Proposition 5

Let B be a topological space locally arcwise connected, and let $a$ be a point of B. In order that a subgroup H of $\pi_1(B, a)$ be admissible, it is necessary and sufficient that there exist a covering $(E, p)$ of B and a point $x\in E_a$ such that $H =p_*(\pi_1(E, x))$.

Let $(E, p)$ be a covering of B and let $x$ be a point of $E_a$; put $H =p_*(\pi_1(E, x))$ and let us prove that this is an admissible subgroup of $\pi_1(B, a)$. Let $b$ be a point of B and let V be a neighbourhood of $b$ such that $E_V=$ $(\overset{-1}{p}(V), p_V)$ is a trivializable covering of V. Let $\gamma \in \varpi_{a,b}(B)$ ; we shall prove that for every element $\delta \in \pi_1(V, b)$, the class of paths $\gamma \delta \gamma^{-1}$ belongs to H.

By III, p. 301, prop. 3, there exists a unique strict homotopy class $\gamma '$ of path with origin $x$ in E such that $p_*(\gamma ') =\gamma$ Let $y$ be the end of $\gamma '$; we have $p(y) =b($III, p. 302, prop. 4). Let then $\delta '$ be the unique class of path with origin $y$ in E such that $p_*(\delta ') =\delta$. Since $E_V$ is trivializable, $\delta '$ is the class of a loop at $y$. Then $\gamma '\delta '(\gamma ')^{-1}$ is the class of a loop at $a$ in E whose image under $p_*$ is the class $\gamma \delta \gamma^{-1}$, which was to be proved.

Conversely, let H be an admissible subgroup of $\pi_1(B, a)$. Let $\lambda_a(B)$ be the quotient of the space $\Lambda_a(B)$ of paths with origin $a$ by the relation of strict homotopy; since two strictly homotopic paths have the same end-point, the end-point mapping $e: \Lambda_a(B)\rightarrow B$ defines, by passing to the quotient, a mapping $\varepsilon :\lambda_a(B)\rightarrow B$. The composition of classes of paths endows the set $\lambda_a(B)$ with a left action of the group $\pi_1(B, a)$. Endow it with the action of the group H deduced by restriction, and denote by $H\backslash \lambda_a(B)$ the set of its orbits.

The mapping $\varepsilon :\lambda_a(B)\rightarrow B$ induces, by passing to the quotient, a mapping $q: H\backslash \lambda_a(B)\rightarrow B$. The composition of classes of paths endows the set $H\backslash \lambda_a(B)$ with a right operation of the groupoid $\varpi (B)$ relative to the mapping $q$.

This operation is without local monodromy. In fact, let $b$ be a point of B and let V be a neighbourhood of $b$ such that $\gamma i_*(\pi_1(V, b))\gamma^{-1}\subset H$ for every class of paths $\gamma$ joining $a$ to $b$ in B, where $i$ denotes the inclusion of V in B. Since B is locally arcwise connected, one may moreover suppose that V is arcwise connected. Let $c\in V$, let $\delta$ be the class in $\pi_1(B, c)$ of a loop at $c$ contained in V, and let $\delta '$ be an element of $\lambda_a(B)$ such that $\varepsilon (\delta ') =c$. Let $\delta ''$ be an element of $\varpi_{c,b}(V)$ and put $\gamma =\delta '\delta ''$. By definition of V, the element $\delta '\delta (\delta ')^{-1}=\gamma ((\delta '')^{-1}\delta \delta '')\gamma^{-1}$ of $\pi_1(B, a)$ belongs to H. Then $H\delta '\cdot \delta = H\delta '$; this proves that $\pi_1(V, c)$ acts trivially on the set $\overset{-1}{q}(c)$.

By III, p. 313, Proposition 3, there exists a unique topology on $H\backslash \lambda_a(B)$ for which $q$ is continuous and the B-space $(H\backslash \lambda_a(B), q)$ is a covering such that the canonical operation of $\varpi (B)$ on this covering is the operation defined above. By assertion b) of Theorem 1 of III, p. 305, the group $q_*(\pi_1(H\backslash \lambda_a(B),H))$ is equal to H.

We shall say that an operation of the group $\pi_1(B, b)$ on a set X is admissible if the kernel of the canonical mapping $\pi_1(B, b)\rightarrow \mathfrak{S}_X$ is an open subgroup of $\pi_1(B, b)$. This amounts to saying that the homomorphism $\pi_1(B, b)\rightarrow \mathfrak{S}_X$ is continuous if the group $\mathfrak{S}_X$ is endowed with the discrete topology. The mapping $\pi_1(B, b)\times X\rightarrow X$ is then continuous, when X is endowed with the discrete topology. Conversely, consider a continuous operation of $\pi_1(B, b)$ on a discrete space X. Let $x$ be a point of X; its stabiliser is an open subgroup H of $\pi_1(B, b)$ by hypothesis. For $\gamma \in \pi_1(B, b)$, the stabiliser of $\gamma \cdot x$ is the subgroup $\gamma H\gamma^{-1}$, so that the subgroup of $\pi_1(B, b)$ fixing each element of the orbit of $x$ is the intersection of the subgroups $\gamma H\gamma^{-1}$, for $\gamma$ ranging over $\pi_1(B, b)$. It is an open subgroup because the open normal subgroups of $\pi_1(B, b)$ form a basis of its topology (III, p. 315, Proposition 4). It follows that a continuous operation of $\pi_1(B, b)$ on a discrete space X is admissible if it is transitive or, more generally, if the set of orbits of the elements of X is finite.

For every discrete group G, every principal covering E of B with group G, and every point $x\in E_b$, let us recall that the mapping $h_{(E,x)}$ from $\pi_1(B, b)$ into G which, to $\gamma \in \pi_1(B, b)$, associates the unique element $g\in G$ such that $x\cdot g=x\cdot \gamma^{-1}$ is a group homomorphism (III, p. 306, prop. 5).