Pulling back the Gromoll-Meyer construction and models of exotic spheres (Q2802134)

The author shows how one can construct exotic 8- and 10- spheres using a generalization of the Gromoll-Meyer techniques [\textit{D. Gromoll} and \textit{W. Meyer}, Ann. Math. (2) 100, 401--406 (1974; Zbl 0293.53015)]. The author's method relies on using explicit maps \(S^{8} \rightarrow S^{7}\) and \(S^{10} \rightarrow S^{7}\) to pull back an \(S^{3}\) bundle over \(S^{7}\) on which \(S^{3}\) acts. The exotic spheres are quotients of these bundles. The author shows how the constructions can be related to other methods of constructing exotic spheres such as Milnor's plumbing construction and the gluing together of disks by exotic diffeomorphisms. The ideas are further used to represent the Kervaire manifolds as quotients of \(O(n)\)-bundles on pull-backs of frame bundles on round spheres.