when is the kth homology group of a space isomorphic to its kth homotopy group?

I'm just thinking about the relationship between homology and homotopy groups of a space. I know that homology is basically an abelianization of the fundamental group (please correct me if I'm wrong). If anyone could please say a few words about my question, I would be appreciative. What/where else could I read about addressing such topics?

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– Ryan Budney May 8 '12 at 13:11
There are two things you could mean by this: you might be asking when the two groups are abstractly isomorphic, or you might be asking whether there is a natural transformation between them, and when it induces an isomorphism. The first question seems to me both difficult and uninteresting as it is not natural; fortunately the second question has a better answer as above. – Qiaochu Yuan May 8 '12 at 15:35
This question actually goes quite deep: one can view the Adams spectral sequence as measuring the failure of the natural map $\pi_*(X) \rightarrow H_*(X)$ to be an isomorphism. – Aaron Mazel-Gee May 11 '12 at 14:54