Precomposition scales the Hopf invariant by degree ($H(\varphi \circ d)=dH(\varphi )$)

Let $\varphi :S^{2q-1}\to S^q$ 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 $d^2$.