Proposition 8

Let G be a connected unrollable topological group and let $\widetilde{G}$ be a universal covering of G. Let V be a connected open neighbourhood of the identity element $e$ of G such that the image of the canonical homomorphism of $\pi_1(V\cdot V, e)$ into $\pi_1(G, e)$ is reduced to the identity element. Let $F(V,\mathbf{r})$ denote the group defined by the generating set V and by the set $\mathbf{r}$ of relators $xyz^{-1}$, where $(x, y, z)$ runs through the set of elements of $V\times V\times V$ such that $xy=z$. Let $j: V\rightarrow F(V,\mathbf{r})$ denote the canonical mapping.

There exists a unique isomorphism $f$ of the group $F(V,\mathbf{r})$ onto $\widetilde{G}$ such that $f\circ j$ is a lifting to $\widetilde{G}$ of the canonical injection of V into G.

The set $V\cdot V$ is connected and open, hence locally arcwise connected. Let $p$ be the projection of $\widetilde{G}$. There exists a continuous section $s$ of $p$ over $V\cdot V$ such that $s(e) =\widetilde{e}$, where $\widetilde{e}$ denotes the identity element of $\widetilde{G}($III, p. 308, prop. 1). Put $\widetilde{V} =s(V)$; the set $\widetilde{V}$ is a connected open neighbourhood of $\widetilde{e}$ in $\widetilde{G}$, and $s$ is a homeomorphism of $V\cdot V$ onto $\widetilde{V}\cdot \widetilde{V}$. The mappings $(x, y)\mapsto s(x)s(y)$ and $(x, y)\mapsto s(xy)$ are liftings to $\widetilde{G}$ of the mapping $(x, y)\mapsto xy$ of $V\times V$ into G which coincide at $(e, e)$; since $V\times V$ is connected, they coincide on $V\times V$ (I, p. 34, cor. 1). Moreover, if $(\widetilde{x},\widetilde{y},\widetilde{z})\in \widetilde{V}\times \widetilde{V}\times \widetilde{V}$, the conditions $\widetilde{x}\widetilde{y}=\widetilde{z}$ and $p(\widetilde{x})p(\widetilde{y}) =p(\widetilde{z})$ are equivalent. The existence of an isomorphism $f: F(V,\mathbf{r})\rightarrow \widetilde{G}$ then follows from corollary 3 (IV, p. 372) of proposition 2.

Let $g$ be an isomorphism of $F(V,\mathbf{r})$ satisfying the conditions of the proposition. The equality $e\cdot e=e$ implies that $eee^{-1}\in \mathbf{r}$, so that $j(e)$ is the identity element of $F(V,\mathbf{r})$. Since V is connected, $g\circ j$ is the unique continuous lifting to $\widetilde{G}$ of the canonical injection of V into G. Consequently, $f$ and $g$ coincide on $j(V)$. Since $j(V)$ generates the group $F(V,\mathbf{r})$, $f=g$.