First integral homology and conjugation coinvariants

In degree one all boundaries to degree zero are zero with trivial coefficients. Degree two has $d[x\mathrm{\mid }y]=[y]-[xy]+[x]$, with $[1]=0$. Thus H1 is the abelian group generated by symbols [x] subject precisely to $[xy]=[x]+[y]$. The assignment $[x]\mapsto x[G,G]$ induces a map to $G_{\mathrm{a}\mathrm{b}}$; conversely $x\mapsto [x]$ is a group homomorphism to an abelian group, kills every commutator, and factors through $G_{\mathrm{a}\mathrm{b}}$. The maps are inverse on generators and commute with group homomorphisms.

For R the isomorphism sends conjugation by f to $r[R,R]\mapsto fr{f}^{-1}[R,R]$. Conjugation by an element of R is trivial on this quotient, so the action factors through F/R. Taking coinvariants adds exactly the relations $fr{f}^{-1}{r}^{-1}=1$. Since $[R,R]\subseteq [F,R]$, the resulting quotient is $R\mathrm{/}[F,R]$. If R=1 it is zero, and if R=F it is $F_{\mathrm{a}\mathrm{b}}$.