I made the following claim over at the SBSeminarSecret Blogging Seminar, and now I'm not sure it's true:
Let f: X --> Y and g: X --> Y be two maps betwen finite CW complexes. If f and g induce the same map on pi_k, for all k, then f and g are homotopic.
Was I telling the truth?
EDIT: Since I didn't say anything about basepoints, I probably should have said that f and g induce the same map
[S^k, X] --> [S^k, Y].
This will also deal better with the situation where X and Y are disconnected. I'd be interested in knowing a result like this either with pointed maps or nonpointed maps. (Although, of course, if you work with pointed maps you have to take X and Y connected, because [S^k, _] can't see anything beyond the number of components in that case.)