A map which is trivial on homology but not on cohomology? Is there a map $f:X\to Y$ of connected CW-complexes which induces the trivial map $f_*=0:H_i(X,\Bbb Z) \to H_i(Y,\Bbb Z)$ for all $i\ge 1$, but with the property that the induced map on cohomology $f^*:H^i(Y,\Bbb Z)\to H^i(X, \Bbb Z)$ is nonzero for at least one $i \ge 1$? 
Note that the universal coefficient theorem does not a priori imply that the induced map is trivial because the splitting $H^i(X, \mathbb{Z}) = H_i(X, \mathbb{Z})^* \oplus \operatorname{Ext}^{i-1}(X, \mathbb{Z})$ is not natural. Since I don't see any other reason against it,
I believe such a counterexample should exist.

you can see exercise 11 of chapter 3.1 of hatcher.let X obtained from$S^n$ by attaching a cell of degree m.you can easily see (by cellular homology and cohomology) the map
$X\to X/S^n$ is trivial on $H_i$ but not on $H^{n+1}$
