= Precomposition scales the Hopf invariant by degree
{title2=$H(\phi\circ d)=d\,H(\phi)$}
Let $\phi:S^{2q-1}\to S^q$ and precompose with a sphere self-map of <topological degree> $d$. The resulting map of mapping cones is the identity on the bottom $q$-cell and has degree $d$ on the top $2q$-cell. Pulling back the defining <cup product> relation for the <Hopf invariant> multiplies its coefficient by $d$. In particular the element $d\eta\in\pi_3(S^2)$ has <Hopf invariant> $d$. Postcomposition by a degree-$d$ map of the target sphere instead multiplies the invariant by $d^2$.
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