If a map $f:X\to Y$ of CW-complexes induces an isomorphism on homotopy groups it is a homotopy equivalence by the Whitehead theorem. In particular it also induces isomorphisms on homology and cohomology.
If now $f$ is just an $n$-equivalence (induces isomorphisms on homotopy up to degree $n$), intuitively I would say it still induces isomorphisms on (co)homology in lower degrees. But does it? Or is there a similar statement?