A minimal Brieskorn 5-sphere in the Gromoll-Meyer sphere and its applications
Abstract
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.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- June 2006
- DOI:
- 10.48550/arXiv.math/0606769
- arXiv:
- arXiv:math/0606769
- Bibcode:
- 2006math......6769D
- Keywords:
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- Mathematics - Differential Geometry;
- Mathematics - Geometric Topology