What can be said about the homotopy groups of a CW-complex in terms of its (co)homology?

One example is the Hurewicz theorem which tells us that (e.g) a CW-cx with only one 0-cell has a nontrivial fundamental group if H_1 is nontrivial. What other examples are there? (The CW-complexes I have in mind always have exactly one 0-cell, but for the sake of a more wide discussion one could assume this is not generally true. However, I am mostly interested in the higher homotopy groups as I already know everything about the fundamental group.)

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The cohomology of $S^n$ doesn't contain much information (nonzero only in degrees 0 and n), but its homotopy groups are really crazy in general, so do you have more specific details about your CW complexes? –  Steven Sam Feb 12 '10 at 7:50
There's a bunch that can be said. To start getting a feel for your question, look at the Adams spectral sequence and, say, the Kan-Thurston theorem. The first will get your hopes up and the second will (partially) dash them. –  Ryan Budney Feb 12 '10 at 7:54

If you don't want to make any assumptions about $\pi_1$, then I think the question is hard. Maybe Hurewicz is most of what you can say. If assume something like simply connected, you can say a lot more. There are many things you can say rationally, some of which were pointed bout my Kevin, but even integrally or mod p there are a lot of techniques. Can you be more precise about the setup you are interested in?