On self-homotopy equivalences of \(S^ 3\)-principal bundles over \(S^ n\) (Q1071348)

A principal \(S^ 3\)-bundle over \(S^ n\) (n\(\geq 6)\) with characteristic class \(\xi \in \pi_{n-1}(S^ 3)\) has the homotopy type \(X=S^ 3 \cup_{\xi} e^ n\cup e^{n+3}\). Let \(\tau\) : \(S^ 6\to S^ 3\) be the Blakers-Massey map. There is an exact sequence \[ 1 \to \epsilon_+(X) \to \epsilon (X)\to^{d}{\mathbb{Z}}_ 2\times {\mathbb{Z}}_ 2 \] where \(\epsilon\) (X) is the group of self-homotopy equivalence classes of X and \({\mathbb{Z}}_ 2\times {\mathbb{Z}}_ 2=aut H_ 3(X)\times aut H_ n(X)\). In the case where \(\tau \circ E^ 3\xi \equiv 0\) mod \(\xi\) \(\circ \pi_{n+2}(S^{n-1})\), d is onto if \(2\xi =0\) and is onto \({\mathbb{Z}}_ 2=<(-1,-1)>\) if \(2\xi\) \(\neq 0\). A sequence for calculating \(\epsilon_+(X)\), up to extension, is also given. For the cases \(E_ 0=S^ 3\times S^ n\) and \(E_{k\tau}=S^ 3\cup e^ 7\cup e^{10}\) (0\(\leq k\leq 6)\), the sequences and, except for \(E_{6\tau}\), the extensions were already known. For general n, a different short exact sequence for \(\epsilon\) (X) is given by \textit{M. Mimura} and \textit{N. Sawashita} [Hiroshima Math. J. 14, 415-424 (1984; Zbl 0559.55010)].