A map which is trivial on homology but not on cohomology?

Take the $n$-sphere and attach a $n+1$-cell with a degree $m$ map. Consider $X \to X/S^{n+1}$ where $X$ is the described space.

You can write down the cellular chain complexes:

$$ \begin{array}{c} \cdots & \to& 0 &\to& \mathbb Z&\stackrel {* m} \to& \mathbb Z & \to &0& \cdots \\ &&\downarrow&&\downarrow&&\downarrow \\ \cdots & \to &0&\to&\mathbb Z& \to&0& \to & 0 &\cdots \end{array} $$

Apply homology, you will kill every possible map (the degrees where the non-trivial groups lie are disjoint, except for the trivial degree).

Apply $hom(-,\mathbb Z)$ and then homology (ie cohomology), you will get on degree $n+1$ the quotient map $\mathbb Z \to \mathbb Z/m$.

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 $H^{n+1}$