Exercise 20, § 5

Let $f:\mathbf{I}\rightarrow \mathbf{R}^2$ be a piecewise affine loop whose restriction to $[0,1[$ is injective; let $C =f(\mathbf{I})$.

a) Prove that $\mathbf{R}^2-$ C has at most two connected components.

b) For $x\in \mathbf{R}^2-$ C and $v\in \mathbf{S}_1$, let $n_v(x)$ be the number of connected components of $(\mathbf{R}^2-C)\cap (x+\mathbf{R}_+v)$. Prove that there exists a unique locally constant mapping $n:\mathbf{R}^2-C\rightarrow \{0,1\}$ such that $n(x)\equiv n_v(x)$ (mod$.2$) for every $v\in \mathbf{S}_1$.

c) Deduce that $\mathbf{R}^2-$ C possesses exactly two connected components and that their boundaries are equal to C.

d) Let $g:\mathbf{I}\rightarrow \mathbf{R}^2$ be a piecewise affine mapping, injective such that $\overset{-1}{g}(C) =\{0,1\}$; put $P =g(\mathbf{I})$ and $D = C\cup P$. Prove that $C-(C\cap P)$ possesses two connected components; let them be $P_1$ and $P_2$. Prove that $\mathbf{R}^2-$ D possesses three connected components and that their boundaries are equal to C, $P_1\cup P$ and $P_2\cup P$.

e) Let G be a finite planar graph; let S be the set of its vertices and A the set of its oriented edges. Let $f:|G| \rightarrow \mathbf{R}^2$ be a continuous injective piecewise affine mapping and let P be its image. Prove that the number of connected components of $\mathbf{R}^2-$ P is equal to 1 + Card(S) $-$ Card(A) + Card($\pi_0(G)$).