I am interested in the following question. Are maps which induce the same homomorphism on homotopy and homology groups homotopic? I am sure the answer is no, however I cannot imagine how to construct counterexamples.



Maps which induce the same homomorphism on homotopy and homology groups are homotopic
Take the composition of a degree one map $f:T^3\to S^3$ with the Hopf map $g:S^3\to S^2$, where $T^3$ is the 3-torus. This composition is trivial on homotopy groups since $T^3$ is aspherical and $\pi\_1S^2=0$. It is trivial on $H\_i$ for $i>0$ since this is true for $g$. If $gf$ were nullhomotopic we could lift a nullhomotopy to a homotopy of $f$ to a map to a circle fiber of $g$, which would imply that $f$ had degree 0, a contradiction. Thus $gf$ induces the same maps on homology and homotopy groups as a constant map, but it isn't homotopic to a constant map. (I forget where I first saw this example, maybe in something of Arnold.)


