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.

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    $\begingroup$ It's an open question. It's not even known if two maps between finite CW-complexes which induce the trivial homomorphisms in stable homotopy groups, is stably nullhomotopic. $\endgroup$ – Fernando Muro Feb 5 '15 at 22:28
  • $\begingroup$ Ooops, thanks Fernando. I did not know. $\endgroup$ – Samarkand Feb 5 '15 at 22:29
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    $\begingroup$ It seems I was wrong! I must have thought too stably. $\endgroup$ – Fernando Muro Feb 5 '15 at 22:44

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.)

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    $\begingroup$ Your answer is so brilliant that I just went blind. $\endgroup$ – Samarkand Feb 8 '15 at 19:30

Phantom maps provide a large class of examples of maps which are not homotopic to the constant map, but which induce the zero map on both homology and homotopy groups. There are uncountably many distinct homotopy classes of phantom maps $\mathbb{C}P^\infty\to S^3$, for example.

Edit: In many cases one can determine whether nontrivial phantom maps can be found between two spaces by investigating rational homotopy invariants. Some great references include Phantom maps and Rational Equivalences by Roitberg and McGibbon, or McGibbon's survey in the Handbook of Algebraic topology.

  • $\begingroup$ James, thank you for your answer. I have read the definitions and it seems that phantom maps are $f:X\rightarrow Y$ such that restriction of $f$ to each skeleton is null-homotopic. So, this definition is of interest when $X$ is infinite dimensional. Is there some theory for finite complexes? For example, when $X$ and $Y$ are 2-dimensional, how can we check that map $f:X\rightarrow Y$ satisfy $0=f_*:\pi_2 X \rightarrow \pi_2 Y$. $\endgroup$ – Samarkand Apr 7 '15 at 21:16

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