Exercise 24, § 5

Let $m$ be an integer $\geqslant 1$ and let $(\theta_j)_{1\leqslant j\leqslant m}$ be a sequence of real numbers such that $0\leqslant \theta_1< \theta_2<\cdots < \theta_n<2\pi$. For $j\in \mathbf{Z}/m\mathbf{Z}$, put $a_j=$ exp($i\theta_k$), where $k$ is the unique element of ${1, . . . , n}$ belonging to the class $j$; we also denote by $Q_j$ the segment $[0, a_j]$ of $\mathbf{B}_2$. Finally put $Q = Q_1\cup \cdots \cup Q_m$.

a) For $j\in \mathbf{Z}/m\mathbf{Z}$, prove that there exists a unique path-connected component $A_j$ of $\mathbf{B}_2-$ Q which contains every point $b\in \mathbf{S}_1$ such that $\mathscr{I}(b/a_j)>0$ and $\mathscr{I}(b/a_{j+1})<0$. Prove that the family $(A_j)$ consists of pairwise disjoint sets and that its union is equal to $\mathbf{B}_2-$ Q.

b) Put $C =\mathbf{B}_2\times [-1,1]$, $N = (0,0,1)$, $S = (0,0,-1)$, and $L = Q\times {0}$. A path in C-L, with origin N and endpoint S, is said to be simple if there exist paths $c_1$ and $c_2$ in $\mathbf{B}_2$, which are composable, such that $c$ is the juxtaposition of the paths given by $t\mapsto (c_1(t),1),t\mapsto (c_1(1),1-2t)$ and $t\mapsto (c_2(t),-1)$. Prove that then $c_1(1)\notin Q$; denote by $A_c$ the path-connected component of $\mathbf{B}_2-$ Q which contains this point.

c) Let $c$ and $c'$ be paths in C-L, with origin N and endpoint S; suppose that they are simple. Prove that they are strictly homotopic if and only if $A_c= A_{c'}$.

d) For every $j\in \mathbf{Z}/m\mathbf{Z}$, let $c_j$ be a loop in C-L, with origin N and endpoint S, which is simple and such that $A_{c_j}= A_j$. Prove that the homomorphism of the free group $F(\mathbf{Z}/m\mathbf{Z})$ into $\pi_1(C-L,N)$ which maps $x_j$ onto $[c_j][c_{j+1}]^{-1}$, for $j\in \mathbf{Z}/m\mathbf{Z}$, is surjective, and that its kernel is the smallest normal subgroup which contains the product $x_1x_2. . . x_m$.