Example 2

Let G be a graph (II, p. 155, definition 1); let S denote the set of vertices of G, A the set of its oriented edges, $o$ and $t$ the origin and term mappings of A into S; for every oriented edge $a\in A$, let $\overline{a}$ denote the opposite oriented edge. Endow the sets S and A with the discrete topology; let X be the sum space of the space S and of the space $\mathbf{I}\times A$, and let $\sim$ be the finest equivalence relation in X for which $(u, a)\sim (1-u, a)$, $(0, a)\sim o(a)$ and $(1, a)\sim t(a)$ for every $u\in \mathbf{I}$ and every oriented edge $a\in A$. The quotient space $|G|= X/\sim$ is called the geometric realization of the graph G. Let $p$ denote the canonical projection of X onto $|G|$.

Let us prove that $|G|$ is locally contractible. Let $s\in S$. Let $X_s$ denote the union of $\{s\}$ and of the sets $[0,1[\times \{a\}$ for $a\in \overset{-1}{o}(s)$ and of the

sets $]0,1]\times \{a\}$ for $a\in \overset{-1}{t}(s)$. Let $U_s$ be the image of $X_s$ in $|G|$; it is an open neighbourhood of $p(s)$ in $|G|$ because $X_s$ is an open saturated neighbourhood of $s$ in X. Let $f$ be the mapping of $X_s\times \mathbf{I}$ into $X_s$ defined, for $u, v\in \mathbf{I}$ and $a\in A$, by the relations

$((1-v)u, a)$ if $0\leqslant u <1$ and $o(a) =s$,

$(1-(1-v)(1-u), a)$ if $0< u\leqslant 1$ and $t(a) =s$.

It is continuous and compatible with the equivalence relation $\sim$. It therefore defines by passing to the quotient a mapping $\varphi_s: U_s\times \mathbf{I}\rightarrow U_s$ which is continuous, since $\mathbf{I}$ is locally compact (I, p. 19, Prop. 10). It is a strong contraction of $U_s$ onto $p(s)$ (III, p. 237, Def. 6).

Let moreover $x= (\tau , a)\in ]0,1[\times A$. Let us denote by $X_x= ]0,1[\times \{a, a\}$ and let $U_x$ be its image in $|G|$ by $p$; it is a neighbourhood of $p(x)$ in $|G|$, homeomorphic to $]0,1[$. The mapping of $X_x\times \mathbf{I}$ into $X_x$ given by $((u, a), v)\mapsto ((1-v)u+v\tau , a)$ and $((u, a), t)\mapsto ((1-v)u+v(1-\tau ), a)$ is continuous. It defines by passing to the quotient a mapping $\varphi_x: U_x\times$ $\mathbf{I}\rightarrow U_x$ which is continuous (loc. cit.) and is a strong contraction at $x$.

Every point of $|G|$ is the image of a point $s\in S$ or of a point of $\mathbf{I}\times A$ of the form $(\tau , a)$ where $0< \tau <1$. It follows that every point of $|G|$ has a neighbourhood contractible at that point. In other words, $|G|$ is locally contractible and, in particular, unravelable (IV, p. 346, Prop. 5).

Let us choose a total order on S and consider the open covering $(U_s)_{s\in S}$ of $|G|$. Let $s$ and $s'$ be distinct elements of S. The intersection $U_s\cap U_{s'}$ is the union of the $p(]0,1[, a)$ where $a$ runs through the set of oriented edges of G whose extremities are $s$ and $s'$. Consequently, the framework of the covering $(U_s)_{s\in S}$ is identified with the quiver G. Moreover, for every $s\in S$, $U_s$ is contractible at $s$, hence $\pi_1(U_s, p(s))$ is reduced to the identity element. It then follows from Cor. 2 of IV, p. 416 that for every $s\in S,\pi_1(|G|, p(s))$ is a free group (Nielsen-Schreier theorem).