Homotopy classification of bilinear maps related to octonion polynomial multiplications (Q968985)

A nonsingular bilinear map \(f : \mathbb{R}^r \times \mathbb{R}^s \to \mathbb{R}^n\) induces a map \(H_f : S^{r+s-1} \to S^n\) via the Hopf construction and an adjoint map \(\widetilde{f} : S^{r-1} \to V_{n,s}\) where \(V_{n,s}\) is the Stiefel manifold. The construction of nonsingular bilinear maps is thus a natural pursuit in homotopy theory with many examples obtained by Adem, Lam, Milgram among others. The main device used for constructing these maps has been the Cayley-Dickson algebras \(\mathbb{A}_n\). The algebra \(\mathbb{A}_n\) has dimension \(2^n\) over \(\mathbb{R}\) and recovers the algebras \(\mathbb{R}\), \(\mathbb{C}\), \(\mathbb{H}\) and \(\mathbb{K}\), the octonion numbers, by taking \(n = 0 ,1, 2, 3,\) respectively. The author describes a matrix representation of the multiplication in \(\mathbb{A}_n\). Using this description, he determines the (nontrivial) homotopy class of the Hopf construction applied to the map \(\phi : \mathbb{K}^2 \times \mathbb{K}^2 \to \mathbb{K}^3\) corresponding to the multiplication of linear polynomials over \(\mathbb{K}\) with an unknown commuting with all coefficients. He also shows the adjoint map \(\widetilde{\phi} : S^{15} \to V_{16,8}\) is nontrivial. This last result leads to an interesting consequence: The map \(\widetilde{\phi}\) generates exactly \(7\) of the \(8\) possible linearly independent vector fields on \(S^{15}.\)