# Nontrivial element in first homology of Hawaiian earring

I am stuck at the following exercise from Hatcher section 3.3:

14. Let $$X$$ be the shrinking wedge of circles in Example 1.25, the subspace of $$\mathbb{R}^2$$ consisting of the circles of radius $$1/n$$ and center $$(1/n, 0)$$ for $$n = 1, 2, \dots$$

(a) If $$f_n : I \to X$$ is the loop based at the origin winding once around the nth circle, show that the infinite product of commutators $$[f_1, f_2] [f_3, f_4]\dots$$ defines a loop in $$X$$ that is nontrivial in $$H_1(X)$$. [Use Exercise 12.]

I am fine with the fact that $$[f_1,f_2][f_3,f_4]\cdots$$ actually defines a loop in $$X$$, but don't know how to prove that it's non-trivial in $$H_1(X)$$.

I tried following the hint. Exercise 12 is the following:

12. As an algebraic application of the preceding problem, show that in a free group $$F$$ with basis $$x_1, \dots, x_{2k}$$, the product of commutators $$[x_1,x_2]\dots[x_{2k-1},x_{2k}]$$ is not equal to a product of fewer than $$k$$ commutators $$[v_i,w_i]$$ of elements $$v_i, w_i \in F$$.

So from the fact that there exists a retraction $$X\to\bigvee_{i=1}^n S^1$$ for all $$n\in \Bbb N$$, we get that the inclusion $$i:\bigvee_{i=1}^n S^1\to X$$ is injective on $$\pi_1$$. Thus, from exercise 12, we can conclude that loops in $$X$$ defined by finite commutators $$[f_1,f_2]\cdots[f_{2k-1},f_{2k}]$$ are not homotopic to loops in $$X$$ expressed by fewer than $$k$$ commutators.

How can we use that to prove the original claim? How do we eventually pass to $$H_1$$? Of course all finite products of commutators are still trivial in $$H_1(X)$$, so I don't really know what to do with the conclusion above.

Since $H_1(X) = \pi_1(X)_{ab} = \pi_1(X) / [\pi_1(X), \pi_1(X)]$, if $f$ were trivial in $H_1(X)$, then it would be conjugate (in $\pi_1(X)$) to a finite product of commutators: $$f = g [u_1, v_1] \dots [u_k, v_k] g^{-1},$$ for some $g, u_i, v_i \in \pi_1(X)$. The retraction $X \to \bigvee_{i=1}^{k+1} S^1$ onto the $(k+1)$st circles induces on the fundamental group a map that sends $f$ to $[x_1, x_2] [x_3,x_4] \dots [x_{2k+1},x_{2k+2}]$, while it sends $g [u_1, v_1] \dots [u_k, v_k] g^{-1}$ to the conjugate of a product of $k$ commutators.
Using the same technique as in Exercise 12, you can use this relation to construct a map $M_k \to M_{k+1}$ of degree $1$, which is a contradiction by Exercise 11 (the conjugate does not really change much). So $f$ cannot be expressed as a conjugate of a finite number of commutators, and so is nontrivial in $H_1(X)$.