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 $\pi\_1(X)$](http://en.wikipedia.org/wiki/Fundamental_group#Relationship_to_first_homology_group).  

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?



What part of the fundamental group is captured by the second homology group?
$H\_2(X)$ is all about $\pi\_1(X)$ and $\pi\_2(X)$. If $\pi\_2(X)$ is trivial (as for knot complements) then it is a functor of $\pi\_1(X)$. 


Let $H\_n(G)$ be $H\_n(BG)$, the homology of the classifying space ($K(G,1)$). If $X$ is path-connected than there is a surjection $H\_2(X)\to H\_2(\pi\_1(X))$ whose kernel is a quotient of $\pi\_2(X)$, the cokernel of a map from $H\_3(\pi\_1(X))$ to the largest quotient of $\pi\_2(X)$ on which the canonical action of $\pi\_1(X)$ becomes trivial.


This $H\_2(G)$ isn't anything like the next piece of the derived series after $H\_1(G)=G^{ab}$, though. For example, if $G$ is abelian then $H\_2(G)$ is the second exterior power of $H\_1(G)$ (EDIT: so it can be nontrivial even though it knows no more than $H\_1(G)$ does), while if $H\_1(G)$ is trivial $H\_2(G)$ is often nontrivial (EDIT: so, even when it does carry some more information than $H\_1(G)$, it is not necessarily derived-series information). 


EDIT: The previous paragraph comes from looking at the integral homology Serre spectral sequence of $X\to K(\pi\_1(X),1)$, where the homotopy fiber is the universal cover $\tilde X$. Since $H\_1\tilde X=0$, the groups $E^\infty\_{p,1}$ are trivial and we get an exact sequence
$$
0\to E^\infty\_{0,2}\to H\_2(X)\to E^\infty\_{2,0}\to 0,
$$
therefore
$$
E^2\_{3,0} \to E^2\_{0,2}\to H\_2(X)\to E^2\_{2,0}\to 0.
$$
Since $H\_2(\tilde X)=\pi\_2(\tilde X)=\pi\_2(X)$, this looks like
$$
H\_3(\pi\_1(X)) \to H\_0(\pi\_1(X);\pi\_2(X))\to H\_2(X)\to H\_2(\pi\_1(X))\to 0.
$$
The place to look for the rest of the derived series would be homology with nontrivial coefficients, for example homology of covering spaces.


