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

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We consider the canonical action of the compact torus T⁴ on the Grassmann manifold G₄,₂ and prove that the orbit space G₄,₂/T⁴ is homeomorphic to the sphere S⁵. We prove that the induced differentiable structure on S⁵ is not the smooth one and describe the smooth and the singular points. We also consider the action of T⁴ on CP⁵ induced by the composition of the second symmetric power T⁴⊂ 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 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 Z ₂⁴. We prove that the orbit space G₄,₂(R)/Z ₂⁴ is homeomorphic to the sphere S⁴ and that the orbit space RP⁵/Z ₂⁴ is homeomorphic to the join RP² S².