Exercise 7, § 5

We take up again the notation of Exercise 6.

a) For every integer $n\geqslant 1$, let $\alpha_n\in \pi_1(P,0)$ be the class of a loop in $C_n$ whose class generates $\pi_1(C_n,0)$. Set then $\beta_n=\alpha_1\alpha_n\alpha^{-1}_1\alpha^{-1}_n$ and $\gamma_n=\beta_n^n$. The sequence $(\gamma_n)$ tends to $\varepsilon_0$ for the admissible topology.

b) Let $n\in \mathbf{N}^*$. For every $c\in \Omega_0(P)$, denote by $\omega_n(c)$ the supremum of the integers $k$ such that there exists a strictly increasing sequence $(t_1, . . . , t_{2k})$ of elements of $\mathbf{I}$ such that $c(t_{2i}) = 0$ and $c(t_{2i-1}) =x_n$ for $i\in \{1, . . . , k\}$. Prove that $\omega_n(c)$ is an integer. Prove that the mapping $\omega_n$ from $\Omega_0(P)$ into $\mathbf{N}$ is upper semi-continuous.

c) For every integer $n\geqslant 1$, let $c_n$ be a path whose class is $\gamma_n$. Prove that $\omega_1(c_n)\geqslant 4n$. Deduce that the sequence $(c_n)$ has no limit in $\mathscr{C}_c(\mathbf{I}; P)$. d) The admissible topology on the group $\pi_1(P,0)$ is separated, nondiscrete, and strictly less fine than the topology of compact convergence. The space P is in particular not unlaceable.