Theorem 4

Let X be a topological space, let G be a discrete group operating properly in X and let $p: X\rightarrow X/G$ be the canonical mapping. Let I be an interval of $\mathbf{R}$, let $f: I\rightarrow X/G$ be a continuous mapping, let $a$ be a point of I and let $x$ be a point of X such that $f(a) =p(x)$. There exists a continuous mapping $\varphi : I\rightarrow X$ such that $p\circ \varphi =f$ and $\varphi (a) =x$.

We first treat the case where I is a closed bounded interval of $\mathbf{R}$.

According to TG, III, p. 29, prop. 3, the space X is separated.

Let $y$ be a point of $X/G$ and let $x\in X$ such that $y=p(x)$. The stabilizer $K_x$ of $x$ is a finite subgroup of G; moreover, there exist neighbourhoods $U_x$ of $x$ in X and $V_y$ of $y$ in $X/G$ such that $U_x$ is stable by $K_x,gU_x\cap U_x=\emptyset$ if $g\notin K_x$ and $p$ induces a homeomorphism of $U_x/K_x$ onto $V_y$ (TG, III, p. 32, proposition 8). Moreover, for every $g\in G,p$ induces a homeomorphism of $gU_x$ onto $V_y$. Since I is compact, there exist integers $m$ and $n$ such that $m\leqslant 0\leqslant n$ and a finite sequence $(a_i)_{m\leqslant i\leqslant n}$ of elements of I such that $a_0=a,I = [a_m, a_n]$, and such that, for every $i\in \{m, . . . , n-1\},f([a_i, a_{i+1}])$ is contained in an open set $V_{y_i}$ of $X/G$ constructed as above.

Soit $x_0$ the unique element of $\overset{-1}{p}(y_0)$ such that $x\in U_{x_0}$. Let $q_0$ be the canonical mapping of $U_{x_0}$ onto $U_{x_0}/K_{x_0}$; by passing to the quotient, the mapping $p$ induces a homeomorphism $i_0$ of $U_{x_0}/K_{x_0}$ onto $V_{y_0}$ such that $i_0\circ q_0=p|U_{x_0}$. The mapping $q_0$ is proper (TG, III, p. 29, prop. 3), open (TG, I, p. 31, example 1) and separated, since X is separated. Its fibres are totally disconnected, since they are finite. According to the corollary, III, p. 286, there exists a continuous mapping $\varphi_0: [a_0, a_1]\rightarrow U_{x_0}$ such that $p\circ \varphi_0=f|[a_0, a_1]$.

In the same way, by induction on the integer $i\in \{0, . . . , n-1\}$, one constructs a point $x_i\in \overset{-1}{p}(y_i)$, a continuous mapping $\varphi_i: [a_i, a_{i+1}]\rightarrow X$ whose image is contained in $U_{x_i}$ such that $p\circ \varphi_i=f|[a_i, a_{i+1}]$ and such that $\varphi_i(a_{i+1}) =\varphi_{i+1}(a_{i+1})$ if $0\leqslant i < n-1$.

Analogously, by decreasing induction on the integer $i\in \{m, . . . ,-1\}$, one constructs a point $x_i\in \overset{-1}{p}(y_i)$, a continuous mapping $\varphi_i: [a_i, a_{i+1}]\rightarrow X$ whose image is contained in $U_{x_i}$ such that $p\circ \varphi_i=$ $f|[a_i, a_{i+1}]$ and such that $\varphi_i(a_{i+1}) =\varphi_{i+1}(a_{i+1})$ if $m\leqslant i <0$.

There exists a unique mapping $\varphi : I\rightarrow$ X which coincides with $\varphi_i$ in $[a_i, a_{i+1}]$ for $m\leqslant i < n$. It is continuous (TG, I, p. 19, prop. 4). It is a continuous lifting of $f$ to X such that $\varphi (a) =x$.

This proves the theorem when I is compact. In the general case, there exist sequences $(a_n)_{n\in\mathbf{N}}$ and $(b_n)_{n\in\mathbf{N}}$ such that $(a_n)$ is stationary with limit inf(I), $(b_n)$ is stationary with limit sup(I)$,a=a_0=b_0$, and such that $(a_n)$ (resp. $(b_n)$) is constant if I has a smallest (resp. greatest) element. From what precedes, there exists for every integer $n\in \mathbf{N}$ a continuous lifting $\varphi_n$ of $f|[a_n, a_{n+1}]$ to X, a continuous lifting $\varphi '_n$ of $f|[b_{n+1}], b_n]$ to X such that $\varphi_0(a_0) =\varphi '_0(b_0) =x$ $\varphi_{n+1}(a_{n+1}) =\varphi_n(a_{n+1}),\varphi '_{n+1}(b_{n+1}) =\varphi '_n(b_{n+1})$. There exists a unique mapping $\varphi : I\rightarrow X$ which coincides with $\varphi_n$ in $[a_n, a_{n+1}]$ and with $\varphi '_n$ in $[b_{n+1}, b_n]$, for every $n\in \mathbf{N}$. It is continuous (loc. cit.) and it is a continuous lifting of $f$ to X such that $phi(a) =x$. The theorem is thus proved.