= Hopf projection of a product smash quotient
{c}
{title2=$\phi=p\circ q$}
Let $a,b\ge2$ with $a+b=2n+1$. Collapse the lower cells of $S^a\times S^b$ to obtain $q:S^a\times S^b\to S^{2n+1}$ and compose with the <Hopf fibration> $p:S^{2n+1}\to\mathbb{CP}^n$. Both factor inclusions are killed by q, so every induced <homotopy group> map is zero; the induced <reduced homology> maps also vanish. If the composite were nullhomotopic, <homotopy lifting property> would deform q into the circle fibre. Every map from the simply connected product to that circle is nullhomotopic, contradicting the degree-one top <homology> map of q.
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