Realization of top-dimensional cohomology by a sphere map
= 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 form
$$
a=f^*u
$$
for 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.