Let $\sigma : B'\times \mathbf{I}\rightarrow B$ be a homotopy joining $f_0$ to $f_1$. Let us denote by $i_0$ and $i_1$ the mappings $x\mapsto (x,0)$ and $x\mapsto (x,1)$ of $B'$ into $B'\times \mathbf{I}$. If $t\in \{0,1\}$, we have $f_t=\sigma \circ i_t$ and the coverings $f_t^*(E)$ and $i^*_t(\sigma^*(E))$ of $B'$ are therefore isomorphic (I, p. 15). Since the topological space $\mathbf{I}$ is locally connected and simply connected (I, p. 127, corollary), the coverings $i^*_0(\sigma^*(E))$ and $i^*_1(\sigma^*(E))$ of $B'$ are isomorphic (I, p. 130, cor. 1 of prop. 8).

Let B and $B'$ be topological spaces, let $f: B\rightarrow B'$ be a homotopy equivalence and let $g: B'\rightarrow B$ be a continuous mapping, inverse to $f$ up to homotopy. Let E be a covering of B. Since $g\circ f$ is homotopic to the identity mapping of B, the covering E is isomorphic to the covering $f^*(g^*(E))$ (prop. 1). If the space $B'$ is simply connected, the covering $g^*(E)$ is trivializable. Consequently, the covering E is trivializable. This proves that the space B is simply connected.

This follows from corollary 3 of prop. 8 of I, p. 130, applied to the simply connected and locally connected topological space $\mathbf{I}$.

$\mathscr{C}_c(\mathbf{I}; E)^{o_E}$ E

$\widetilde{p}p$

$\mathscr{C}_c(\mathbf{I}; B)^{o_B}$ B is a cartesian square.

The topological space $\mathbf{I}$ is locally connected, locally compact and simply connected. The proposition thus follows from prop. 9 of I, p. 131, applied with $T =\mathbf{I}$ and $t= 0$.

This follows from I, p. 71, cor. 2 of prop. 1.

With the notation of proposition 3, we have $\Lambda_a(B) =o^{-1}_B(a)$ and $\Lambda_x(E) =\overset{-1}{o_{E}}(x)$. The corollary then follows from I, p. 10, cor. of prop. 4.

We shall say that a continuous mapping $p: E\rightarrow B$ satisfies the path-lifting property if, for every path $c:\mathbf{I}\rightarrow$ B and every point $x$ of E lying over $c(0)$, there exists a path $c'$ in E with origin $x$ which lifts $c$. Such a path is then unique if $p$ is étale and separated (I, p. 34, cor. 1 of prop. 11). Let E be a covering and $p$ its projection; the mapping $p$ is étale, separated and has the path-lifting property (corollary 2). For a partial converse, cf. III, p. 315, corollary.

Let $\sigma :\mathbf{I}\times \mathbf{I}\rightarrow$ B be a strict homotopy joining $c_0$ to $c_1$. For $s\in \mathbf{I}$, let $c'_s$ be the unique path with origin $x$ in E which lifts the path $t\mapsto \sigma (t, s)$. For $(t, s)\in \mathbf{I}\times \mathbf{I}$, put $\sigma '(t, s) =c'_s(t)$; we have $p\circ \sigma '=\sigma$. The mapping $\sigma '$ is constant on $\{0\} \times \mathbf{I}$; by construction, it is continuous on $\mathbf{I}\times \{s\}$ for every $s\in \mathbf{I}$. It is therefore continuous, by I, p. 37, cor. 1 of th. 1. In particular, the mapping from $\mathbf{I}$ into $E_b$ defined by $s\mapsto \sigma '(1, s)$ is a continuous lifting of the constant path with image $b$; it is therefore constant. This proves that the mapping $\sigma '$ is a strict homotopy joining $c'_0$ to $c'_1$.

The condition is obviously sufficient. Conversely, let $c'_1$ be a loop of E at $x$. Put $c_1=p\circ c'_1$ and suppose that the loops $c$ and $c_1$ are strictly homotopic. By Prop. 4, the path $c'$ has the same end point as the path $c'_1$. It is therefore a loop at $x$.

Let $c$ be a loop in B at $p_1(x_1) =p_2(x_2)$ and, for $i\in \{1,2\}$, let $c'_i$ be the path with origin $x_i$ in $E_i$ lifting $c$. The fibre product over B $E = E_1\times_BE_2$ is a covering of B (I, p. 72, cor. 3 of Prop. 2), and the path $c':t\mapsto$ $(c'_1(t), c'_2(t))$ is the unique path with origin $x$ in E which lifts $c$. In order that $c'$ should be a loop, it is necessary and sufficient that $c'_1$ and $c'_2$ should be so. Corollary 3 thus follows from Corollary 2.

a) The orbit of $x$ for the canonical operation of the group $\pi_1(B, b)$ is the intersection of $E_b$ and the arcwise connected component of $x$ in E. In particular, if the space E is arcwise connected, this operation is transitive.

b) The stabilizer (A, I, p. 51) of $x$ is the subgroup $p_*(\pi_1(E, x))$ of $\pi_1(B, b)$.

c) For every $\gamma \in \pi_1(B, b)$, we have $p_*(\pi_1(E, x)) =$ Int($\gamma$ )$(p_*(\pi_1(E, x\cdot$ $\gamma )))$.

d) If the space B is connected and the covering E is Galois, the subgroup $p_*(\pi_1(E, x))$ is normal in $\pi_1(B, b)$.

Assertion a) is immediate. Assertion b) follows from Cor. 2 of Prop. 4 (III, p. 303). Assertion c) follows from b) and from Proposition 2 of A, I, p. 52. Finally, suppose that B is connected and that E is a Galois covering of B. For every $\gamma \in \pi_1(B, b)$, there exists a B-automorphism $g$ of E such that $g(x) =x\cdot \gamma ($I, p. 102, th. 2, c)). We have $p=p\circ g$, whence $p_*(\pi_1(E, x)) =p_*(\pi_1(E, x\cdot \gamma ))$. Assertion d) therefore follows from c).

For every $g\in G$, one has $h_{(E,x\cdot g)}=$ Int($g^{-1}$)$\circ h_{(E,x)}$.

Let E be a principal covering of B with group G, let $x$ be a point of $E_b$, and let us denote by $p$ the projection of E. For every $g\in G$, the mapping $y\mapsto y\cdot g$ is a B-automorphism of E. Hence, for every $g\in G$, every $y\in E_b$, and every $\gamma \in \pi_1(B, b)$, one has the relation $(y\cdot g)\cdot \gamma = (y\cdot \gamma )\cdot g($cf. III, p. 305, relation (4)). Consequently, for $\gamma , \delta \in \pi_1(B, b)$, one has

which proves that $h_{(E,x)}$ is a group homomorphism.

Its kernel is the stabilizer of $x$ for the canonical operation of $\pi_1(B, b)$, hence is equal to $p_*(\pi_1(E, x))$ by Theorem 1 of III, p. 305. The mapping $g\mapsto x\cdot g$ is a bijection of G onto $E_b$. The image of the homomorphism $h_{(E,x)}$ is therefore the set of $g\in G$ such that $x\cdot g$ belongs to the orbit of $x$ for this operation. If E is connected, $\pi_1(B, b)$ operates transitively on $E_b($loc. cit.) and the homomorphism $h_{(E,x)}$ is surjective.

Let $g\in G$; for every $\gamma \in \pi_1(B, b)$, one has

whence $h_{(E,x\cdot g)}(\gamma ) =g^{-1}h_{(E,x)}(\gamma )g$. This completes the proof of the proposition.