The group of self-homotopy equivalences of \(S^ 2\)-bundles over \(S^ 4\). I (Q1083065)

The group \({\mathcal E}(X_ m)\) of self-homotopy equivalence classes of the \(S^ 2\)-bundles \(X_ m=S^ 2 \cup_{m\eta} e^ 4\cup e^ 6\) over \(S^ 4\) is determined up to an exact sequence \(\pi_ 6(X_ m)\to {\mathcal E}(X_ m)\to G_ m\to 1\). Here \(\chi (X_ m)=m\rho \in \pi_ 3(SO_ 3)\) is the characteristic class, where \(\rho\) is the class of the double covering \(S^ 3\to {\mathbb{R}}P^ 3=SO_ 3\). The groups \(\pi_ 6(X_ m)\) and \(G_ m\) are calculated for each m. For other values of p and q, the group \({\mathcal E}(X)\) is already known for many other spaces which are \(S^ p\)-bundles over \(S^ q\).