Exercise 23, § 5

Let $u$ and $v$ be two points of $\mathbf{S}_1$ such that $\mathscr{I}(v/u)>0$; one defines a family $(a_j)_{j\in\mathbf{Z}/4\mathbf{Z}}$ by $a_1=u,a_2=v,a_3=-u$ and $a_4=-v$. Let $Q_1$ and $Q_2$ denote the two segments $[a_1, a_3]$ and $[a_2, a_4]$ in the unit disk $\mathbf{B}_2$ and put $Q = Q_1\cup Q_2$.

a) Prove that, for every $j\in \mathbf{Z}/4\mathbf{Z}$, there exists a unique arcwise connected component $A_j\in \pi_0(\mathbf{B}_2-$ Q) whose closure contains $a_j$ and $a_{j-1}$. Prove that the union of the family $(A_j)$ is equal to $\mathbf{B}_2-$ Q.

Put $C =\mathbf{B}_2\times [-1,1]$, $N = (0,0,1)$ and $S = (0,0,-1)$. Let $f: [-1,1]\rightarrow$ $\mathbf{R}_+$ be a continuous mapping such that $f(-1) =f(1) = 0$ and $0< f(t)<1$ for all $t\in ]-1; 1[$. Let $L_1$ and $L_2$ denote the images of the mappings from $[-1,1]$ into C given by $t\mapsto (ta_1,0)$ and $t\mapsto (ta_2, f(t))$ respectively; finally put $L = L_1\cup L_2$.

b) Prove that C-L is arcwise connected.

c) A path $c$ in C-L, with origin N and endpoint S, is said to be frank if there exist juxtaposed paths $c_1$ and $c_2$ in $\mathbf{B}_2$ such that $c$ is the juxtaposed path 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 arcwise connected component of $\mathbf{B}_2-$ Q which contains this point.

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

e) For every $j\in \mathbf{Z}/4\mathbf{Z}$, let $c_j$ be a path in C-L, with origin N and endpoint S, frank and such that $A_{c_j}= A_j$. Prove the relation

in the group $\pi_1(C-L,N)$.

f) Prove that the canonical homomorphism of the free group on two generators $x, y$ into $\pi_1(C-L,N)$ which maps $x$ and $y$ onto $[c_1][c_2]^{-1}$ and $[c_2][c_3]^{-1}$ respectively is a group isomorphism. (Apply the van Kampen theorem to the covering of C - L formed by the eight closed sets bounded by the three vector planes of $\mathbf{R}^3$ containing two of the points of the set $\{a_1, a_2,N\}.$)