On the group of self-homotopy equivalences of principal \(S^ 3\)-bundles over spheres (Q2266257)

A principal \(S^ 3\)-bundles over \(S^ n\) (n\(\geq 5)\) with characteristic class \(f\in \pi_{n-1}(S^ 3)\) has the homotopy type \(E_ f=S^ 3\cup_{f}e^ n\cup e^{n+3}\). Let \(\omega\) be a generator of \(\pi_ 6(S^ 3)\cong {\mathbb{Z}}_{12}\). The authors obtain an exact sequence \[ 0\to \pi_{n+3}(E_ f)\to \epsilon (E_ f)\to \epsilon (S^ 3\cup_{f}e^ n)\to 1 \] in the cases where \(\omega \circ S^ 3f\in f_* \pi_{n+2}(S^{n-1}):\) in particular every homotopy equivalence of the n-skeleton extends, in this case, to one of the whole space. The extensions are not given. For the cases \(E_ 0=S^ 3\times S^ n\) and \(E_{k\omega}=S^ 3\cup e^ 7\cup e^{10}\) (0\(\leq k\leq 6)\) the sequences and, except for \(E_{6\omega}\), the extensions were already known. Some calculations are given for \(\epsilon (S^ 3\cup_{f}e^ n)\), and \(\epsilon (E_ f)\) is given up to extension in the cases where f is the generator of \(\pi_ 7(S^ 3)\), \(\pi_ 8(S^ 3)\) or \(\pi_ 9(S^ 3)\).