TITLE: What part of the fundamental group is captured by the second homology group?
QUESTION [43 upvotes]: Let $X$ be a connected CW complex. One can ask to what extent $H_\ast(X)$ determines $\pi_1(X)$. For example, it determines its abelianization, because the Hurewicz Theorem implies that $H_1(X)$ is isomorphic to the abelianization of $\pi_1(X)$.
I'm thinking about invariants of 2-knots which can be extracted from have to do with the second homology of (covers of) their complements, and I'm therefore very much interested in the answer to the following question:
What part of the fundamental group is detected by $H_2(X)$?
In particular, is there an obvious map from $H_2(X)$ (or from part of it) into $\pi_1(X)$? Where in the derived series of $\pi_1(X)$ would the image of $H_2(X)$ live?

REPLY [6 votes]: Maybe it is worth to point out something that is at the origin of the results mentioned in several answers to the question. Hopf in one of his famous papers (Fundamentalgruppe und zweite Bettische Gruppe, Comment. Math. Helv. 14 (1940), 257-309) gives a characterization of the cokernel of the Hurewicz map in degree 2. His result states that for an arcwise-connected locally finite simplicial complex $X$, the Hurewicz map $h_2:\pi_2(X)\to H_2(X)$ has cokernel:
$$ H_2(X)/h_2(\pi_2(X))\simeq (R\cap [F,F])/[F,R], $$
where $\pi_1(X)\simeq F/R$ is any presentation of the fundamental group. 
In the same paper, he observes first that for any finitely generated group $G$ with a presentation $F/R$ the abelian group:
$$ G^*_1=(R\cap [F,F])/[F,R] $$
only depends on $G$, and not on the particular presentation $F/R$. 
In our modern day language, we would write Hopf's result as stating that:
$$ H_2(X)/h_2(\pi_2(X))\simeq H_2(\pi_1(X)), $$
which is the form that appears in some of the answers above.