Self-homotopy equivalences of the total spaces of a sphere bundle over a sphere
The group \({\mathcal E}(X)\) of homotopy classes of self-homotopy equivalences is calculated here up to two extensions in the case where \(X=S^ m\cup e^ n\cup e^{m+n}\) is an \(S^ m\) bundle over \(S^ n\) \((3<m+1<n<2m- 2):\) the extensions are not given. The group \({\mathcal E}(X)\) was known in the case \(m=3\) and \(n=7\) for rank 2 H-spaces and others: for the Stiefel manifolds \(V_{p,2}=S^{p-2}\cup e^{p-1}\cup e^{2p-3}\) and \(W_{p,2}=S^{2p-3}\cup e^{2p-1}\cup e^{4p-3}\) (most values of p) up to extension: and, up to one extension in special cases for \(m=3\). Rather than making two successive applications of the standard technique using the Barcus-Barratt sequence, the proof uses a modification. The extensions are based on the sequences \(1\to {\mathcal E}_+(X)\to {\mathcal E}(X)\to^{H_*}aut H_*(X)\) and \(0\to H_{\xi}\to {\mathcal E}_+(X)\to G_{\xi}\to \{1\}\) where \({\mathcal E}_+(X)\) is the kernel of \(H_*\), \(G_{\xi}\) is a subgroup of a quotient of \(\pi_ n(S^ m)\), and \(H_{\xi}\) is a quotient of \(\pi_{m+n}(X)\).
homotopy classes of self-homotopy equivalences
scientific article; zbMATH DE number 3895934