 Realization of top-dimensional cohomology by a sphere map

Let $X$ be a CW complex of dimension at most $n$, with $n\ge 1$. Every class $a\in H^n(X;\mathbb{Z})$ is $f^{\ast }u$ for a continuous map $f:X\to S^n$ and an orientation generator $u\in H^n(S^n;\mathbb{Z})$.

Choose a cellular cocycle representing $a$ and collapse the $(n-1)$-skeleton to a point. The quotient is a wedge sum of one $n$-dimensional sphere for every $n$-cell. map each sphere to $S^n$ with mapping degree equal to the chosen cellular coefficient. Composing with the quotient gives the required continuous map. There are no higher-dimensional cells whose attachments could obstruct it; the weak topology of a CW complex also permits an infinite wedge.

The restriction $n\ge 1$ is essential. On a three-point discrete space, an integral degree-zero cohomology class can take the three values $0,1,2$. pullback along a map to $S^0$ can take at most the two values of the chosen class on $S^0$, so it cannot realize that class.