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


[Link](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-15/issue-2/Maps-which-induce-the-zero-map-on-homotopy/pjm/1102995805.full)


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?



Maps inducing zero on homotopy groups but are not null-homotopic
Consider ordinary singular cohomology with varying coefficients. You can look at the short exact sequence of abelian groups:


$$0 \to \mathbb{Z}/2 \to \mathbb{Z}/4 \to \mathbb{Z}/2 \to 0$$


This gives rise, for any space X, to a short exact sequence of chain complexes:


$$0 \to C^i(X;\mathbb{Z}/2) \to C^i(X;\mathbb{Z}/4) \to C^i(X;\mathbb{Z}/2) \to 0$$


and hence you get a long exact sequence in cohomology. Thus we get an interesting boundary map known as the [Bockstein](http://en.wikipedia.org/wiki/Bockstein_homomorphism)


$$H^i(X; \mathbb{Z}/2) \to H^{i+1}(X; \mathbb{Z}/2).$$


This is natural in X and so is represented by a (homotopy class of) map(s) of Eilenberg-Maclane spaces:


$$K(i, \mathbb{Z}/2) \to K(i+1, \mathbb{Z}/2)$$


This map is necessarily zero on homotopy groups. To show that this map is not null-homotopy, you just need to find a space for which the Bockstein is non-trivial. There are lots of examples of this. Rather then explain one, I suggest you look up "Bockstein homomorphism" in a standard algebraic topology reference, e.g. Hatcher's book. 


