Exercise 33, § 5

[^1] Let L be a subspace of$\surd \mathbf{R}^3$ as defined in Exercise$\surd$ 23, where one takes $u= (1-i)/$ 2 and $v= (1 +i)/$ 2. Let $a$ and $r$ be real numbers such that $0<2r < a <1-2r$. We put $p_1= (0,-a),p_2= (0,0)$ and $p_3= (0, a);q_1= (-1,-a),q_2= (-1,0),q_3= (-1, a);q'_1= (1,-a)$, $q'_2= (1,0),q'_3= (1, a)$. Let K be the union of the sets $p_1+rL,p_2+rL$, $p_3+rL$ and of the segments $[q_1, p_1-rv]$, $[q_2, p_2-ru]$, $[q_3, p_3-ru]$, $[q'_1, p_1+ru]$, $[q'_2, p_3+ru]$, $[q'_3, p_3+rv]$.

Let $(x_n)_{n\in\mathbf{Z}}$ be a strictly increasing family of real numbers such that $_n$lim$_{\rightarrow -\infty}x_n=-1$ and lim$_{n\rightarrow+\infty}x_n= 1$ ; for $n\in \mathbf{Z}$, we put $z_n=y_n= 1- |x_n|$. For $n\in \mathbf{Z}$, we denote by $f_n$ the mapping from $[-1,1]^3$ into $\mathbf{R}^3$ given by

and we put $A_n=f_n(K)$ and $B_n=f_n([-1,1]^3)$. Let A be the union of the family $(A_n)_{n\in\mathbf{Z}}$.

a) Prove that A is homeomorphic to $[0,1]$. b) For every natural number $m$, put

Prove that $\mathbf{R}^3-A'_m$ is arcwise connected and that its fundamental group is generated by elements $a_n, b_n, c_n$ (for $n\in \mathbf{Z}$ such that $-m\leqslant n\leqslant m$) subject to the relations $b_{-m}=c_{-m}a_{-m},c_{n+1}a_{n+1}=c_nc_{n+1},c_{n+1}b_n=a_nc_{n+1}$, $b_nc_{n+1}=b_{n+1}b_n$ (for $n\in \mathbf{Z}$ such that $-m\leqslant n < m$) and $c_ma_m=b_m$.

c) Prove that $\mathbf{R}^3-$ A is arcwise connected and that its fundamental group is generated by elements $a_n, b_n, c_n$ (for $n\in \mathbf{Z}$) subject to the relations

for $n\in \mathbf{Z}$.

d) Prove that there exists a homomorphism from $\pi_1(\mathbf{R}^3-$ A) into the group $\mathfrak{S}_5$ which maps the class of $c_n$ onto the permutation $(1,2,3,4,5)\mapsto$ $(2,3,4,5,1)$ if $n$ is odd and onto the permutation $(1,2,3,5,4)\mapsto (4,3,5,2,1)$ if $n$ is even. Deduce that $\mathbf{R}^3-$ A is not simply arcwise connected.

[^1]: This example is due to E. Artin and R. Fox, « Some wild cells and spheres in three-dimensional space », Annals of Math. 49 (1948), p. 979–990.