,i}$ by passing to the vertices is the injection $A_i\rightarrow A_j$ and since the $A_i$ cover X, the mapping induced by $\rho$ by passing to the vertices is bijective.

Let $a$ and $b$ be points of X and let $c$ be a path joining $a$ to $b$ in X. The image of $c$ is a quasi-compact subset of X (TG, I, p. 62, Theorem 2), since $\mathbf{I}$ is compact. There therefore exists an element $i\in I$ such that the image of $c$ is contained in $A_i$. Consequently, the mapping induced by $\rho$ by passing to the sets of arrows is surjective.

Let $i\in I$, let $a$ and $b$ be points of $A_i$, let $c,c'$ be paths joining $a$ to $b$ in $A_i$; let $h$ be a strict homotopy joining $c$ to $c'$ in X. Since $\mathbf{I}\times \mathbf{I}$ is compact, $h(\mathbf{I}\times \mathbf{I})$ is a quasi-compact subset of X (loc. cit.) and there exists an element $i\in I$ such that the image of $h$ is contained in $A_i$. The paths $c$ and $c'$ are strictly homotopic in $A_i$; fortiori, the path classes $[c]$ and $[c']$ have the same image in lim$\longrightarrow \varpi (A_i)$. Consequently, $\rho$ is injective.

lim$\longrightarrow_{i\in J}\pi_1(A_i, a)\rightarrow \pi_1(X, a)$

is bijective.