TITLE: Is the Hurewicz theorem ever used to compute abelianizations?
QUESTION [32 upvotes]: 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?

REPLY [13 votes]: The mapping class group of a smooth manifold $M$ is the group of all its self diffeomorphisms up to isotopy, i.e. $\pi_0(\operatorname{Diff}(M))\cong \pi_1(B\operatorname{Diff}(M))$.
A large portion of geometric topology is concerned with gaining a better understanding of the homotopy type of $B\operatorname{Diff}(M)$, in particular of its cohomology as the latter is the ring of characteristic class of smooth $M$ bundles.
In the last 15 years, initiated by Madsen and Weiss' solution of the Mumford conjecture, the understanding of the homology of $B\operatorname{Diff}(M)$ has fundamentally improved which led, among others, to calculations of $H_1(B\operatorname{Diff}(M))\cong \pi_0(\operatorname{Diff}(M))^{ab}$ for some $M$, e.g. by Galatius and Randal-Williams in the following paper.
Abelian quotients of mapping class groups of highly connected manifolds, Mathematische Annalen
June 2016, Volume 365, Issue 1, pp 857–879