Exercise 5, § 2

Let $P$ be the Hawaiian earring (III, p. 336, exerc. 6, whose notation we use again). For $n\in \mathbf{N}^*$, let $\alpha_n\in \pi_1(P, o)$ be the class of a loop in $C_n$ whose class generates $\pi_1(C_n,0)$.

a) Prove that the mapping Hom($\pi_1(P,0),\mathbf{Z}$)$\rightarrow \mathbf{Z}^{\mathbf{N}},\varphi \mapsto (\varphi (\alpha_n))_{n\in\mathbf{N}}$ is an injective homomorphism of groups and that its image is equal to $\mathbf{Z}^{(\mathbf{N})}$.[^1]

b) The group $\pi_1(P,0)$ is not a free group; the quotient of $\pi_1(P,0)$ by its derived group is not a free abelian group.

c) Let A be the Hawaiian archipelago (III, p. 338, exerc. 10). Prove that every homomorphism of $\pi_1(A, o)$ into $\mathbf{Z}$ is trivial.

[^1]: See B. de Smit, « The fundamental group of the Hawaiian earring is not free », International Journal of Algebra and Computation, Vol. 2, No. 1 (1992), p. 33–37.