# Connecting group ring, abelianization

For reasons arising in algebraic topology, I'm wanting to better understand the relations between two functors from groups to abelian groups, $\mathbb{Z}[\cdot]$ and $\operatorname{ab}$; group ring and abelianization.

Specifically, we can extend the homomorphism $G\to \operatorname{ab}(G)$ to an additive map $\mathbb{Z}[G]\to \operatorname{ab}(G)$, and this map is clearly natural. It seems too vague to ask "what should I do with this next?", so instead the question is

What references are there that consider this natural transformation?

For instance, I'd be immensely pleased if this is a good way to start a free resolution of $\operatorname{ab}$ without forgetting too much about $G$, and to learn what's the natural way to continue... but I'm not picky! You'll understand when I complain that automatic search-engines aren't too helpful in looking for anonymous natural transformations.

Thanks.

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I'm curious to hear more about how this relates to algebraic topology. – David White Aug 15 '11 at 14:19
Oh, well, if $X$ is a good-enough space, and $G=\pi_1(X)$, then $\operatorname{ab}G=H_1(X)$ (Poincaré!), while $Z[G]=H_0 (\Omega X)$. We can do better: there's another endofunctor $h$ on spaces such that the two group functors in the question correspond to the connected components of $h\Omega$ and $\Omega h$, respectively; one can even deduce a natural transformation $h\Omega\to \Omega h$ that lifts my "anonymous" transformation... but I'm wanting to focus on the algebraic side of this for now. – some guy on the street Aug 15 '11 at 17:05