## Abstract

We consider the canonical action of the compact torus T^{4} on the complex Grassmann manifold G_{4,2} and prove that the orbit space G_{4,2}/T^{4} is homeomorphic to the sphere S^{5}. We prove that the induced map from G4;2 to the sphere S5 is not smooth and describe its smooth and singular points. We also consider the action of T^{4} on ℂP^{5} induced by the composition of the second symmetric power representation of T^{4} in T^{6} and the standard action of T^{6} on ℂP^{5} and prove that the orbit space ℂP^{5}/T^{4} is homeomorphic to the join ℂP^{2}*S^{2}. The Plücker embedding G_{4,2} ⊂ ℂP^{5} is equivariant for these actions and induces the embedding ℂP^{1}*S^{2} ⊂ ℂP^{2}*S^{2} for the standard embedding ℂP^{1} ⊂ ℂP^{2}. All our constructions are compatible with the involution given by the complex conjugation and give the corresponding results for the real Grassmannian G^{4,2}(ℝ) and the real projective space ℝP^{5} for the action of the group ℤ^{4} _{2}. We prove that the orbit space G_{4,2}(ℝ)/ℤ^{4} _{2} is homeomorphic to the sphere S^{4} and that the orbit space ℝP^{5}/ℤ^{4} _{2} is homeomorphic to the join ℝP^{2}*S^{2}.

Original language | English |
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Pages (from-to) | 237-273 |

Number of pages | 37 |

Journal | Moscow Mathematical Journal |

Volume | 16 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Apr 2016 |

Externally published | Yes |

## Keywords

- Complex projective space
- Grassmann manifold
- Orbit
- Space
- Torus action

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