Proposition 7

Let B be a connected topological space, locally connected by arcs, and let $b$ be a point of B. Endow the group $\pi_1(B, b)$ with the admissible topology.

a) Let G be a discrete topological group. For every continuous homomorphism $f:\pi_1(B, b)\rightarrow$ G, there exist a principal covering E of B with group G and a point $x$ of $E_b$ such that $h_{(E,x)}=f$.

b) For every discrete topological space F endowed with an admissible right action of the group $\pi_1(B, b)$, there exists a covering E of B such that the $\pi_1(B, b)$-sets F and $E_b$ are isomorphic.

Let us prove a). Let $f$ be a continuous homomorphism from $\pi_1(B, b)$ into a discrete group G. Its kernel is an open invariant subgroup K of $\pi_1(B, b)$; let H denote the group $\pi_1(B, b)/K$ and $\overline{f}: H\rightarrow G$ the group homomorphism deduced from $f$ by passing to the quotient. By III, p. 316, prop. 5, there exists a connected covering $E'$ of B such that the subgroup K is the stabilizer of every point of the fibre $E'_b$; the covering $E'$ is principal with group $H =\pi_1(B, b)/K$. Let $x'$ be a point of $E'_b$; the homomorphism $h_{(E',x')}:\pi_1(B, b)\rightarrow H$ is surjective with kernel K (III, p. 306, prop. 5). There is therefore a unique group homomorphism $\varphi : H\rightarrow G$ such that $\varphi \circ h_{(E',x')}=f$. The associated covering $E = E'\times^HG$ of B is principal with group G and one has $h_{(E,x)}=\varphi \circ h_{(E',x')}=f$ (III, p. 307, example 2). This proves assertion a).

Let us prove b). Let F be a set endowed with an admissible right action of the group $\pi_1(B, b)$. Let $f:\pi_1(B, b)\rightarrow \mathfrak{S}_F$ denote this action and endow the group $\mathfrak{S}_F$ with the discrete topology. By a), there exist a covering E of B, principal with group $\mathfrak{S}_F$, and a point $x\in E$ such that the homomorphism $h_{(E,x)}$ is equal to $f$. The canonical action of the group $\pi_1(B, b)$ on the fiber over $b$ of the associated covering $E\times^{\mathfrak{S}_F}F$ of B is identified with the action of $\pi_1(B, b)$ on F (III, p. 306, Example 1).