Exercise 3, § 1

Let X be a topological space and let $x$ be a point of X.

Let $\mathscr{R}$ be a category of coverings of X and $\mathscr{E}$ a category of sets (cf. II, p. 160, example 2).

(a) Suppose that for every covering E of X which is an object of $\mathscr{R}$, the fibre $E_x$ of E at $x$ is an object of $\mathscr{E}$; we then put $\Phi_x(E) = E_x$. For every morphism $f: E'\rightarrow E$ of coverings of B which are objects of $\mathscr{R}$, we denote by $\Phi_x(f)$ the mapping from $E'_x$ into $E_x$ deduced from $f$ by passing to subsets. Prove that $\Phi_x$ is a functor from the category $\mathscr{R}$ into the category $\mathscr{E}$.

(b) Let $G(x)$ be the set of families $\varphi = (\varphi_E)$ where, for every object E of $\mathscr{R}$, $\varphi_E$ is a permutation of the set $\Phi_x(E)$ such that for every arrow $f: E'\rightarrow E$ of $\mathscr{R}$, one has $f_*\circ \varphi_{E'}=\varphi_E\circ f_*$ (« automorphisms of the functor $\Phi_x$ ».)

The relation given by $\varphi \cdot \psi = (\varphi_E\circ \psi_E)$ for $\varphi , \psi \in G(x)$ equips the set $G(x)$ with a group structure. For every object E of $\mathscr{R}$, we denote by $G(x)_E$ the set of the $\varphi \in G(x)$ such that $\varphi_E=$ Id$_{\Phi_x(E)}$; it is a subgroup of $G(x)$.

(c) Prove that there exists a unique topological group structure on $G(x)$ for which the groups $G(x)_E$ form a basis of neighbourhoods of the identity element.

(d) Let T be a simply connected subspace of X. If $x, y$ are points of T, the topological groups $G(x)$ and $G(y)$ are isomorphic.

(e) Prove that the lifting of paths defines a continuous group homomorphism from $\pi_1(X, x)$ to $G(x)$. This homomorphism is not necessarily surjective (exerc. 1 ; I, p. 146, exerc. 6) nor injective (exerc. 2). It is an isomorphism if X is path-connected and if every connected covering of X is isomorphic to an object of $\mathscr{R}$.