Exercise 4, § 2

Let X be a topological space which is connected and locally arcwise connected. Let $x$ be a point of X which possesses a countable fundamental system of neighbourhoods. Let S be a set and let $\varphi :\pi_1(X, x)\rightarrow \mathbf{Z}^{(S)}$ be a surjective group homomorphism.

(a) Prove that S is countable.

(b) Suppose that X is compact; prove that S is finite.

(c) Suppose that $\pi_1(X, x)$ is a free abelian group; prove that X is unlaceable.

(d) Suppose that $\pi_1(X, x)$ is a free group; prove that X is unlaceable. (Use the fact that the intersection of the subgroups of the descending central sequence of a free group is reduced to the identity element.)