TITLE: Maps inducing zero on homotopy groups but are not null-homotopic
QUESTION [21 upvotes]: Today my fellow grad student asked me a question, given a map f from X to Y, assume $f_*(\pi_i(X))=0$ in Y, when is f null-homotopic?
I search the literature a little bit, D.W.Kahn 
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.pjm/1102995805
And M.Sternstein has worked on this, and Sternstein even got a necessary and sufficient condition, for suitable spaces. 
http://www.jstor.org/stable/pdfplus/2037939.pdf
However, his condition is a little complicated for me as a beginner. Right now I just wanted a counter example of a such a map. Kahn in his paper said one can have many such examples using Eilenberg Maclance spaces. Well, we can certainly show a lot of map between E-M spaces induce zero map on homopoty groups just by pure group theoretic reasons, but I can not think of a easy example when you can show that map, if it exists, is not null-homotopic. Could someone give me some hint?
or, maybe even some examples arising from manifolds?

REPLY [14 votes]: I realize this is a very old question.  Nonetheless, here is a large, easy,  and well-known class of examples.
Let $X$ be a path connected CW-complex, and consider the diagonal map $\delta_X\colon X\to X\wedge X$ into the smash product (i.e., the composite of $X\xrightarrow{\text{diag}} X\times X \xrightarrow{\text{quot}} X\wedge X$).  
This map is often non-null: for instance, 
$$
\widetilde{H}^*(X)\otimes \widetilde{H}^*(X)\xrightarrow{\text{Kunneth}} \widetilde{H}^*(X\wedge X)\xrightarrow{\delta_X} \widetilde{H}^*(X)
$$
is exactly the cup-product, so any $X$ with non-trivial cup-product in positive degrees must have non-null $\delta_X$.
On the other hand, $\delta_X$ is always trivial on homotopy groups: any $f\colon S^k\to X$ fits in the commutative square
$$\begin{array}{ccc}
S^k & \xrightarrow{\delta_{S^k}} & S^k\wedge S^k
\\
\downarrow & & \downarrow
\\
X & \xrightarrow{\delta_X} & X\wedge X
\end{array}
$$
and $\delta_{S^k}\sim *$ for $k\geq 1$.  In fact, this idea proves that any composite $\Sigma Y\to X\xrightarrow{\delta_X} X\wedge X$ is null, or equivalently that $\Omega(\delta_X)\colon \Omega X\to \Omega(X\wedge X)$ is null-homotopic for any $X$.