Proposition 10

Let B be a topological space, let $(E, p)$ be a principal covering of B of group G and let X be a covering of B. Suppose that E and X are connected and nonempty. The following properties are equivalent:

(i) The covering X is associable to the principal covering E;

(ii) There exists a subgroup H of G such that X is B-isomorphic to $E/H$;

(iii) There exists a surjective B-morphism $h: E\rightarrow X$;

(iv) For every point $(y, x)$ of $E\times_BX$, there exists a B-morphism $r: E\rightarrow X$ such that $r(y) =x$.

Suppose these conditions satisfied and let H be a subgroup of G such that X is B-isomorphic to $E/H$. The covering X is Galois if and only if the subgroup H is normal in G.

(i)$\Rightarrow$(ii) : Let F be a discrete G-set such that the covering $E\times^GF$ is B-isomorphic to X. The set F is nonempty and the space $E\times^GF$ is connected, hence the group G operates transitively in F (proposition 9). Then the space $E\times^GF$ is B-isomorphic to $E/H$, where H is the subgroup of G fixing a point of F (I, p. 106, example 4).

(ii)$\Rightarrow$(iii) : In fact, the canonical surjection of E onto $E/H$ is a surjective B-morphism.

(iii)$\Rightarrow$(iv) : Let $(y, x)$ be a point of $E\times_BX$. Since the mapping $h$ is surjective, there exists a point $y'$ of E such that $h(y') =x$. The points $y$ and $y'$ have the same projection in B. Since the covering E is principal with group G, there exists $g\in G$ such that $y\cdot g=y'$. The mapping $r: E\rightarrow X$ defined by $z\mapsto h(z\cdot g)$ is a B-morphism and one has $r(y) =x$.

(iv)$\Rightarrow$(i) : Since X is not empty and the projection of E onto B is surjective, the space $E\times_BX$ is not empty. Hence there exists a B-morphism $r$ of E into X. The mapping (Id$_E, r$)$: E\rightarrow E\times_BX$ is a section of the covering $p^*(X)$ of E. Since the space E is connected, it follows from Corollary 2 of proposition 1 of I, p. 69 that the covering $p^*(X)$ is trivializable, which proves that the covering X is associable to the Galois covering $p$ (proposition 8).

Suppose now that these conditions are satisfied. We denote by X the B-space $E/H$, by $q$ its projection, and by $h: E\rightarrow E/H$ the canonical mapping.

If the group H is normal in G, then X is a principal covering with group $G/H$ (I, p. 106, example 4). Since X and B are connected and nonempty, X is a Galois covering of B (I, p. 102, theorem 2). Conversely, suppose that $E/H$ is a Galois covering of B and let K = Aut$_B(E/H)^{\circ}$. One defines a mapping $\alpha : E\times G\rightarrow K$ by associating to $(t, g)\in E\times G$ the unique element $k$ of K such that $h(t\cdot g) =h(t)\cdot k$. It is continuous because it is obtained by composing the continuous mapping $(t, g)\mapsto (h(t), h(t\cdot g))$ of $E\times G$ into $X\times_BX$ with the canonical trivialization (I, p. 95, remark 1) $X\times_BX\rightarrow X\times K$ of $q^*(X)$. Since E is connected and K is a discrete group, the continuous mapping $t\mapsto \alpha (t, g)$ is constant, for every $g\in G$; we denote by $\alpha (g)$ its value. If $t\in E,g\in G$ and $g'\in H$, one then has

It follows that $g^{-1}g'g$ belong to H and hence that H is normal in G.