A minimal Brieskorn 5-sphere in the Gromoll-Meyer sphere and its applications

We recognize the Gromoll-Meyer sphere Sigma^7 as the geodesic join of a simple closed geodesic and a minimal subsphere Sigma^5, which can be equivariantly identified with the Brieskorn sphere W^5_3. As applications we in particular determine the full isometry group of Sigma^7, classify all closed subgroups that act freely, determine the homotopy type of the corresponding orbit spaces, identify the Hirsch-Milnor involution in dimension 5 with the Calabi involution of W^5_3, and obtain explicit formulas for diffeomorphisms between the Brieskorn spheres W^5_3 and W^13_3 with standard Euclidean spheres.