 Realization of top-dimensional cohomology by a sphere map

If $X$ is a CW complex of dimension at most $n$, every class $a\in H^n(X;\mathbb{Z})$ has the formfor some map $f:X\to S^n$ and generator $u\in H^n(S^n;\mathbb{Z})$. Cellular obstruction theory constructs $f$ on the top cells; there are no cells of larger dimension that could obstruct the extension.