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

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

2006-06-29

arxiv.org/abs/math/0606769v1

Mathematics

Differential Geometry

Scientific paper

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.

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