Exercise 9, § 5

Let us take up again the notations of exercise 6. For every integer $n\in \mathbf{N}^*$, let $c_n:\mathbf{I}\rightarrow C_n$ be the loop at the origin defined by $t\mapsto \frac{1}{2n}(1-$ cos(2$\pi t$), sin(2$\pi t$)).

(a) For every sequence $\alpha \in \{0,1\}^{\mathbf{N}^*}$, let $c_{\alpha}$ be the mapping defined by $c_{\alpha}(0) = 0$ by $c_{\alpha}(t) =\alpha_nc_n(2^nt-1)$ for every integer $n\geqslant 1$ and every $t\in \mathbf{I}$ such that $2^{-n}\leqslant$ $t\leqslant 2^{1-n}$. Show that it is a loop at the origin. Show that the mapping $\alpha \mapsto [c_{\alpha}]$ from $\{0,1\}^{\mathbf{N}^*}$ into $\pi_1(P,0)$ is injective.

(b) For every integer $n\geqslant 1$, let $q_n$ be the mapping from P into $C_n$ given by $q_n(x) =x$ if $x\in C_n$ and $q_n(x) = 0$ otherwise. It is continuous. Show that the group homomorphism $\psi :\pi_1(P,0)\rightarrow \prod_{n\geqslant 1}\pi_1(C_n,0)$ given by $\psi (\gamma ) = ((q_n)_*(\gamma ))_n$ admits a section.