The Hurewicz theorem tells us that if $X$ is a path-connected space then $H_1(X, \, \mathbb{Z})$ is isomorphic to the abelianisation of $\pi_1(X)$. This gives a potential method for computing the abelianisation of a (sufficiently nice) group $G$: realise it as the fundamental group of a space $X$ and then compute $H_1(X, \, \mathbb{Z})$ by your favourite means.
Is this method ever used in practice? Are there nice examples of abelianisations which are easily (best?) computed in this way?
Best Answer
Yes, the fundamental group of the Hawaiian earring $\pi_1(\mathbb{H},b_0)$ is an important group which is sometimes called the free sigma product $\#_{\mathbb{N}}\mathbb{Z}$. Its is often defined in purely algebraic terms as a group of "transfinite words" in countably many letters. In many ways this group behaves like the non-abelian version of the Specker group $\prod_{\mathbb{N}}\mathbb{Z}$ and it is the key to the homotopy classification of 1-dimensional Peano continua. However, it's abelianization is not $\prod_{\mathbb{N}}\mathbb{Z}$; it is a good deal more complicated. The abelianization was first described by Eda and Kawamura in the following paper:
The Singular Homology Group of the Hawaiian Earring, Journal of the London Mathematical Society, 62 no. 1 (2000) 305–310.
The proof explicitly uses the Hawaiian earring and the authors mention in Remark 2.7 that
One way to represent the isomorphism class of this abelianization is as $$Ab(\#_{\mathbb{N}}\mathbb{Z})\cong \prod_{\mathbb{N}}\mathbb{Z}\oplus \prod_{\mathbb{N}}\mathbb{Z}/\bigoplus_{\mathbb{N}}\mathbb{Z}$$
Similar methods have been used to identify abelienizations of fundamental groups of other 1-dimensional spaces, e.g. the Menger curve.