Alternatives to Hurewicz Theorem for non-simply connected spaces For a topological space $X$, the Hurewicz Theorem relates the homology groups of $X$ with its homotopy groups. However, for non-simply connected spaces the Hurewicz theorem doesn't say much. I'd like to know if there are other theorems relating the homology and homotopy groups of a topological space even if the relation is very weak. 
 A: After the Hurewicz theorem, I'd say the next result along those lines deals with minimal models in rational homotopy theory. It still works best in the simply-connected (or at least nilpotent case), but the results are stronger. The basic idea is that we can construct a certain graded differential $\mathbb{Q}$-algebra $A_{PL}(X)$, the Sullivan minimal model, from the space $X$; and then we can read off the rational homotopy groups $\pi_*(X) \otimes \mathbb{Q}$ from it. (In a rough sense, $A_{PL}(X)$ is the smallest graded differential algebra whose cohomology is just $H^*(X, \mathbb{Q})$.) This leads to a very quick computation of the rational homotopy groups of $S^n$, for example, since its cohomology is so simple; and because the (integral) homotopy groups of $S^n$ are ridiculously complicated, this result is about the best that can be expected. The details of the construction of $A_{PL}(X)$ can get a bit involved, but Felix, Halperin, and Thomas' "Rational Homotopy Theory" goes into the details (and is a very well-written book).
A: Not directly with homology, but with cohomology. There are a natural bijection $T: <X,K(G,n)>\to H^n(X,G)$ for all CW-complexes $X$ and all $n>0$, with $G$ an abelian group.[ $<X,K(G,n)>$ is the base point preserving homotopy classes of maps]. $T([f])=f*(\alpha)$ where $\alpha\in H^n(K(G,n),G)$ be a  certain class. And now you know how to relate homology with co homology, i.e, by Universal Coefficient theorem (https://en.wikipedia.org/wiki/Universal_coefficient_theorem). This fundamental fact plays very important role in obstruction theorey as well. For more details see Hatcher section $4.3$.
