Gromoll–Meyer sphere

Brieskorn sphere In \mathbb{C}^5 consider the complex variety: : a^2+b^2+c^2+d^3+e^5=0. A description of the Gromoll–Meyer sphere is the intersection of the above variety with a small sphere around the origin. Lie group biquotient The first symplectic group \operatorname{Sp}(1) (isomorphic to \operatorname{SU}(2)) acts on the second symplectic group \operatorname{Sp}(2) (isomorphic to \operatorname{Spin}(5)) with the embedding \operatorname{Sp}(1)\hookrightarrow\operatorname{Sp}(2), q\mapsto\operatorname{diag}(q,q) and multiplication from the left as well as the embedding \operatorname{Sp}(1)\hookrightarrow\operatorname{Sp}(2), q\mapsto\operatorname{diag}(q,1) and multiplication from the right. A description of the Gromoll–Meyer sphere is the biquotient space: : \operatorname{Sp}(1)\backslash\operatorname{Sp}(2)/\operatorname{Sp}(1). == Properties ==

• It is the only seven-dimensional exotic sphere which can be expressed as a biquotient of a compact Lie group. • It can be expressed as a S^3-fiber bundle over S^4 and hence is a Milnor sphere. Such bundles also include the quaternionic Hopf fibration, whose total space is the ordinary S^7. • It generates the seventh Kervaire–Milnor group \Theta_7\cong\mathbb{Z}_{28}. == Literature ==

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