Theorem 2

Let B be a connected topological space, let E be a covering of B, connected and nonempty. The following properties are equivalent:

(i) The covering E is Galois;

(ii) There exists a discrete topological group G and a continuous right operation of G on E making E into a principal covering with group G;

(iii) The covering $E\times_BE$ of E defined by the projection pr$_1$ is trivializable.

When these conditions are satisfied, the canonical mapping $G\rightarrow$ Aut$_B(E)^{\circ}$ defined by the operation of G is a group isomorphism.

If, moreover, the space B is locally connected, the preceding properties are equivalent to the following property:

(i$'$) There exists a point $b$ of B such that the operation of the group Aut$_B(E)$ on the fibre $E_b$ is transitive.

The implication (i)$\Rightarrow$(ii) results from proposition 5 and the implication (ii)$\Rightarrow$(iii) from remark 1 of I, p. 95. Let us prove (iii)$\Rightarrow$(i). Let us denote by $p: E\rightarrow B$ the projection of the covering E. Since B is connected, this covering has a degree, and this degree is not zero since E is not empty. Let $b$ be a point of B. Let us prove that the operation of Aut$_B(E)$ on the fibre $E_b$ is transitive. From the above, this fibre is not empty. Let $x$ and $x'$ be points of $E_b$. The B-morphisms $h: E\rightarrow E$ such that $h(x) =x'$ correspond bijectively to the continuous sections $s$ of the mapping pr$_1: E\times_BE\rightarrow E$ such that $s(x) = (x, x')$ (I, p. 9, prop. 3). Under assumption (iii), such a section exists (I, p. 70, cor. 2) and is unique (I, p. 34, cor. 1 of prop. 11) since the space E is connected. Thus, for every pair $(x, x')\in E\times_BE$, there exists a unique B-morphism $h: E\rightarrow E$ such that $h(x) =x'$. If $h': E\rightarrow E$ is the unique B-morphism such that $h'(x') =x$, one has $h'(h(x)) =x$ and $h(h'(x')) =x'$, whence $h'\circ h=$ Id$_E$ and $h\circ h'=$ Id$_{E'}$. This proves that $h$ is a B-automorphism of E. Hence the covering E is galoisian.

We have proved that conditions (i), (ii), (iii) are equivalent. Suppose them satisfied. Let $\delta : G\rightarrow$ Aut$_B(E)$ be the mapping which to $g\in G$ associates the B-automorphism $x\mapsto x\cdot g$ of E. The mapping $\delta$ is a group homomorphism of G into Aut$_B(E)^{\circ}$. Since G operates freely in E, the mapping $\delta$ is injective. Let $h\in$ Aut$_B(E)$, let $x\in$ E and let $g$ be the unique element of G such that $x\cdot g=h(x)$. The B-morphisms $h$ and $\delta (g)$ coincide at $x$, hence everywhere, since the space E is connected (I, p. 34, cor. 1 of prop. 11), which proves that $\delta$ is an isomorphism.

Now suppose the space B locally connected and let us prove, under assumption (i$'$), that the covering E is galoisian. Let $E'$ be the quotient space of E by the right operation of Aut$_B(E)^{\circ}$ and let us denote by $q: E\rightarrow E'$ the canonical mapping. It makes E into a surjective covering of $E'($I, p. 98, example 2); the mapping $p: E\rightarrow B$ defines by passing to the quotient a continuous mapping $p': E'\rightarrow B$ such that $p'\circ q=p$. Since the space B is locally connected, the B-space $(E', p')$ is a covering (I, p. 81, prop. 7). Under assumption (i$'$), there exists a point $b$ of B such that the fibre $E'_b$ has exactly one element; since the space B is connected, the same is then true for every point $b$ of B (I, p. 74, prop. 4), which proves that E is a galoisian covering of B.