Exercise 25, § 5

On pose $C =\mathbf{B}_2\times [-1,1]$. Let K be a closed subspace of $\mathbf{R}^3$ and let P be its image by the projection given by $(x, y, z)\mapsto (x, y)$. Suppose that there exists a finite set I, a family $(p_i)_{i\in I}$ of points of $\mathbf{R}^2$, a family $(r_i)_{i\in I}$ of strictly positive real numbers and a family $(m_i)_{i\in I}$ of natural integers satisfying the following properties, where $\varphi_i$ denotes the mapping from $\mathbf{R}^3$ into $\mathbf{R}^3$ given by $x\mapsto (p_i,0) +r_ix$ and $C_i=\varphi_i(C)$ :

– The sets $C_i$ are pairwise disjoint ;

– For all $i\in I$, the set $L_i=\overset{-1}{\varphi_{i}}(C_i\cap K)$ is the subspace L defined in exercise 23 if $m_i= 0$ and in exercise 24 (where one puts $m=m_i$) if $m_i\geqslant 1$.

– The set $K'= K-\bigcup_i(\mathring{C}_i\cap K)$ is the union of a finite family of closed segments pairwise disjoint in the plane $\mathbf{R}^2\times \{0\}$.

Let $N = (x_N, y_N, z_N)$ and $S = (x_S, y_S, z_S)$ be points of $\mathbf{R}^3$ such that $z_N>$ sup($r_i$)$_{i\in I}$ and $z_S<-$ sup($r_i$)$_{i\in I}$.

a) A path $c$ in $\mathbf{R}^3-$K with origin N and end S is said to be frank if there exist paths $c_1$ and $c_2$ in $\mathbf{R}^2$, juxtaposed, such that $c$ is the juxtaposed path of the paths given by $t\mapsto (c_1(t), z_N),t\mapsto (c_1(1),(1-t)z_N+tz_S)$, and $t\mapsto (c_2(t), z_S)$.

Prove that then the point $c_1(1)$ does not belong to P ; denote by $A_c$ the path-connected component of $\mathbf{R}^2-$ P which contains this point. Prove that two frank paths $c$ and $c'$ are strictly homotopic if and only if $A_c= A_{c'}$.

For every path-connected component $\alpha$ of $\mathbf{R}^2-$ P, fix a path $c(\alpha )$ in $\mathbf{R}^2-$ K, of origin N and end S, which is frank and such that $A_{c(\alpha)}=\alpha$.

(b) Let $i\in I$ such that $m_i= 0$, so that $\overset{-1}{\varphi_{i}}(K\cap C_i)$ is the set L defined in Exercise 23, whose notations we take up again; one has $\overset{-1}{\varphi_{i}}(P\times \{0\} \cap$ $C_i) = Q\times \{0\}$. For every $j\in \mathbf{Z}/4\mathbf{Z}$, denote by $\alpha_{i,j}$ the path-connected component of $\mathbf{R}^2-$ P which contains the image by $\varphi_i$ of the connected component $A_j$ of $\mathbf{B}_2-$ P. Prove that one has $[c(\alpha_{i,1})][c(\alpha_{i,4})]^{-1}= [c(\alpha_{i,2})][c(\alpha_{i,3})]^{-1}$ in $\pi_1(\mathbf{R}^3-K,N)$.

(c) Let $\omega$ be an element of $\pi_0(\mathbf{R}^2-$ P). Let $\lambda_{\omega}: F(\pi_0(\mathbf{R}^2-P))\rightarrow \pi_1(\mathbf{R}^3-K,N)$ be the unique group homomorphism which, for $\alpha \in \pi_0(\mathbf{R}^2-$ P), maps the element $x_{\alpha}$ onto $[c(\alpha )][c(\omega )]^{-1}$.

Prove that the homomorphism $\lambda_{\omega}$ is surjective and that its kernel is the smallest normal subgroup of $F(\pi_0(\mathbf{R}^2-$ P)) which contains the element $x_{\omega}$ and the elements $x^{-1}_{\alpha_{i,1}}x_{\alpha_{i,2}}x^{-1}_{\alpha_{i,3}}x_{\alpha_{i,4}}$, for $i\in I$ such that $m_i= 0$.

(d) For every segment $s$ belonging to $\pi_0(K')$, we choose elements $\alpha_s$ and $\alpha '_s$ of $\pi_0(\mathbf{R}^2-$ P) such that the set of connected components of $\mathbf{R}^2-$ P whose closure contains $s$ is equal to $\{\alpha_s, \alpha '_s\}$. Let $\mu: F(\pi_0(K'))\rightarrow \pi_1(\mathbf{R}^3$ $K,N)$ be the unique group homomorphism which, for $s\in \pi_0(K')$, maps the element $x_s$ onto $[c(\alpha_s)][c(\alpha '_s)]^{-1}$. Prove that $\mu$ is surjective. For $i\in I$ such that $m_i= 0$ and $j\in \{1,2,3,4\}$, denote by $s_{i,j}$ the unique segment of $\pi_0(K')$ such that $\varphi_i(a_j)$ meets $s_{i,j}$. Prove that there exist elements $u_i, v_i, w_i$ of $\{-1,1\}$ such that $\mu(x_{s_{i,4}}) =\mu(x^{w_i}_{s_{i,2}})$ and $\mu(x_{s_{i,3}}) =\mu(x^{-u_i}_{s_{i,2}}x^{v_i}_{s_{i,1}}x^{u_i}_{s_{i,2}})$. Prove that the kernel of $\mu$ is the smallest normal subgroup of $F(\pi_0(K'))$ which contains the elements $x_{s_{i,3}}x^{-u_i}_{s_{i,2}}x^{-v_i}_{s_{i,1}}x^{u_i}_{s_{i,2}}$ and $x_{s_{i,4}}x^{-w_i}_{s_{i,2}}$ for $i\in I$.

(e) We set

1

$k=$ Card($\pi_0(K)$)$-$ Card($\{i\in I|m_i>0\}$) $+\sum m_i$.

Prove that $k\in \mathbf{N}$. Prove that the quotient of $\pi_1(\mathbf{R}^3-$ K) by its derived subgroup is isomorphic to $\mathbf{Z}^k$.