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 <a href="http://en.wikipedia.org/wiki/Fundamental_group#Relationship_to_first_homology_group">$H_1(X)$ is isomorphic to the abelianization of $\pi_1(X)$</a>.<br>
I'm thinking about invariants of 2-knots which <strike>can be extracted from</strike> 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:
<blockquote>What part of the fundamental group is detected by $H_2(X)$?</blockquote>
In particular, is there an obvious map from $H_2(X)$ (or from part of it) into $\pi_1(X)$?<br> Where in the derived series of $\pi_1(X)$ would the image of $H_2(X)$ live?