Scholium

Let G be a connected and unknottable topological group, let $e$ be its identity element. Let $\widetilde{G}$ be a topological group simply connected by arcs, with identity element $\widetilde{e}$, and $p:\widetilde{G}\rightarrow G$ a group homomorphism which makes $\widetilde{G}$ into a universal covering of G. Let us denote by N the kernel of $p$.

The right translation $\boldsymbol{\delta }_h$ (resp. the left translation) by an element $h\in N$ is a G-automorphism of the principal covering $\widetilde{G}$, and the mapping $h\mapsto \boldsymbol{\delta }_h$ is an isomorphism of the group N onto the automorphism group of this principal covering.

For every subgroup K of N, the mapping $p':\widetilde{G}/K\rightarrow$ G deduced from $p$ is a Galois covering of G, and Aut$_G(G/K)$ is identified with the group $N/K$. When, as above, the groups N and $\pi_1(G)$ are identified, the group $p'_*(\pi_1(\widetilde{G}/K))$ is identified with the subgroup K of N. Moreover, $\widetilde{G}$ is a covering of $\widetilde{G}/K$, because G is locally connected (I, p. 81, prop. 7).

Conversely, every nonempty connected covering E of G is G-isomorphic to a covering of this type (I, p. 113, th. 3 and I, p. 111, prop. 10). Consider in fact a point $x$ of the fibre $E_e$ and let $f$ be the unique homomorphism of $\widetilde{G}$ into E which maps $\widetilde{e}$ to $x$. Endowed with $f,\widetilde{G}$ is a Galois covering of E; the subgroup Aut$_E(\widetilde{G})$ of Aut$_G(\widetilde{G})$ is identified with $f^{-1}(x)$ and is therefore a subgroup of N. It follows that $f$ induces a G-isomorphism of $\widetilde{G}/f^{-1}(x)$ onto E. By transport of structure, there results a topological group structure on E for which $x$ is the identity element and for which the mapping $f$ is a surjective homomorphism. The projection of the G-space E is then a group homomorphism, and the composition law of E is therefore the composition law defined by prop. 4 of IV, p. 375.

Let $(E, q)$ and $(E', q')$ be connected coverings of G, let $x$ be a point of $E_e$ and $x'$ a point of $E'_e$. In order that there exist a G-morphism of E into $E'$, it is necessary and sufficient that $p_*(\pi_1(E, x))$ be contained in $p'_*(\pi_1(E', x'))$. If this condition is satisfied, there then exists a unique G-morphism $f: E\rightarrow E'$ such that $g(x) =x'($III, p. 311, cor. 2 of prop. 1). If E and $E'$ are endowed with the group composition laws for which $q$ and $q'$ are homomorphisms and $x$ and $x'$ identity elements, the mapping $g$ is a group homomorphism. In fact, $g$ is identified with the canonical homomorphism $\widetilde{G}/p_*(\pi_1(E, x))\rightarrow$ $\widetilde{G}/p'_*(\pi_1(E', x'))$.