Harmonic maps from spheres to spheres (Q1373487)

The authors give new examples of elements from the homotopy groups of spheres which are harmonically representable. If \(p,q\geq 3\), one knows that every orthogonal multiplication \(g:\mathbb{R}^p \times \mathbb{R}^q \to \mathbb{R}^n\) gives rise, via the Hopf construction on the associated map, \(S^{p-1} \times S^{q-1} \to S^{n-1}\) to a map \(f:S^{p+ q-1} \to S^n\), which has a harmonic representative by the Ratto-Ding theorem [\textit{W.-Y. Ding}, Int. J. Math. 5, No. 6, 849-860 (1994; Zbl 0822.58011)]. The main results concern the precise description of \([f]\in \pi_{p+ q-1}S^n\), for various particular choices of \(g\).