TITLE: A question on Hawaiian earring group
QUESTION [8 upvotes]: I have asked this question in MSE but have not got any satisfactory answer, so I am asking it here. Any idea on how to approach this problem will be highly appreciated.
Consider the Hawaiian earring. Suppose $f_n$'s are the loops representing the circles of radius $1/n$ centered at $(1/n, 0)$ for $n=1,2\cdots$. Suppose the following holds in the fundamental group
$$\langle [f_1, f_2][f_3, f_4]\cdots\rangle = \langle [g_1, g_2]\cdots [g_{2k-1},g_{2k}]\rangle $$
for some loops $g_1,g_2,\ldots , g_{2k}$,
where $\langle\cdot\rangle$ denotes the homotopy class and $[a,b]=aba^{-1}b^{-1}$ denotes the commutator. Then can we say that 
$$\langle [f_1, f_2]\cdots [f_{2n-1}, f_{2n}]\rangle = \langle [g_1, g_2]\cdots [g_{2k-1},g_{2k}]\rangle$$
for all sufficiently large $n$, or for some $n>k$ ?
I had this question while I was reading the paper here (see page 76, last paragraph), where the above has been mentioned (in some other equivalent form) without proof.

REPLY [4 votes]: This question seems to assume that the infinite product loop $[f_1,f_2][f_3,f_4][f_5,f_6]\dots$ is null-homologous in the Hawaiian earring, which is false. In fact, it is false by Katusya Eda's $0$-form Lemma, which says that a loop $\alpha$ in the Hawaiian earring is null-homologous if and only if the reduced representative of $\alpha$ factors into a finite concatenation $\prod_{i=1}^{2n}\alpha_i$ where there is an inverse pairing among the factors. Formally, this means $\{1,2,\dots ,2n\}$ splits into the disjoint union of two $n$-element sets $A,B$ and there is a bijection $\phi:A\to B$ such that for each $i\in A$, $\alpha_i$ is a reparameterization of the reverse of $\alpha_{\phi(i)}$.
The $0$-form Lemma can be found as Lemma 3.6 in Singular homology groups of one-dimensional Peano continua. Another nice generalization for the Hawaiian earring is Lemma 4.3 in Cotorsion-free groups from a topological point of view by Eda and Fischer. The double induction of this proof is rather subtle and is a thing of beauty.