= Realization of top-dimensional cohomology by a sphere map
Let $X$ be a <CW complex> of <dimension> at most $n$, with $n\geq1$. Every class $a\in H^n(X;\mathbb Z)$ is $f^*u$ 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 $n\geq1$ 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.
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