Exercise 7, § 5

Let $G$ be a graph, $S$ the set of its vertices and $A$ the set of its oriented edges. We denote by $|G|$ its geometric realization; we identify $S$ with a subset of $|G|$ and denote by $p:\mathbf{I}\times A\rightarrow |G|$ the canonical mapping.

(a) Prove that the space $|G|$ is compact (resp. locally compact) if and only if the graph $G$ is finite (resp. locally finite).

(b) Prove that the canonical mapping from $S$ into $|G|$ induces a bijection of $\pi_0(G)$ onto $\pi_0(|G|)$.

(c) Let $\varpi (|G|)_S$ denote the full subgroupoid of $\varpi (|G|)$ whose set of vertices is $S$. Prove that there exists a unique isomorphism of groupoids $\varpi (G)\rightarrow \varpi (|G|)_S$ which is the identity on the set of vertices and which applies the class of an oriented edge $a\in A$ to the class of the path $t\mapsto p(t, a)$

(d) Suppose that the graph $G$ is connected and finite; let $m= 1 +$ Card(S) $-$ Card(A). Prove that the Poincaré group of $|G|$ is a free group on $m$ generators.