Topology and geometry of the canonical action of T4 on the complex Grassmannian G4,2 and the complex projective space CP5

Moscow Mathematical Journal. 2016. Vol. 16. No. 2. P. 237–273.

Buchstaber V.M., Terzic S.

We consider the canonical action of the compact torus T4 on the complex Grassmann manifold G4,2 and prove that the orbit space G4,2/T4 is homeomorphic to the sphere S5. 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 T4 on ℂP5 induced by the composition of the second symmetric power representation of T4 in T6 and the standard action of T6 on ℂP5 and prove that the orbit space ℂP5/T4 is homeomorphic to the join ℂP2*S2. The Plücker embedding G4,2 ⊂ ℂP5 is equivariant for these actions and induces the embedding ℂP1*S2 ⊂ ℂP2*S2 for the standard embedding ℂP1 ⊂ ℂP2. All our constructions are compatible with the involution given by the complex conjugation and give the corresponding results for the real Grassmannian G4,2(ℝ) and the real projective space ℝP5 for the action of the group ℤ4 2. We prove that the orbit space G4,2(ℝ)/ℤ4 2 is homeomorphic to the sphere S4 and that the orbit space ℝP5/ℤ4 2 is homeomorphic to the join ℝP2*S2.

Keywords: Torus Actions, Grassmann manifold, Complex projective space, Orbit, Space