Cohomology classification of self maps of sphere bundles over spheres (Q1922671)

Let \(E\) denote the \(q\)-sphere bundle over the \(n\)-sphere determined by an element of \(\pi_{n-1} (\text{SO} (q+1))\). Any self-map of \(E\) determines an automorphism of \(\widetilde H^* (E)\), modulo torsion. The purpose of this work is to determine what automorphisms of \(\widetilde H^* (E)\), modulo torsion, arise in this way. The results express the answers in terms of numerical conditions on the multiples of the generators obtained by applying a possible self-map. Special attention is given to the cases of real, complex and quaternionic Stiefel manifolds of 2-frames. A final section looks at the self-maps of the suspension of \(E\).