Exercise 3, § 2

[^1] Let X be a topological space and let $x$ be a point of X.

Let G be a group and let $\varphi :\pi_1(X, x)\rightarrow G$ be a group homomorphism. Let $(U_n)_{n\in\mathbf{N}}$ be a decreasing sequence of subspaces of X whose intersection is reduced to the point $x$. For every $n$, let $G_n$ be the set of elements of G which are images by $\varphi$ of the class of a loop contained in $U_n$. a) If G is countable, the sequence $(G_n)$ is stationary. (Reduce to the case where $(G_n: G_{n+1})\geqslant n+ 1$ for every $n$ and consider an enumeration $(\gamma_n)_n$ of G; construct by induction elements $\lambda_n\in G_n$ such that, denoting $w_n=\lambda_1. . . \lambda_n$, one has $\gamma_i\notin w_nG_{n+1}$ for $1\leqslant i\leqslant n$. Now prove that the intersection $\bigcap_iw_iG_{i+1}$ is not empty.)

b) Suppose that G is abelian and that its identity element is the only indefinitely divisible element of G. Then $\bigcap_nG_n$ is reduced to the identity element. (Let $a$ be an element of the intersection; for every $n$, let $\gamma_n\in \pi_1(U_n, x)$ whose image by $\varphi$ is equal to $a$. Construct by induction path classes $\lambda_n\in \pi_1(U_n, x)$ such that $\lambda_n=\gamma_n\lambda^d_{n+1}$, where $d$ is a fixed integer $\geqslant 2$. Now prove that $b= (d-1)\varphi (\lambda_1) +a$ is indefinitely divisible. Deduce that $a= 0$.) c) More generally, if G is abelian and if every indefinitely divisible element of G is torsion, then the intersection of the $G_n$ is formed of torsion elements.

[^1]: The results of this exercise and of the following one are taken from the article by J. W. Cannon and G. R. Conner, « On the fundamental groups of one-dimensional spaces », Topology and its Applications 153 (2006), p. 2648–2672.