show/hide this revision's text 3 corrected Hurewitz to Hurewicz

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 Hurewitz 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?
show/hide this revision's text 2 added 45 characters in body

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 Hurewitz 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?
show/hide this revision's text 1

What part of the fundamental group is captured by the second homology group?

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 Hurewitz 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 the second homology 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?