## Return to Question

3 added 80 characters in body

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.)

2 added 488 characters in body

I made the following claim over at the SBSeminar, 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.)

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# Whitehead for maps

I made the following claim over at the SBSeminar, 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?