= Cohomological obstruction to a section of the twistor bundle
{title2=$S^2\longrightarrow\mathbb{CP}^3\longrightarrow S^4$}
If the displayed <fiber bundle> had a section $s$, its projection $p$ would satisfy $s^*p^*=1$. The <cohomology ring of complex projective space> is $\mathbb Z[h]/(h^4)$ with $|h|=2$. For a generator $u\in H^4(S^4;\mathbb Z)$, write $p^*u=mh^2$. But $s^*h=0$ since $H^2(S^4;\mathbb Z)=0$, giving $u=s^*p^*u=0$. This contradiction proves that no section exists, without needing the value of $m$.
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