Exercise 19, § 5

Let G be a graph; denote by S the set of its vertices, by A that of its oriented edges and by $|G|$ its geometric realization. Denote by $j: S\rightarrow |G|$ and $p:\mathbf{I}\times A\rightarrow |G|$ the canonical mappings. Suppose that the sets S and A are finite.

We say that a mapping $f:|G| \rightarrow \mathbf{R}^n$ is affine (resp. piecewise affine) if, for every edge $a\in A$, the mapping $t\mapsto f\circ p(a, t)$ is affine (resp. piecewise affine). (Cf. III, p. 331, exerc. 6.)

(a) Prove that there exists an injective and piecewise affine mapping of $|G|$ into $\mathbf{R}^3$. (Begin by proving that there exists an integer $n\geqslant 1$ and an injective and piecewise affine mapping of $|G|$ into $\mathbf{R}^n.$) (b) For every continuous injective mapping $f:|G| \rightarrow \mathbf{R}^n$, prove that there exists a continuous injective mapping $g:|G| \rightarrow \mathbf{R}^n$ which is piecewise affine and homotopic to $f$.

(c) Suppose that the mapping $(o, t): A\rightarrow S\times S$ is injective. Let $f$ be an injective map from S into $\mathbf{R}$. Prove that there exists a unique affine mapping F from $|G|$ into $\mathbf{R}^3$ which maps every vertex $s\in S$ onto the point $(f(s), f(s)^2, f(s)^3)$ of $\mathbf{R}^3$. Prove that F is injective.

We say that the graph G is planar if there exists a continuous injective mapping of $|G|$ into $\mathbf{R}^2$.

(d) Suppose that the graph G is planar and that the mapping $(o, t): A\rightarrow$ $S\times S$ is injective. Let $f:|G| \rightarrow \mathbf{R}^2$ be a continuous injective mapping. Prove that there exists an injective affine mapping $g:|G| \rightarrow \mathbf{R}^2$ which is homotopic to $f$.