Proposition 8

Let X be a topological space and let G be a discrete group operating properly in X. Let us denote by $p$ the canonical projection of X onto $X/G$.

a) Suppose that X is connected by arcs and that the group G is generated by the stabilizers of the points of X. Then the canonical homomorphism $\pi_1(X, x)\rightarrow \pi_1(X/G, p(x))$ is surjective for every point $x$ of X. In particular, $X/G$ is simply connected by arcs if X is.

b) If the space X is unloopable, the space $X/G$ is unloopable.

a) The set N of the elements $g\in G$ for which there exists a path $c_g:\mathbf{I}\rightarrow X$ such that $c_g(0) =x,c_g(1) =g\cdot x$ and such that the loop $p\circ c_g$ in $X/G$ is strictly homotopic to a constant loop is a subgroup of G. If $g\in G$ and if there exists a point $y$ of X such that $g\cdot y=y$, let us choose a path $c:\mathbf{I}\rightarrow X$ joining $x$ to $y$; then the path $c'=c*$ $\overline{(g\cdot c)}$ satisfies $c'(0) =x,c'(1) =g\cdot x$ and $[p\circ c'] = [p\circ c][p\circ (g\cdot c)]^{-1}=e_{p(x)}$, whence $p\circ c'$ is strictly homotopic to a constant loop. Since G is generated by the stabilizers of the points of X, it follows that N = G.

Let $c$ be a loop in $X/G$ at $p(x)$. By Theorem 4 of III, p. 287, there exists a path $\widetilde{c}:\mathbf{I}\rightarrow X$ lifting $c$ such that $\widetilde{c}(0) =x$. Since $p(\widetilde{c}(1)) =c(1) =p(x)$, there exists $g\in G$ such that $\widetilde{c}(1) =g\cdot x$. Choose a path $c_g:\mathbf{I}\rightarrow X$ joining $x$ to $g\cdot x$ and such that $p\circ c_g$ is strictly homotopic to a constant loop. Then the path $c'=\widetilde{c}*\overline{c_g}$ is a loop at $x$ in X such that $[p\circ c'] = [c]$. This shows that the homomorphism $\pi_1(X, x)\rightarrow \pi_1(X/G, p(x))$ is surjective. The other assertion follows immediately.

Let us now prove assertion b). The space $X/G$ is locally arcwise connected (III, p. 261, Prop. 8).

Let $x$ be a point of X and let $K_x$ be its stabilizer in G. Let $U_1$ be an open neighbourhood of $x$, contained in U, such that the image of the canonical homomorphism $\pi_1(U_1, x)\rightarrow \pi_1(X, x)$ is reduced to the identity element. By Lemma 1 (IV, p. 349), there exists a neighbourhood U of $x$ in X, contained in $U_1$, stable under $K_x$, such that $g\cdot U\cap U =\emptyset$ if $g\notin K_x$ and such that the canonical mapping $p$ induces a homeomorphism of $U/K_x$ onto a neighbourhood V of $p(x)$ in $X/G$.

Let $c$ be a loop at $x$ in V; by Theorem 4 (III, p. 287), applied to the topological space U and the group $K_x$, there exists a path $\widetilde{c}:\mathbf{I}\rightarrow U$ such that $\widetilde{c}(0) =x$ and which lifts $c$. Necessarily $\widetilde{c}(1) =x$, so that $\widetilde{c}$ is a loop at $x$ in U. By assumption, $\widetilde{c}$ is strictly homotopic to a constant loop in X; consequently, the same is true of $c$, and the canonical homomorphism $\pi_1(V, p(x))\rightarrow \pi_1(X/G, p(x))$ is trivial. This proves that $X/G$ is unloopable if X is.

5) In the Euclidean plane $\mathbf{R}^2$, let P be the topological space equal to the union of the circles with center $(1/n,0)$ passing through the origin (for $n\in \mathbf{N}^*$). The space P is compact, connected and locally arcwise connected, but is not unloopable. The group $\pi_1(P,0)$, endowed with the admissible topology, is separated and non-discrete (III, p. 337, Exerc. 7).