Exercise 7, § 4

Let X be a delaceable topological space and let G be a discrete group operating properly in X; we denote by $m: G\times X\rightarrow$ X the operation of G, pr$_2: G\times X\rightarrow X$ the second projection, and $f: X\rightarrow X/G$ the canonical surjection.

a) Exercise prove that the morphism of groupoids $\varpi (f)$ is surjective.

b) Let H be a groupoid and let $\varphi :\varpi (X)\rightarrow H$ be a morphism of groupoids such that $\varphi \circ \varpi (m) =\varphi \circ \varpi$(pr$_2$).

Let $c_1$ and $c_2$ be paths in X such that the paths $f\circ c_1$ and $f\circ c_2$ in $X/G$ are strictly homotopic. Exercise prove that one has $\varphi ([c_1]) =$ $\varphi ([c_2])$.

c) Exercise prove that there exists a unique morphism of groupoids $\varphi '$ from $\varpi (X/G)$ into H such that $\varphi =\varphi '\circ \varpi (f)$. Deduce a new proof of theorem 3 of IV, p. 403.