Exercise 3, § 5

Let $C$ be the circle of diameter $((0,0),(0,1))$ in $\mathbf{R}^2$ and let P be the union of the $2^nC$, for $n\in \mathbf{Z}$. Let E be the quotient of the space $P\times \mathbf{R}$ by the equivalence relation which is the least fine one for which $(x, t)\sim (2x, t+ 1)$ for every $(x, t)\in$ $P\times \mathbf{R}$. Denote by $p$ the canonical surjection from $P\times \mathbf{R}$ onto E. Let $a$ be the point $(0,0)$ of P and put $e=p(a,0)$.

a) Show that $p$ makes $P\times \mathbf{R}$ a Galois covering with group $\mathbf{Z}$. Show that $p_*(\pi_1(P\times \mathbf{R},(a,0)))$ is an open normal subgroup of $\pi_1(E, e)$, with quotient $\mathbf{Z}$.

b) Let $\varphi : P\rightarrow P$ be the mapping defined by $\varphi (x) =x$ if $x\in C$ and $\varphi (x) = (0,0)$ otherwise. Show that it is continuous.

c) Show that the kernel of the homomorphism $\varphi_*=\pi_1(\varphi ,(a,0))$ is a subgroup of $\pi_1(P\times \mathbf{R},(a,0))$ which is open for the topology of compact convergence.

d) Let $u:\mathbf{I}\rightarrow C$ be the loop given by $t\mapsto \frac{1}{2}(1-$ cos(2$\pi t$), sin(2$\pi t$)) and let $c\in \Omega_e(E)$ be the loop given by $t\mapsto (u(t),0)$. Show that the image of $\varphi_*$ is an open subgroup of $\pi_1(E, e)$ which does not contain the class of the loop $c$.

e) Let $c_n:\mathbf{I}\rightarrow E$ be the loop defined by $c_n(t) =p(2^{-n}u(t),0)$ and let $d:\mathbf{I}\rightarrow E$ be the loop given by $t\mapsto p(a, t)$. Show that $d^nc_n\overline{d}^n$ is strictly homotopic to $c$. Deduce that the class $[c]$ is contained in every admissible subgroup of $\pi_1(E, e)$.

f) Show that the admissible topology on $\pi_1(E, e)$ is strictly less fine than the topology of compact convergence.